Interviews with the Abel Prize Laureates 2003–2016

Home , Andrey Kolmogorov, Fermat Prize, Grigory Margulis, Isaac Newton Institute, Isadore Singer, Israel Gelfand

Martin aussen and Christian Skau, Editors Interviews with the Abel Prize Laureates 2003–2016

Martin Raussen Christian Skau Editors

The Abel Prize was established in 2002 by the Norwegian Ministry of Education and Research. It has been awarded annually to mathematicians in recognition of pioneering scientific achievements. Interviews with the Since the first occasion in 2003, Martin aussen and Christian Skau have had the opportunity to conduct extensive interviews with the Abel Prize Laureates laureates. The interviews were broadcast by Norwegian television; moreover, they have appeared in the membership journals of several mathematical societies. 2003–2016 The interviews from the period 2003 – 2016 have now been collected in this edition. They highlight the mathematical achievements of the laureates in a historical perspective and they try to unravel the way in Martin Raussen which the world’s most famous mathematicians conceive and judge Christian Skau their results, how they collaborate with peers and students, and how they perceive the importance of mathematics for society. Editors

ISBN 978-3-03719-177-4

www.ems-ph.org

Raussen/Skau | Fonts: Helvetica Neue | Farben: Pantone 116, Pantone 287, Cyan | RB 15.1 mm

Interviews with the Abel Prize Laureates 2003–2016

Martin Raussen Christian Skau Editors Editors:

Martin Raussen Christian Skau Department of Mathematical Sciences Department of Mathematical Sciences Aalborg University Norwegian University of Science and Technology Skjernvej 4A 7491 Trondheim 9220 Aalborg Norway Denmark E-mail: [email protected] E-mail: [email protected]

2010 Mathematics Subject Classification: 01A70. 01A60, 01A61, 01A80, 00A35

Key words: Abel prize, laureates, interviews, history of mathematics, appreciation of mathematics

ISBN 978-3-03719-177-4

The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliogra- phy, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch.

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained.

Contact address:

European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum SEW A21 8092 Zürich Switzerland

E-mail: [email protected] Homepage: www.ems-ph.org

Typeset by Sylvia Lotrovsky, Basel, and Christoph Eyrich, Berlin Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 9 8 7 6 5 4 3 2 1 Table of Contents

Preface...... vii

Introduction...... ix

Abel Prize 2003: Jean-Pierre Serre...... 1

Abel Prize 2004: Sir Michael Francis Atiyah and Isadore M. Singer...... 11

Abel Prize 2005: Peter D. Lax...... 31

Abel Prize 2006: Lennart Carleson...... 47

Abel Prize 2007: Srinivasa S. R. Varadhan...... 61

Abel Prize 2008: John Griggs Thompson and Jacques Tits...... 79

Abel Prize 2009: Mikhail Gromov...... 97

Abel Prize 2010: John Tate...... 123

Abel Prize 2011: John Milnor...... 141

Abel Prize 2012: Endre Szemerédi...... 161

Abel Prize 2013: Pierre Deligne...... 183

Abel Prize 2014: Yakov G. Sinai...... 201

Abel Prize 2015: John F. Nash, Jr. and Louis Nirenberg...... 219

Abel Prize 2016: Sir Andrew J. Wiles...... 245

An Imaginary Interview with Niels Henrik Abel...... 267

Preface

The Abel Prize is an international prize established in 2002 by the Norwegian Min- istry of Education and Research and is administered through an annual grant by the Norwegian Academy of Science and Letters. It is awarded every year in recognition of pioneering scientific achievements in mathematics. The year 2002 marked the bicentennial of Nils Henrik Abel’s birth and the prize was established in honour of his exceptional mathematical heritage. The Abel Prize is also meant to promote the field of mathematics and to make the prize winners and their work known to the public. Every year from 2003 until 2016, the Abel Prize laureates have been interviewed by Martin Raussen and Christian Skau in connection with the award ceremony. These interviews have been broadcast on Norwegian national television (and often repeated) and can be streamed from the Abel Prize website, www.abelprize.no. Transcripts of the interviews have been published in the EMS Newsletter and the Notices of the AMS. This book is a collection of the interviews, with a new introduction by the interviewers. The Abel Board and the Norwegian Academy of Science and Letters would here- by like to express our gratitude to Martin Raussen and Christian Skau for conducting the Abel interviews and for preparing their transcripts for publication in this book, thus making them available to a broad audience in printed form that has proven its durability. Autobiographical pieces by the laureates and descriptions of their work by fellow mathematicians are published by Springer in volumes entitled “The Abel Prize”, edited by Helge Holden and Ragni Piene.

Ole M Sejersted Kristian Ranestad President Chairman Norwegian Academy of Science and Letters Abel Board

vii

Introduction

How it all started: the first interview in 2003

The Abel Prize was established by the Norwegian government in 2002 on the occasion of the 200th anniversary of Niels Henrik Abel's birth. Its main aim was to recognise contributions to mathematics of extraordinary depth and influence. An Abel Committee consisting of five prominent mathematicians had the difficult task of selecting the first Abel laureate for 2003. In March of that year, its chair Erling Størmer announced the committee’s decision to award the first Abel Prize to Jean- Pierre Serre, Emeritus Professor at the Collège de France in Paris, to be honoured “for playing a key role in shaping the modern form of many parts of mathematics, including topology, algebraic geometry and number theory”. The Abel Prize, having been recently established, was not yet well known to mathematicians (and even less so to the public at large). The Abel Board tried to ad- vertise the prize by various means, including using Norwegian diplomacy. Among many other initiatives, the Norwegian embassy in Denmark approached the edi- torial board of Matilde, a small membership journal of the Danish Mathematical Society; the editors were asked whether they would be willing to cover the prize ceremony and help in making the prize more well known in Denmark. Martin Raus- sen had previously interviewed a number of well known Danish mathematicians for Matilde. After some reflection, he gathered his courage and asked whether it would be possible to have an interview with the laureate. To his surprise, arrangements were made quickly and an interview was granted by the organisers and by Professor Serre. Very shortly after this, the Norwegian Mathematical Society came up with the same idea – an interview to be conducted with the laureate. The reply from the organisers was negative as an arrangement had already been made with a mathematician from Denmark! Luckily, the society’s president at the time, Kristian Seip, did not take this as the end of the story; he called Martin Raussen, proposing that the interview become a joint venture with Christian Skau, a Norwegian mathematician with a keen interest in Abel and his mathematics. The two of us did not know each other at the time but it turned out that this “forced marriage” would be the beginning of a long and fruitful partnership. We did not have a lot of time to prepare for our first Abel interview. As a point of departure, we had, of course, the prize citation. We both knew bits and pieces about certain aspects of Serre’s work and its importance but neither of us had the

ix x Introduction expertise to fully appreciate the scope of his lifetime contribution (or, for that matter, the contributions of the Abel laureates after him). However, a scheme for the preparation was constructed on the fly (see below). After introducing ourselves to Professor Serre by mail and briefly communicating about the topics to be covered, we were ready to start our first Abel interview at the Hotel Continental in Oslo. The interview was recorded and when one of our Norwegian colleagues watched it, she commented that we appeared like schoolboys! It is true we were nervous con- fronted with a mathematician of Serre’s stature and wit! With substantial help from Professor Serre, the interview was edited and appeared first in Matilde and then in several other journals of mathematical societies: the Newsletter of the European Mathematical Society, the Nieuw Archief voor Wiskunde, the Notices of the Ameri- can Mathematical Society and others. We felt that the result was worth the efforts and we were motivated and eager to continue!

Preparing, conducting and editing the interviews

To understand the process leading to a completed Abel interview, one needs to keep in mind that it has a double purpose, aimed at two entirely different audi- ences. One of these is the Norwegian public at large, who have the opportunity to watch an edited version of the recording (with Norwegian subtitles) on a science channel (Kunnskapskanalen) on Norwegian TV (NRK). We are still surprised that so many people follow our interviews on TV! Most of the recordings can be found on the website, www..no. The second audience consists of the readers of various journals for members of mathematical societies, notably the Newsletter of the Euro- pean Mathematical Society and the Notices of the American Mathematical Society (and these readers are mainly mathematicians, of course). As a consequence, an interview needs to contain both general questions and comments and also more mathematically advanced ones (although these are sometimes omitted from the TV version). Every year, the Abel laureate is announced in the middle of March at a public ceremony (in recent years, this has been streamed live on the Abel Prize website), the venue being the beautiful villa belonging to the Norwegian Academy of Sci- ence and Letters. In most cases, the laureate is then contacted by phone, receives congratulations and is asked a few questions. The citation for the prize-winner, a biography and some explanations for the public and for the press are shared on the web almost instantaneously. This is the time when the interviewers’ preparations start, roughly two months before the interview is to take place. We start by sorting out the intelligence. Are there “popular” articles in journals or on the internet explaining the work of the prize-winner? Are there even interviews with the laureate conducted by others? Are there colleagues who either know the laureate or who are, at least, familiar with his1

1 So far, all Abel laureates have been male. Introduction xi research areas in mathematics? Next in line is the first contact with the prize-winner, asking permission for an interview. So far, all have given their consent. The first written account is an interview guide that is sent to the laureate. This is one or two pages long, containing keywords and expressions referring to topics that we would like to touch on during the interview. As one may guess, some of them are very specific to a laureate whilst others reappear from year to year. Every year, we give the laureate the opportunity to edit these interview guides; only a few have done so. The two of us contact each other a few times and start dividing the preparation between us; the outcome usually depends on which of us is closer to the laureate’s work. We meet each other face-to-face at a hotel in Oslo on the evening before the interview takes place and we put the finishing touches to our interview plan the next morning. For the first interviews in the series, we made very detailed plans of who would ask what. As we have become more experienced, we have been able to relax a bit. We still prepare questions but now we let the interviews develop at their own pace and ask our questions whenever they seem appropriate, even if they are not pre- planned. Laureates do not always answer as you expect them to; they come up with a keyword or a line of thought that naturally leads to a follow-up question. Our task is to let the laureate do the talking and explaining; we as interviewers keep in the background, only “feeding” the prize-winner with suggestive questions. So far, the average age of an Abel laureate at the time of their award is 76 years. Taking this into consideration, they have to fulfil many arduous duties during the Abel week: handing out prizes for mathematics schoolteachers and pupils; wreath-laying at the Abel monument; attending a dinner and party at the Academy of Science and Letters (for mathematicians) and a banquet at Akershus Castle (given by the Norwegian government, in the presence of royalty); attending an audience at the royal castle and often also at the embassy of the laureate’s country of origin; attending the prize ceremony itself at the Aula of the University of Oslo, preceded by a rehearsal and succeeded by a reception, with a shorter and rather more personal interview and the Abel Prize lectures, including a lecture by the laureate; and finally a visit to one of the Norwegian universities outside the capital! Fortunately, “our” interview is one of the first of the laureate’s duties! In recent years, a nice tradition has been established. We have lunch with the laureate right before the interview. This allows us to become acquainted and to go through the overall setup for the interview in a relaxed atmosphere. Sometimes ideas for further questions also arise from an interesting remark by the laureate. The interview then takes place and is recorded; usually it lasts between one and one and a half hours. A couple of days later, editing begins, firstly for the television programme. The written version of the interview requires some special editing. The oral version may contain repetitions, half-formulated sentences and even small language errors. Arranging some of the text in a different order may make more sense. Of course, the prize-winner has a say as well. Sometimes, we have an important additional question that we did not ask during the interview or a few additions or xii Introduction clarifications come up when a written version becomes available. This work needs to be finished around six weeks after the interview in order to meet the deadline for the Newsletter of the European Mathematical Society. A very nice collaboration with Sylvia Fellmann from the EMS Publishing House working on the layout for the Newsletter has developed over the years; only very few corrections have to be made to the proofs that all participants receive. The interview appears first in the online version of the EMS Newsletter and then the EMS members receive it in their printed version of the Newsletter. It has become customary that the interview is reproduced in the Notices of the AMS at the beginning of the following year, thus reaching a far wider audience. As an aside, it should be noted that most of the interviews have also been translated into Chinese and published in a Chinese journal! The interviews in this collection are reprinted almost as they appeared in theEMS Newsletter and in the Notices. We have used the opportunity to correct a few minor errors, and we have added the life spans of mathematicians and other personalities mentioned during the conversations.

A rewarding experience

When we had the first interview with Jean-Pierre Serre, neither of us imagined that we would continue this for so long! One of us, Martin Raussen, has now decided to step down – with regrets; the other, Christian Skau, is going to continue with a new partner. As one can imagine, it is an immense privilege to get the chance to talk to the world’s most famous mathematicians! Although these encounters are of short duration, you develop an enormous respect for the personalities of the laureates, for their very special talents and for the breadth of their mathematical background and knowledge. It has also been very rewarding to register their reflections concerning the process of mathematical exploration, the perseverance they have needed to make progress and sometimes the sudden bolts of inspiration they have encountered. The personalities and the mathematical areas these mathematicians excel in are different but their experiences have a lot in common with each other. Needless to say, participation in the Abel week is an additional bonus. You meet with colleagues from Scandinavia and from around the world. The beautiful award ceremony in the university Aula is an attraction in itself. The Abel lectures on the following day provide intellectual stimulation and there are delightful dinners and parties to participate in: at the Academy of Science and Letters and, more formally, at a banquet given by the government in honour of the laureate at Akershus Castle.

An imaginary interview with Niels Henrik Abel

As an appendix, we present an imaginary interview with Niels Henrik Abel. We have tried to imagine an interview taking place shortly before his all-to-early death, following a pattern and a scheme similar to that used when interviewing the Abel Introduction xiii

Prize laureates. As for the questions and the answers, a lot of historical evidence has been considered. We hope that we have been able to convey an authentic and, at the same time, vivid impression of this mathematician par excellence, who has made such revolutionary contributions to mathematics and whose achievements we honour every time a prize is awarded in his name.

Acknowledgements

We wish to express our sincere thanks to

- The Abel laureates 2003–2016 for their cooperation. - The hairsc of the Abel Board and the Abel Committee for their continuous support. - The Norwegian Academy of Science and Letters for allowing us to use the award citations and photos. The copyrights for these remain with the Academy. - Anne-Maire Astad and Trine Gerlyng from the office of the Norwegian Academy of Science for assistance with all practical arrangements. - The recording team from UniMedia and Kunnskapskanalen for their professional and very helpful cooperation. - Sylvia Fellmann from the EMS publishing house for the skilful layout of the Abel interviews – both in the EMS Newsletter and in this book. - The editors of the EMS Newsletter and the Notices of the AMS for publishing the Abel interviews over all these years.

Aalborg and Trondheim, January 2017

Martin Raussen and Christian Skau

Abel Prize 2003: Jean-Pierre Serre

The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2003 to

Jean-Pierre Serre Collège de France, Paris, France

“for playing a key role in shaping the modern form of many parts of mathematics, including topology, algebraic geometry and number theory.”

The first Abel Prize has been awarded to Jean-Pierre Serre, one of the great mathematicians of our time. Serre is an Emeritus Professor at the Collège de France in Paris. He has made profound contributions to the progress of mathematics for over half a century, and continues to do so. Serre’s work is of extraordinary breadth, depth and influence. He has played a key role in shaping the modern form of many parts of mathematics, including:

- Topology, which treats the question: what remains the same in geometry even when the length is distorted? - Algebraic geometry, which treats the question: what is the geometry of solutions of Abel Laureate Jean-Pierre Serre. polynomial equations? (Photo: Anne Lise Flavik) - Number theory, the study of basic properties of numbers. For example prime numbers and the solution of polynomial equations as in Fermat’s Last Theorem.

Serre developed revolutionary algebraic methods for studying topology, and in particular studied the transformations between spheres of higher dimensions. He is

* 15.9.1926

1 2 Abel Prize 2003 responsible for a spectacular clarification of the work of the Italian algebraic geometers by introducing and developing the right algebraic machinery for determing when their geometric construction worked. This powerful technique of Serre, with its new language and viewpoint, ushered in a golden age for algebraic geometry. For the past four decades Serre’s magnificent work and vision of number theory have been instrumental in bringing that subject to its current glory. This work connects and extends in many ways the mathematical ideas introduced by Abel, in particular his proof of the impossibility of solving the 5th degree equation by radicals, and his analytic techniques for the study of polynomial equations in two variables. Serre’s research has been vital in setting the stage for many of the most celebrated recent breakthroughs, including the proof by Wiles of Fermat’s Last Theorem. Although Serre’s effort has been directed to more conceptual mathematics, his contributions have connection to important applications. The practical issues of finding efficient error-correcting codes and of public-key cryptography, both make use of solutions of polynomial equations (specifically over finite fields) and Serre’s work has substantially deepened our understanding of this topic.

Jean-Pierre Serre receives the prize from King Harald of Norway. (Photo: Arash A. Nejad) Jean-Pierre Serre 3

Topology

First, we congratulate you on winning the first Abel Prize. You started your career with a thesis that centred on algebraic topology. This was then (at least in France) a very new discipline and not a major area. What made you choose this topic? I was participating in the Cartan Seminar, on Algebraic Topology. But Cartan [1904–2008] did not suggest research topics to his students: they had to find one themselves; after that he would help them. This is what happened to me. I found that Leray’s [1906–1998] theory (about fibre spaces and their spectral sequence) could be applied to many more situations than was thought possible, and that such an extension could be used to compute homotopy groups.

The methods and results that you created in your thesis revolutionised homotopy theory and shaped it in its modern look… They certainly opened up lots of possibilities. Before my thesis, homotopy groups of spheres were almost entirely terra incognita; one did not even know that they are finitely generated! One interesting aspect of the method I introduced was its algebraic character. In particular, one could make “local” computations, where the word “local” here is taken as in number theory: relative to a given prime number.

Is it true that one of the crucial points in this story was to identify something that looks like a fibre space without it being on the nose? Indeed, to apply Leray’s theory I needed to construct fibre spaces which did not exist if one used the standard definition. Namely, for every space X, I needed a fibre space E with base X and with trivial homotopy (for instance contractible). But how to get such a space? One night in 1950, on the train bringing me back from our summer vacation, I saw it in a flash: just take for E the space of paths on X (with fixed origin a), the projection E → X being the evaluation map: path → extremity of the path. The fibre is then the loop space of (X, a). I had no doubt: this was it! So much so that I even waked up my wife to tell her… (Of course, I still had to show that E → X deserves to be called a “fibration”, and that Leray’s theory applies to it. This was purely technical, but not completely easy.) It is strange that such a simple construction had so many consequences.

Work themes and work style

This story about your sudden observation is reminiscent of Poincaré’s [1854– 1912] flash of insight when stepping into a tramway: this is told in Hadamard’s booklet The psychology of invention in the mathematical field. Do you often rely on sudden inspiration or would you rather characterise your work style as systematic? Or is it a mixture? 4 Abel Prize 2003

There are topics to which I come back from time to time (l-adic representations, for instance), but I do not do this in a really systematic way. I rather follow my nose. As for flashes, like the one Hadamard [1865–1963] described, I have had only two or three in more than 50 years. They are wonderful … but much too rare!

These flashes come after a long effort, we guess? I would not use the word “effort” in that case. Maybe a lot of thinking. It is not the conscious part of the mind which does the job. This is very well explained in Little- wood’s [1885–1977] charming book A Mathematician’s Miscellany.

Most of your work, since the ‘topology years’, has been devoted to number theory and algebraic geometry. You see, I work in several apparently different topics, but in fact they are all related to each other. I do not feel that I am really changing. For instance, in number theory, group theory or algebraic geometry, I use ideas from topology, such as cohomology, sheaves and obstructions. From that point of view, I especially enjoyed working on l-adic representations and modular forms: one needs number theory, algebraic geometry, Lie groups (both real and l-adic), q-expansions (combinatorics style) … A wonderful mélange.

Do you have a geometric or an algebraic intuition and way of thinking – or both? I would say algebraic, but I understand the geometric language better than the purely algebraic one: if I have to choose between a Lie group and a bi-algebra, I choose the Lie group! Still, I don’t feel I am a true geometer, such as Bott [1923– 2005], or Gromov. I also like analysis, but I can’t pretend to be a true analyst either. The true analyst knows at first sight what is “large”, “small”, “probably small” and “provably small” (not the same thing). I lack that intuitive feeling: I need to write down pedestrian estimates.

You have had a long career and have worked on many different subjects. Which of your theories or results do you like most? Which are most important to you? A delicate question. Would you ask a mother which of her children she prefers? All I can say is that some of my papers were very easy to write, and some others were truly difficult. In the first category, there is FAC (“faisceaux algébriques cohérents”). When I wrote it, I felt that I was merely copying a text which already existed; there was almost no effort on my part. In the “difficult” category, I remember a paper on open subgroups of profinite groups, which gave me so much trouble that, until the very end, I was not sure whether I was proving the theorem or making a counter-example! Another difficult one was the paper dedicated to Manin where I made some very precise (and very daring) conjectures on “modular” Galois representations (mod p); this one was even painful; after I had finished it, I was so exhausted that I stopped publishing for several years. Jean-Pierre Serre 5

On the pleasure side, I should mention a paper dedicated to Borel [1923–2003], on tensor products of group representations in characteristic p. I had been a group theory lover since my early twenties, and I had used groups a lot, and even proved a few theorems on them. But the theorem on tensor products, obtained when I was in my late sixties, was the first one I really enjoyed. I had the feeling that Group Theory, after a 40 years courtship, had consented to give me a kiss.

You have been active in the mathematical frontline for more than 50 years. Hardy made the often quoted remark that “Mathematics is a young man’s game”. Isn’t that wrong – aren’t you a perfect counterexample? Not a perfect one: have you noticed that most of the quotations of the Abel Prize are relative to things I had done before I was 30? What is true is that people of my generation (such as Atiyah, Borel, Bott, Shi- mura,…) keep working longer than our predecessors did (with a few remarkable exceptions such as Élie Cartan [1869–1951], Siegel [1896–1981], Zariski [1899– 1986]). I hope we shall continue.

Relations to mathematical history

Since you’ve won the Abel Prize, we’d like to ask some questions with a background in Abel’s time. The algebraic equations that Abel and Galois studied, coming from the transformation theory of elliptic functions, turned out to be very important much later for the arithmetic theory of elliptic curves. What are your comments on this remarkable fact, especially in connection with your own contribution to this theory? Yes, elliptic curves are very much in fashion (with good reasons, ranging from Lang- land’s program to cryptography). In the 60s and 70s I spent a lot of time studying their division points (a.k.a. Tate modules) and their Galois groups. A very entertain- ing game: one has to combine information coming from several different sources: Hodge-Tate decompositions, tame inertia, Frobenius elements, finiteness theorems à la Siegel,… I like that.

Hermite [1822–1901] once said that Abel had given mathematicians something to work on for the next 150 years. Do you agree? I dislike such grand statements as Hermite’s. They imply that the person who speaks knows what will happen in the next century. This is hubris.

In the introduction of one of his papers Abel writes that one should strive to give a problem a form such that it is always possible to solve it – something which he claims is always possible. And he goes on, saying that by presenting a problem in a well-chosen form the statement itself will contain the seeds of its solution. 6 Abel Prize 2003

An optimistic point of view! Grothendieck [1928–2014] would certainly share it. As for myself, I am afraid it applies only to algebraic questions, not to arithmetic ones. For instance, what would Abel have said about the Riemann hypothesis? That the form in which it is stated is not the good one?

The role of proofs

When you are doing mathematics, can you know that something is true even before you have the proof? Of course, this is very common. But one should distinguish between the genuine goal (say, the modularity of elliptic curves, in the case of Wiles), which one feels is surely true, and the auxiliary statements (lemmas, etc), which may well be untracta- ble (as happened to Wiles in his first attempt) or even downright false (as happened similarly to Lafforgue).

Do proofs always have a value in themselves? What about, for example, the proof of the four-colour theorem? We are entering a grey area: computer-aided proofs. They are not proofs in the standard sense that they can be checked by a line by line verification. They are especially unreliable when they claim to make a complete list of something or other. [I remember receiving in the 90s such a list for the subgroups of a given index of some discrete group. The computer had found, let us say, 20 of them. I was familiar with these groups, and I easily found “by hand” about 30 such. I wrote to the authors. They explained their mistake: they had made part of the computation in Japan, and another part in Germany, but they had forgotten to do some intermediate part… Typical!] On the other hand, computer-aided proofs are often more convincing than many standard proofs based on diagrams which are claimed to commute, arrows which are supposed to be the same, and arguments which are left to the reader.

What about the proof of the classification of the finite simple groups? You are pushing the right button. For years, I have been arguing with group theorists who claimed that the “Classification Theorem” was a “theorem”, i.e. had been proved. It had indeed been announced as such in 1980 by Gorenstein, but it was found later that there was a gap (the classification of “quasi-thin” groups). When- ever I asked the specialists, they replied something like: “Oh no, it is not a gap, it is just something which has not been written, but there is an incomplete unpublished 800 pages manuscript on it”. For me, it was just the same as a “gap”, and I could not understand why it was not acknowledged as such. Fortunately, Aschbacher and Smith have now written a long manuscript (more than 1200 pages) in order to fill in the gap. When this will have been checked by other experts, it will be the right moment to celebrate. Jean-Pierre Serre 7

But if a proof is 1200 pages long, what use is it? As a matter of fact, the total length of the proof of the classification is much more than 1200 pages; about 10 times more. But that is not surprising: the mere statement of the theorem is itself extremely long, since, in order to be useful, it has to include the detailed description, not only of the Chevalley groups, but also of the 26 sporadic groups. It is a beautiful theorem. It has many very surprising applications. I don’t think that using it raises a real problem for mathematicians in other fields: they just have to make clear what part of their proof depends on it.

Important mathematical problems

Do you feel that there are core or mainstream areas in mathematics – are some topics more important than others? A delicate question. Clearly, there are branches of mathematics which are less important; those where people just play around with a few axioms and their logical dependences. But it is not possible to be dogmatic about this. Sometimes, a neglected area becomes interesting, and develops new connections with other branches of mathematics. On the other hand, there are questions which are clearly central for our understanding of the mathematical world: the Riemann hypothesis and the Langlands program are two obvious cases. There is also the Poincaré conjecture – which may well stop being a conjecture, thanks to Perelman!

Do you have more information, or a hunch, about the correctness of the proof? Hunch? Who cares about hunches? Information? Not really, but I have heard that people at IHES and MIT are very excited about this sketch of proof. An interesting aspect of Perelman’s method is that it uses analysis, for what is a purely topological problem. Very satisfying.

We have already moved a little into the future with our discussion of the Poincaré conjecture. Which important mathematical problems would you like to see solved in the near future? And do you agree with the primary importance of the Clay Millennium Prize Problems? Ah, the million dollars Clay problems! A strange idea: giving so much money for one problem … Jean-Pierre Serre at the Prize lecture. but how can I criticise it, just after (Photo: Arash A. Nejad) 8 Abel Prize 2003 having received the Abel prize? Still, I feel there is some risk involved, namely that people would shy from discussing their partial results, as already happened ten years ago with Fermat’s theorem. As for the choice of questions made by the Clay Institute, I feel it is very good. The Riemann hypothesis and the Birch & Swinnerton–Dyer conjecture are rightly there. The Hodge conjecture, too; but for a different reason: it is not clear at all whether the answer will be yes or no; what will be very important will be to decide which (I am hoping, of course, that it will not turn out to be undecidable…). The P = NP question belongs to the same category as Hodge, except that there would be many more applications if the answer turned out to be “yes”.

Can you think of any other problems of the same stature? I already told you that the Langlands program is one of the major questions in mathematics nowadays. It was probably not included in the Clay list because it is very hard to formulate with the required precision.

Besides your scientific merits, you are also known as a master expositor, as we witnessed during your lecture today. Thanks. I come from the South of France, where people like to speak; not only with their mouth, but with their hands, and in my case with a piece of chalk. When I have understood something, I have the feeling that anybody else can understand it too, and it gives me great pleasure to explain it to other mathematicians, be they students or colleagues. Another side of the coin is that wrong statements make me almost physically sick. I can’t bear them. When I hear one in a lecture I usually interrupt the speaker, and when I find one in a preprint, a paper or in a book I write to the author (or, if the author happens to be myself, I make a note in view of a next edition). I am not sure this habit of mine has made me very popular among lec- turers and authors…

Accessibility and importance of mathematics

Mathematics witnesses an explosion of subjects and disciplines, making it difficult to master even the minor disciplines. On the other hand – as you demonstrated today in your lecture – it is very important that disciplines cross-fer- tilise each other. How can young mathematicians, in particular, cope with this explosion of knowledge and come up with something new?

1 An interview with J-P. Serre, The Mathematical Intelligencer 8 (1986), 8–13. Consult also: “Jean-Pierre Serre”, in Wolf Prize in Mathematics Vol. II (eds. S. S. Chern and F. Hirzebruch), World Sci. Publ. Co. (2001) 523–551; “Jean-Pierre Serre, medalla Fields” by Pilar Bayer, La Gaceta 4 (1) (2001), 211–247. Jean-Pierre Serre 9

Oh yes, I have already been asked that question in my Singapore interview, reproduced by the Intelli- gencer1. My answer is that, when one is truly interested in a specific question, there is usually very little in the existing literature which is relevant. This means you are on your own. As for the feeling of “explosion” of mathemat- From left to right: Martin Raussen, Christian Skau and Jean- Pierre Serre. ics, I am convinced that Abel felt the same way when he started working, after Euler, Lagrange, Legendre and Gauss. But he found new questions and new solutions. It has been the same ever since. There is no need to worry.

Another current problem is that many young and talented people – and also public opinion leaders – don't think that mathematics is very exciting. Yes. Sadly enough, there are many such examples. A few years ago, there was even a French minister of Research who was quoted as saying that mathematicians are not useful any more, since now it is enough to know how to punch a key on a computer. (He probably believed that keys and computer programs grow on trees…) Still, I am optimistic about young people discovering, and being attracted by, mathematics. One good aspect of the Abel festivities is the Norwegian Abel competitions, for high school students.

Sports and literature

Could you tell us about your interests besides mathematics? Sports! More precisely: skiing, ping-pong, and rock climbing. I was never really good at any of them (e.g. when I skied, I did not know how to slalom, so that I would rather go “schuss” than trying to turn); but I enjoyed them a lot. As luck has it, a consequence of old age is that my knees are not working any more (one of them is even replaced by a metal-plastic contraption), so that I had to stop doing any sport. The only type of rock-climbing I can do now is a vicarious one: taking friends to Fontainebleau and coaxing them into climbing the rocks I would have done ten years ago. It is still fun; but much less so than the real thing. 10 Abel Prize 2003

Other interests: - movies (“Pulp Fiction” is one of my favourites – I am also a fan of Altman [1925– 2006], Truffaut [1932–1984], Rohmer [1920–2010], the Coen brothers…); - chess; - books (of all kinds, from Giono [1895–1970] to Böll [1917–1985] and to Kawabata [1899–1972], including fairy tales and the “Harry Potter” series).

Prof. Serre, thank you for this interview on behalf of the Danish and the Nor- wegian Mathematical Societies. Abel Prize 2004: Sir Michael Francis Atiyah and Isadore M. Singer

The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2004 to

Sir Michael Francis Atiyah and Isadore M. Singer University of Edinburgh Massachusetts Institute of Technology

“for their discovery and proof of the index theorem, bringing together topology, geometry and analysis, and their outstanding role in building new bridges between mathematics and theoretical physics.”

Abel Laureate Sir Michael Francis Atiyah. Abel Laureate Isadore M. Singer. (Photo: Anne Lise Flavik) (Photo: Anne Lise Flavik)

* 22.4.1929 * 3.5.1924

11 12 Abel Prize 2004

The second Abel Prize has been awarded jointly to Michael Francis Atiyah and Isa- dore M. Singer. The Atiyah-Singer index theorem is one of the great landmarks of twentieth-century mathematics, influencing profoundly many of the most important later developments in topology, differential geometry and quantum field theory. Its authors, both jointly and individually, have been instrumental in repairing a rift between the worlds of pure mathematics and theoretical particle physics, initiating a cross-fertilization which has been one of the most exciting developments of the last decades. We describe the world by measuring quantities and forces that vary over time and space. The rules of nature are often expressed by formulas involving their rates of change, so-called differential equations. Such formulas may have an “index”, the number of solutions of the formulas minus the number of restrictions which they impose on the values of the quantities being computed. The index theorem calculates this number in terms of the geometry of the surrounding space. A simple case is illustrated by a famous paradoxical etching of M. C. Escher, “Ascending and Descending”, where the people, going uphill all the time, still manage to circle the castle courtyard. The index theorem would have told them this was impossible! The Atiyah-Singer index theorem was the culmination and crowning achievement of a more than one-hundred-year-old evolution of ideas, from Stokes’s theorem, which students learn in calculus classes, to sophisticated modern theories like Hodge’s theory of harmonic integrals and Hirzebruch’s signature theorem. The problem solved by the Atiyah-Singer theorem is truly ubiquitous. In the forty years since its discovery, the theorem has had innumerable applications, first in mathematics and then, beginning in the late 1970s, in theoretical physics: gauge theory, instantons, monopoles, string theory, the theory of anomalies etc. At first, the applications in physics came as a complete surprise to both the mathematics and phsysics communities. Now the index theorem has become an integral part of their cultures. Atiyah and Singer, together and individually, have been tireless in their attempts to explain the insights of physicists to mathematicians. At the same time, they brought modern differential geometry and analysis as it applies to quantum field theory to the attention of physicists and suggested new directions in physics itself. This cross-fertilization continues to be fruitful for both sciences. Michael Francis Atiyah and Isadore M. Singer are among the most influential mathematicians of the last century and are still working. With the index theorem they changed the landscape of mathematics. Over a period of twenty years they worked together on the index theorem and its ramifications. Atiyah and Singer came originally from different fields of mathematics: Atiyah from algebraic geometry and topology, Singer from analysis. Their main contributions in their respective areas are also highly recognized. Atiyah’s early work on meromorphic forms on algebraic varieties and his important 1961 paper on Thom complexes are such examples. Atiyah’s pioneering work together with Friedrich Hir- zebruch on the development of the topological analogue of Grothendieck’s K-theory Sir Michael Francis Atiyah and Isadore M. Singer 13 had numerous applications to classical problems of topology and turned out later to be deeply connected with the index theorem. Singer established the subject of triangular operator algebras (jointly with Richard V. Kadison). Singer’s name is also associated with the Ambrose-Singer holonomy theorem and the Ray-Singer torsion invariant. Singer, together with Henry P. McKean, pointed out the deep geometrical information hidden in heat kernels, a discovery that had great impact.

Isadore Singer and Sir Michael Atiyah receive the Abel Prize from King Harald. (Photo: Knut Falch) 14 Abel Prize 2004

The Index Theorem

First, we congratulate both of you for having been awarded the Abel Prize 2004. This prize has been given to you for “the discovery and the proof of the Index Theorem connecting geometry and analysis in a surprising way and your outstanding role in building new bridges between mathematics and theoretical physics”. Both of you have an impressive list of fine achievements in mathematics. Is the Index Theorem your most important result and the result you are most pleased with in your entire careers? ATIYAH First, I would like to say that I prefer to call it a theory, not a theorem. Actually, we have worked on it for 25 years and if I include all the related topics, I have probably spent 30 years of my life working in this area. So it is rather obvious that it is the best thing I have done.

SINGER I too, feel that the Index Theorem was but the beginning of a high point that has lasted to this very day. It’s as if we climbed a mountain and found a plateau we’ve been on ever since.

We would like you to give us some comments on the history of the discovery of the Index Theorem.1 Were there precursors, conjectures in this direction already before you started? Were there only mathematical motivations or also physical ones? ATIYAH Mathematics is always a continuum, linked to its history, the past – nothing comes out of zero. And certainly the Index Theorem is simply a continuation of work that, I would like to say, began with Abel [1802–1829]. So of course there are precursors. A theorem is never arrived at in the way that logical thought would lead you to believe or that posterity thinks. It is usually much more accidental, some chance discovery in answer to some kind of question. Eventually you can rationalize it and say that this is how it fits. Discoveries never happen as neatly as that. You can rewrite history and make it look much more logical, but actually it happens quite differently.

SINGER At the time we proved the Index Theorem we saw how important it was in mathematics, but we had no inkling that it would have such an effect on physics some years down the road. That came as a complete surprise to us. Perhaps it should not have been a surprise because it used a lot of geometry and also quantum mechanics in a way, à la Dirac [1902–1984].

You worked out at least three different proofs with different strategies for the Index Theorem. Why did you keep on after the first proof? What different insights did the proofs give?

1 More details were given in the laureates’ lectures. Sir Michael Francis Atiyah and Isadore M. Singer 15

ATIYAH I think it is said that Gauss had ten different proofs for the law of quadratic reciprocity. Any good theorem should have several proofs, the more the better. For two reasons: usually, different proofs have different strengths and weaknesses, and they generalize in different directions – they are not just repetitions of each other. And that is certainly the case with the proofs that we came up with. There are different reasons for the proofs, they have different histories and backgrounds. Some of them are good for this application, some are good for that application. They all shed light on the area. If you cannot look at a problem from different directions, it is probably not very interesting; the more perspectives, the better!

SINGER There isn’t just one theorem; there are generalizations of the theorem. One is the Index Theorem for a family of elliptic operators using K-theory; another is the heat equation proof which makes the formulas that are topological, more geometric and explicit. Each theorem and proof has merit and has different applications.

Collaboration

Both of you contributed to the Index Theorem with different expertise and vi- sions – and other people had a share as well. Could you describe this collaboration and the establishment of the result a little closer? SINGER Well, I came with a background in analysis and differential geometry, and Sir Michael’s expertise was in algebraic geometry and topology. For the purposes of the Index Theorem, our areas of expertise fit together hand in glove. Moreover, in a way, our personalities fit together, in that “anything goes”: Make a suggestion – and whatever it was, we would just put it on the blackboard and work with it; we would both enthusiastically explore it; if it didn’t work, it didn’t work. But often enough, some idea that seemed far-fetched did work. We both had the freedom to continue without worrying about where it came from or where it would lead. It was exciting to work with Sir Michael all these years. And it is as true today as it was when we first met in ’55 – that sense of excitement and “anything goes” and “let’s see what happens”.

ATIYAH No doubt: Singer had a strong expertise and background in analysis and differential geometry. And he knew certainly more physics than I did; it turned out to be very useful later on. My background was in algebraic geometry and topology, so it all came together. But of course there are a lot of people who contributed in the background to the build-up of the Index Theorem – going back to Abel, Riemann [1826–1866], much more recently Serre, who got the Abel prize last year, Hirze- bruch [1927–2012], Grothendieck [1928–2014] and Bott [1923–2005]. There was lots of work from the algebraic geometry side and from topology that prepared the ground. And of course there are also a lot of people who did fundamental work in analysis and in the study of differential equations: Hörmander [1931–2012], Niren- 16 Abel Prize 2004 berg… In my lecture I will give a long list of names2; even that one will be partial. It is an example of international collaboration; you do not work in isolation, neither in terms of time nor in terms of space – especially in these days. Mathematicians are linked so much, people travel around much more. We two met at the Institute at Princeton. It was nice to go to the Arbeitstagung in Bonn every year, which Hirze- bruch organised and where many of these other people came. I did not realize that at the time, but looking back, I am very surprised how quickly these ideas moved…

Collaboration seems to play a bigger role in mathematics than earlier. There are a lot of conferences, we see more papers that are written by two, three or even more authors – is that a necessary and commendable development or has it drawbacks as well? ATIYAH It is not like in physics or chemistry where you have 15 authors because they need an enormous big machine. It is not absolutely necessary or fundamental. But particularly if you are dealing with areas which have rather mixed and interdis- ciplinary backgrounds, with people who have different expertise, it is much easier and faster. It is also much more interesting for the participants. To be a mathematician on your own in your office can be a little bit dull, so interaction is stimulating, both psychologically and mathematically. It has to be admitted that there are times when you go solitary in your office, but not all the time! It can also be a social activity with lots of interaction. You need a good mix of both, you can’t be talking all the time. But talking some of the time is very stimulating. Summing up, I think that it is a good development – I do not see any drawbacks.

SINGER Certainly computers have made collaboration much easier. Many mathematicians collaborate by computer instantly; it’s as if they were talking to each other. I am unable to do that. A sobering counterexample to this whole trend is Perelman’s results on the Poincaré conjecture: He worked alone for ten to twelve years, I think, before putting his preprints on the net.

ATIYAH Fortunately, there are many different kinds of mathematicians, they work on different subjects, they have different approaches and different personalities – and that is a good thing. We do not want all mathematicians to be isomorphic, we want variety: different mountains need different kinds of techniques to climb.

SINGER I support that. Flexibility is absolutely essential in our society of mathematicians.

2 Among those: Newton [1643–1727], Gauss, Cauchy [1871–1946], Laplace [1749–1827], Abel, Jacobi [1804–1851], Riemann, Weierstrass [1815–1897], Lie [1842–1899], Picard [1856–1941], Poincaré [1854–1912], Castelnuovo [1865–1952], Enriques [1871–1946], Severi [1879–1961], Hilbert [1862–1943], Lefschetz [1884–1972], Hodge [1903–1975], Todd [1911–2007], Leray [1906–1998], Cartan [1904–2008], Serre, Kodaira [1915–1997], Spencer, Dirac, Pontryagin [1908–1988], Chern [1911–2004], Weil [1906–1998], Borel [1923–2003], Hirzebruch, Bott, Eilenberg [1913–1998], Grothendieck, Hörmander, Nirenberg. Sir Michael Francis Atiyah and Isadore M. Singer 17

Perelman’s work on the Poincaré conjecture seems to be another instance where analysis and geometry apparently get linked very much together. It seems that geometry is profiting a lot from analytic perspectives. Is this linkage between different disciplines a general trend – is it true, that important results rely on this interrelation between different disciplines? And a much more specific question: What do you know about the status of the proof of the Poincaré conjecture? SINGER To date, everything is working out as Perelman says. So I learn from Lott’s seminar at the University of Michigan and Tian’s seminar at Princeton. Although no one vouches for the final details, it appears that Perelman’s proof will be validated. As to your first question: When any two subjects use each other’s techniques in a new way, frequently, something special happens. In geometry, analysis is very important; for existence theorems, the more the better. It is not surprising that some new [at least to me] analysis implies something interesting about the Poincaré conjecture.

ATIYAH I prefer to go even further – I really do not believe in the division of mathematics into specialities; already if you go back into the past, to Newton and Gauss… Although there have been times, particularly post-Hilbert, with the axiomatic approach to mathematics in the first half of the twentieth century, when people began to specialize, to divide up. The Bourbaki trend had its use for a particular time. But this is not part of the general attitude to mathematics: Abel would not have distinguished between algebra and analysis. And I think the same goes for geometry and analysis for people like Newton. It is artificial to divide mathematics into separate chunks, and then to say that you bring them together as though this is a surprise. On the contrary, they are all part of the puzzle of mathematics. Sometimes you would develop some things for their own sake for a while e.g. if you develop group theory by itself. But that is just a sort of temporary convenient division of labour. Fundamentally, mathematics should be used as a unity. I think the more examples we have of people showing that you can usefully apply analysis to geometry, the better. And not just analysis, I think that some physics came into it as well: Many of the ideas in geometry use physical insight as well – take the example of Riemann! This is all part of the broad mathematical tradition, which sometimes is in danger of being overlooked by modern, younger people who say “we have separate divisions”. We do not want to have any of that kind, really.

SINGER The Index Theorem was in fact instrumental in breaking barriers between fields. When it first appeared, many oldtimers in special fields were upset that new techniques were entering their fields and achieving things they could not do in the field by old methods. A younger generation immediately felt freed from the barriers that we both view as artificial.

ATIYAH Let me tell you a little story about Henry Whitehead [1904–1960], the topologist. I remember that he told me that he enjoyed very much being a topolo- 18 Abel Prize 2004 gist: He had so many friends within topology, and it was such a great community. “It would be a tragedy if one day I would have a brilliant idea within functional analysis and would have to leave all my topology friends and to go out and work with a different group of people.” He regarded it to be his duty to do so, but he would be very reluctant. Somehow, we have been very fortunate. Things have moved in such a way that we got involved with functional analysts without losing our old friends; we could bring them all with us. Alain Connes was in functional analysis, and now we interact closely. So we have been fortunate to maintain our old links and move into new ones – it has been great fun.

Mathematics and physics

We would like to have your comments on the interplay between physics and mathematics. There is Galilei’s [1564–1642] famous dictum from the beginning of the scientific revolution, which says that the Laws of Nature are written in the language of mathematics. Why is it that the objects of mathematical creation, satisfying the criteria of beauty and simplicity, are precisely the ones that time and time again are found to be essential for a correct description of the external world? Examples abound, let us just mention group theory and, yes, your Index Theorem! SINGER There are several approaches in answer to your questions; I will discuss two. First, some parts of mathematics were created in order to describe the world around us. Calculus began by explaining the motion of planets and other moving objects. Calculus, differential equations, and integral equations are a natural part of physics because they were developed for physics. Other parts of mathematics are also natural for physics. I remember lecturing in Feynman’s [1918–1988] seminar, trying to explain anomalies. His postdocs kept wanting to pick coordinates in order to compute; he stopped them saying: “The Laws of Physics are independent of a coordinate system. Listen to what Singer has to say, because he is describing the situation without coordinates.” Coordinate-free means geometry. It is natural that geometry appears in physics, whose laws are independent of a coordinate system. Symmetries are useful in physics for much the same reason they’re useful in mathematics. Beauty aside, symmetries simplify equations, in physics and in mathematics. So physics and math have in common geometry and group theory, creating a close connection between parts of both subjects Secondly, there is a deeper reason if your question is interpreted as in the title of Eugene Wigner’s [1902–1995] essay “The Unreasonable Effectiveness of Math- ematics in the Natural Sciences”3. Mathematics studies coherent systems which I will not try to define. But it studies coherent systems, the connections between

3 Comm. Pure App. Math. 13(1), 1960. Sir Michael Francis Atiyah and Isadore M. Singer 19 such systems and the structure of such systems. We should not be too surprised that mathematics has coherent systems applicable to physics. It remains to be seen whether there is an already developed coherent system in mathematics that will describe the structure of string theory. [At present, we do not even know what the symmetry group of string field theory is.] Witten has said that 21st century mathematics has to develop new mathematics, perhaps in conjunction with physics intuition, to describe the structure of string theory.

ATIYAH I agree with Singer’s description of mathematics having evolved out of the physical world; it therefore is not a big surprise that it has a feedback into it. More fundamentally: to understand the outside world as a human being is an attempt to reduce complexity to simplicity. What is a theory? A lot of things are happening in the outside world, and the aim of scientific inquiry is to reduce this to as simple a number of principles as possible. That is the way the human mind works, the way the human mind wants to see the answer. If we were computers, which could tabulate vast amounts of all sorts of information, we would never develop theory – we would say, just press the button to get the answer. We want to reduce this complexity to a form that the human mind can understand, to a few simple principles. That’s the nature of scientific inquiry, and mathematics is a part of that. Mathematics is an evolution from the human brain, which is responding to outside influences, creating the machinery with which it then attacks the outside world. It is our way of trying to reduce complexity into simplicity, beauty and elegance. It is really very fundamental, simplicity is in the nature of scientific inquiry – we do not look for complicated things. I tend to think that science and mathematics are ways the human mind looks and experiences – you cannot divorce the human mind from it. Mathematics is part of the human mind. The question whether there is a reality independent of the human mind, has no meaning – at least, we cannot answer it.

Is it too strong to say that the mathematical problems solved and the techniques that arose from physics have been the lifeblood of mathematics in the past; or at least for the last 25 years? ATIYAH I think you could turn that into an even stronger statement. Almost all mathematics originally arose from external reality, even numbers and counting. At some point, mathematics then turned to ask internal questions, e.g. the theory of prime numbers, which is not directly related to experience but evolved out of it. There are parts of mathematics where the human mind asks internal questions just out of curiosity. Originally it may be physical, but eventually it becomes something independent. There are other parts that relate much closer to the outside world with much more interaction backwards and forward. In that part of it, physics has for a long time been the lifeblood of mathematics and inspiration for mathematical work. There are times when this goes out of fashion or when parts of mathematics evolve purely internally. Lots of abstract mathematics does not directly relate to the outside world. 20 Abel Prize 2004

It is one of the strengths of mathematics that it has these two and not a single lifeblood: one external and one internal, one arising as response to external events, the other to internal reflection on what we are doing. SINGER Your statement is too strong. I agree with Michael that mathematics is blessed with both an external and internal source of inspiration. In the past several decades, high energy theoretical physics has had a marked influence on mathematics. Many mathematicians have been shocked at this unexpected development: new ideas from outside mathematics so effective in mathematics. We are delighted with these new inputs, but the “shock” exaggerates their overall effect on mathematics.

Newer developments

Can we move to newer developments with impact from the Atiyah–Singer In- dex Theorem? I.e., string theory and Edward Witten on the one hand and on the other hand non-commutative geometry represented by Alain Connes. Could you describe the approaches to mathematical physics epitomized by these two protagonists? ATIYAH I tried once in a talk to describe the different approaches to progress in physics like different religions. You have prophets, you have followers – each prophet and his followers think that they have the sole possession of the truth. If you take the strict point of view that there are several different religions, and that the intersection of all these theories is empty, then they are all talking nonsense. Or you can take the view of the mystic, who thinks that they are all talking of different aspects of reality, and so all of them are correct. I tend to take the second point of view. The main “orthodox” view among physicists is certainly represented by a very large group of people working with string theory like Edward Witten. There are a small number of people who have different philosophies, one of them is Alain Connes, and the other is Roger Penrose. Each of them has a very specific point of view; each of them has very interesting ideas. Within the last few years, there have been non-trivial interactions between all of these. They may all represent different aspects of reality and eventually, when we under stand it all, we may say “Ah, yes, they are all part of the truth”. I think that that will happen. It is difficult to say which will be dominant, when we finally understand the picture – we don’t know. But I tend to be open-minded. The problem with a lot of physicists is that they have a tendency to “follow the leader”: as soon as a new idea comes up, ten people write ten or more papers on it and the effect is that everything can move very fast in a technical direction. But big progress may come from a different direction; you do need people who are exploring different avenues. And it is very good that we have people like Connes and Penrose with their own independent line from different origins. I am in favour of diversity. I prefer not to close the door or to say “they are just talking nonsense”.

SINGER String theory is in a very special situation at the present time. Physicists have found new solutions on their landscape – so many that you cannot expect to Sir Michael Francis Atiyah and Isadore M. Singer 21 make predictions from string theory. Its original promise has not been fulfilled. Nevertheless, I am an enthusiastic supporter of super string theory, not just because of what it has done in mathematics, but also because as a coherent whole, it is a marvellous subject. Every few years new developments in the theory give additional insight. When that happens, you realize how little one understood about string theory previously. The theory of D-branes is a recent example. Often there is mathematics closely associated with these new insights. Through D-branes, K-theory entered string theory naturally and reshaped it. We just have to wait and see what will happen. I am quite confident that physics will come up with some new ideas in string theory that will give us greater insight into the structure of the subject, and along with that will come new uses of mathematics. Alain Connes’ program is very natural – if you want to combine geometry with quantum mechanics, then you really want to quantize geometry, and that is what noncommutative geometry means. Non-commutative geometry has been used effectively in various parts of string theory explaining what happens at certain singularities, for example. I think it may be an interesting way of trying to describe black holes and to explain the Big Bang. I would encourage young physicists to understand non-commutative geometry more deeply than they presently do. Physicists use only parts of non-commutative geometry; the theory has much more to offer. I do not know whether it is going to lead anywhere or not. But one of my projects is to try and redo some known results using non-commutative geometry more fully.

If you should venture a guess, which mathematical areas do you think are going to witness the most important developments in the coming years? ATIYAH One quick answer is that the most exciting developments are the ones which you cannot predict. If you can predict them, they are not so exciting. So, by definition, your question has no answer. Ideas from physics, e.g. quantum theory, have had an enormous impact so far, in geometry, some parts of algebra, and in topology. The impact on number theory has still been quite small, but there are some examples. I would like to make a rash prediction that it will have a big impact on number theory as the ideas flow across mathematics – on one extreme number theory, on the other physics, and in the middle geometry: the wind is blowing, and it will eventually reach to the farthest extremities of number theory and give us a new point of view. Many problems that are worked upon today with old-fashioned ideas will be done with new ideas. I would like to see this happen: it could be the Riemann hypothesis, it could be the Langlands program or a lot of other related things. I had an argument with Andrew Wiles where I claimed that physics will have an impact on his kind of number theory; he thinks this is nonsense but we had a good argument. I would also like to make another prediction, namely that fundamental progress on the physics/mathematics front, string theory questions etc., will emerge from a much more thorough understanding of classical four-dimensional geometry, of Einstein’s Equations etc. The hard part of physics in some sense is the non-linearity of Einstein’s Equations. Everything that has been done at the moment is circum- 22 Abel Prize 2004 venting this problem in lots of ways. They haven’t really got to grips with the hardest part. Big progress will come when people by some new techniques or new ideas really settle that. Whether you call that geometry, differential equations or physics depends on what is going to happen, but it could be one of the big breakthroughs. These are of course just my speculations.

SINGER I will be speculative in a slightly different way, though I do agree with the number theory comments that Sir Michael mentioned, particularly theta functions entering from physics in new ways. I think other fields of physics will affect mathematics – like statistical mechanics and condensed matter physics. For example, I predict a new subject of statistical topology. Rather than count the number of holes, Betti-numbers, etc., one will be more interested in the distribution of such objects on noncompact manifolds as one goes out to infinity. We already have precursors in the number of zeros and poles for holomorphic functions. The theory that we have for holomorphic functions will be generalized, and insights will come from condensed matter physics as to what, statistically, the topology might look like as one approaches infinity.

Continuity of mathematics

Mathematics has become so specialized, it seems, that one may fear that the subject will break up into separate areas. Is there a core holding things together? ATIYAH I like to think there is a core holding things together, and that the core is rather what I look at myself; but we tend to be rather egocentric. The traditional parts of mathematics, which evolved – geometry, calculus and algebra – all centre on certain notions. As mathematics develops, there are new ideas, which appear to be far from the centre going off in different directions, which I perhaps do not know much about. Sometimes they become rather important for the whole nature of the mathematical enterprise. It is a bit dangerous to restrict the definition to just whatever you happen to understand yourself or think about. For example, there are parts of mathematics that are very combinatorial. Sometimes they are very closely related to the continuous setting, and that is very good: we have interesting links between combinatorics and algebraic geometry and so on. They may also be related to e.g. statistics. I think that mathematics is very difficult to constrain; there are also all sorts of new applications in different directions. It is nice to think of mathematics having a unity; however, you do not want it to be a straitjacket. The centre of gravity may change with time. It is not necessarily a fixed rigid object in that sense, I think it should develop and grow. I like to think of mathematics having a core, but I do not want it to be rigidly defined so that it excludes things which might be interesting. You do not want to exclude somebody who has made a discovery saying: “You are outside, you are not doing mathematics, you are playing around”. You never know! That particular discovery might be the mathematics of the next century; you have got to be careful. Very often, when new Sir Michael Francis Atiyah and Isadore M. Singer 23 ideas come in, they are regarded as being a bit odd, not really central, because they look too abstract.

SINGER Countries differ in their attitudes about the degree of specialization in mathematics and how to treat the problem of too much specialization. In the United States I observe a trend towards early specialization driven by economic considerations. You must show early promise to get good letters of recommen- dations to get good first jobs. You can’t afford to branch out until you have established yourself and have a secure position. The realities of life force a narrowness in perspective that is not inherent to mathematics. We can counter too much specialization with new resources that would give young people more freedom than they presently have, freedom to explore mathematics more broadly, or to explore connections with other subjects, like biology these days where there are lots to be discovered. When I was young the job market was good. It was important to be at a major university but you could still prosper at a smaller one. I am distressed by the coer- cive effect of today’s job market. Young mathematicians should have the freedom of choice we had when we were young.

The next question concerns the continuity of mathematics. Rephrasing slightly a question that you, Prof. Atiyah, are the origin of, let us make the following Gedanken Experiment: If, say, Newton or Gauss or Abel were to reappear in our midst, do you think they would understand the problems being tackled by the present generation of mathematicians – after they had been given a short refresher course? Or is present day mathematics too far removed from traditional mathematics? ATIYAH The point that I was trying to make there was that really important progress in mathematics is somewhat independent of technical jargon. Important ideas can be explained to a really good mathematician, like Newton or Gauss or Abel, in conceptual terms. They are in fact coordinate-free, more than that, technology-free and in a sense jargon-free. You don’t have to talk of ideals, modules or whatever – you can talk in the common language of scientists and mathematicians. The really important progress mathematics has made within 200 years could easily be understood by people like Gauss and Newton and Abel. Only a small refresher course where they were told a few terms – and then they would immediately understand. Actually, my pet aversion is that many mathematicians use too many technical terms when they write and talk. They were trained in a way that if you do not say it 100 percent correctly, like lawyers, you will be taken to court. Every statement has to be fully precise and correct. When talking to other people or scientists, I like to use words that are common to the scientific community, not necessarily just to mathematicians. And that is very often possible. If you explain ideas without a vast amount of technical jargon and formalism, I am sure it would not take Newton, Gauss and Abel long – they were bright guys, actually! 24 Abel Prize 2004

SINGER One of my teachers at Chicago was André Weil, and I remember his saying: “If Riemann were here, I would put him in the library for a week, and when he came out he would tell us what to do next.”

Communication of mathematics

Next topic: Communication of mathematics. Hilbert, in his famous speech at the International Congress in 1900, in order to make a point about mathematical communication, cited a French mathematician who said: “A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street”. In order to pass on to new generations of mathematicians the collective knowledge of the previous generation, how important is it that the results have simple and elegant proofs? ATIYAH The passing of mathematics on to subsequent generations is essential for the future, and this is only possible if every generation of mathematicians understands what they are doing and distils it out in such a form that it is easily understood by the next generation. Many complicated things get simple when you have the right point of view. The first proof of something may be very complicated, but when you understand it well, you readdress it, and eventually you can present it in a way that makes it look much more understandable – and that’s the way you pass it on to the next generation! Without that, we could never make progress – we would have all this messy stuff. Mathematics does depend on a sufficiently good grasp, on understanding of the fundamentals so that we can pass it on in as simple a way as possible to our successors. That has been done remarkably successfully for centuries. Otherwise, how could we possibly be where we are? In the 19th century, people said: “There is so much mathematics, how could anyone make any progress?” Well, we have – we do it by various devices, we generalize, we put all things together, we unify by new ideas, we simplify lots of the constructions – we are very successful in mathematics and have been so for several hundred years. There is no evidence that this has stopped: in every new generation, there are mathematicians who make enormous progress. How do they learn it all? It must be because we have been successful communicating it.

SINGER I find it disconcerting speaking to some of my young colleagues, because they have absorbed, reorganized, and simplified a great deal of known material into a new language, much of which I don’t understand. Often I’ll finally say, “Oh; is that all you meant?” Their new conceptual framework allows them to encompass succinctly considerably more than I can express with mine. Though impressed with the progress, I must confess impatience because it takes me so long to understand what is really being said. Has the time passed when deep and important theorems in mathematics can be given short proofs? In the past, there are many such examples, e.g., Abel’s one-page Sir Michael Francis Atiyah and Isadore M. Singer 25 proof of the addition theorem of algebraic differentials or Goursat’s [1858–1936] proof of Cauchy’s integral theorem.

ATIYAH I do not think that at all! Of course, that depends on what foundations you are allowed to start from. If we have to start from the axioms of mathematics, then every proof will be very long. The common framework at any given time is constantly advancing; we are already at a high platform. If we are allowed to start within that framework, then at every stage there are short proofs. One example from my own life is this famous problem about vector fields on spheres solved by Frank Adams [1930–1989] where the proof took many hundreds of pages. One day I discovered how to write a proof on a postcard. I sent it over to Frank Adams and we wrote a little paper which then would fit on a bigger postcard. But of course that used some K-theory; not that complicated in itself. You are always building on a higher platform; you have always got more tools at your disposal that are part of the lingua franca which you can use. In the old days you had a smaller base: If you make a simple proof nowadays, then you are allowed to assume that people know what group theory is, you are allowed to talk about Hilbert space. Hilbert space took a long time to develop, so we have got a much bigger vocabulary, and with that we can write more poetry.

SINGER Often enough one can distil the ideas in a complicated proof and make that part of a new language. The new proof becomes simpler and more illuminating. For clarity and logic, parts of the original proof have been set aside and discussed separately.

ATIYAH Take your example of Abel’s Paris memoir: His contemporaries did not find it at all easy. It laid the foundation of the theory. Only later on, in the light of that theory, we can all say: “Ah, what a beautifully simple proof!” At the time, all the ideas had to be developed, and they were hidden, and most people could not read that paper. It was very, very far from appearing easy for his contemporaries.

Individual work style

I heard you, Prof. Atiyah, mention that one reason for your choice of mathematics for your career was that it is not necessary to remember a lot of facts by heart. Nevertheless, a lot of threads have to be woven together when new ideas are developed. Could you tell us how you work best, how do new ideas arrive? ATIYAH My fundamental approach to doing research is always to ask questions. You ask “Why is this true?” when there is something mysterious or if a proof seems very complicated. I used to say – as a kind of joke – that the best ideas come to you during a bad lecture. If somebody gives a terrible lecture, it may be a beautiful result but with terrible proofs, you spend your time trying to find better ones, you do not listen to the lecture. It is all about asking questions – you simply have to have 26 Abel Prize 2004 an inquisitive mind! Out of ten questions, nine will lead nowhere, and one leads to something productive. You constantly have to be inquisitive and be prepared to go in any direction. If you go in new directions, then you have to learn new material. Usually, if you ask a question or decide to solve a problem, it has a background. If you understand where a problem comes from then it makes it easy for you to understand the tools that have to be used on it. You immediately interpret them in terms of your own context. When I was a student, I learned things by going to lectures and reading books – after that I read very few books. I would talk with people; I would learn the essence of analysis by talking to Hörmander or other people. I would be asking questions because I was interested in a particular problem. So you learn new things because you connect them and relate them to old ones, and in that way you can start to spread around. If you come with a problem, and you need to move to a new area for its solution, then you have an introduction – you have already a point of view. Interacting with other people is of course essential: if you move into a new field, you have to learn the language, you talk with experts; they will distil the essentials out of their experience. I did not learn all the things from the bottom upwards; I went to the top and got the insight into how you think about analysis or whatever.

SINGER I seem to have some built-in sense of how things should be in mathematics. At a lecture, or reading a paper, or during a discussion, I frequently think, “that’s not the way it is supposed to be.” But when I try out my ideas, I’m wrong 99% of the time. I learn from that and from studying the ideas, techniques, and procedures of successful methods. My stubbornness wastes lots of time and energy. But on the rare occasion when my internal sense of mathematics is right, I’ve done something different.

Both of you have passed ordinary retirement age several years ago. But you are still very active mathematicians, and you have even chosen retirement or visiting positions remote from your original work places. What are the driving forces for keeping up your work? Is it wrong that mathematics is a “young man’s game” as Hardy [1877–1947] put it? ATIYAH It is no doubt true that mathematics is a young man’s game in the sense that you peak in your twenties or thirties in terms of intellectual concentration and in originality. But later you compensate that by experience and other factors. It is also true that if you haven’t done anything significant by the time you are forty, you will not do so suddenly. But it is wrong that you have to decline, you can carry on, and if you manage to diversify in different fields this gives you a broad coverage. The kind of mathematician who has difficulty maintaining the momentum all his life is a person who decides to work in a very narrow field with great depths, who e.g. spends all his life trying to solve the Poincaré conjecture – whether you succeed or not, after 10–15 years in this field you exhaust your mind; and then, it may be too late to diversify. If you are the sort of person that chooses to make restrictions to yourself, to specialize in a field, you will find it harder and harder – because the only Sir Michael Francis Atiyah and Isadore M. Singer 27 things that are left are harder and harder technical problems in your own area, and then the younger people are better than you. You need a broad base, from which you can evolve. When this area dries out, then you go to that area – or when the field as a whole, internationally, changes gear, you can change too. The length of the time you can go on being active within mathematics very much depends on the width of your coverage. You might have contributions to make in terms of perspective, breadth, interactions. A broad coverage is the secret of a happy and successful long life in mathematical terms. I cannot think of any counter example.

SINGER I became a graduate student at the University of Chicago after three years in the US army during World War II. I was older and far behind in mathematics. So I was shocked when my fellow graduate students said, “If you haven’t proved the Riemann Hypothesis by age thirty, you might as well commit suicide.” How infantile! Age means little to me. What keeps me going is the excitement of what I’m doing and its possibilities. I constantly check [and collaborate!] with younger colleagues to be sure that I’m not deluding myself – that what we are doing is interesting. So I’m happily active in mathematics. Another reason is, in a way, a joke. String theory needs us! String theory needs new ideas. Where will they come from, if not from Sir Michael and me?

ATIYAH Well, we have some students…

SINGER Anyway, I am very excited about the interface of geometry and physics, and delighted to be able to work at that frontier.

History of the EMS

You, Prof. Atiyah, have been very much involved in the establishment of the European Mathematical Society around 1990. Are you satisfied with its development since then? ATIYAH Let me just comment a little on my involvement. It started an awful long time ago, probably about 30 years ago. When I started trying to get people interested in forming a European Mathematical Society in the same spirit as the Euro- pean Physical Society, I thought it would be easy. I got mathematicians from different countries together and it was like a mini-UN: the French and the Germans wouldn’t agree; we spent years arguing about differences, and – unlike in the real UN – where eventually at the end of the day you are dealing with real problems of the world and you have to come to an agreement sometime; in mathematics, it was not absolutely essential. We went on for probably 15 years, before we founded the EMS. On the one hand, mathematicians have much more in common than politicians, we are international in our mathematical life, it is easy to talk to colleagues 28 Abel Prize 2004 from other countries; on the other hand, mathematicians are much more argumentative. When it comes to the fine details of a constitution, then they are terrible; they are worse than lawyers. But eventually – in principle – the good will was there for collaboration. Fortunately, the timing was right. In the mean- From left to right: Isadore M. Singer, Sir Michael Francis time, Europe had solved Atiyah, Martin Raussen and Christian Skau. (Photo: Knut Falch) some of its other problems: the Berlin Wall had come down – so suddenly there was a new Europe to be involved in the EMS. This very fact made it possible to get a lot more people interested in it. It gave an opportunity for a broader base of the EMS with more opportunities and also relations to the European Commission and so on. Having been involved with the set-up, I withdrew and left it to others to carry on. I have not followed in detail what has been happening except that it seems to be active. I get my Newsletter, and I see what is going on. Roughly at the same time as the collapse of the Berlin Wall, mathematicians in general – both in Europe and in the USA – began to be more aware of their need to be socially involved and that mathematics had an important role to play in society. Instead of being shut up in their universities doing just their mathematics, they felt that there was some pressure to get out and get involved in education, etc. The EMS took on this role at a European level, and the EMS congresses – I was involved in the one in Barcelona – definitely made an attempt to interact with the public. I think that these are additional opportunities over and above the old-fashioned role of learned societies. There are a lot of opportunities both in terms of the geography of Europe and in terms of the broader reach. Europe is getting ever larger: when we started we had discussions about where were the borders of Europe. We met people from Georgia, who told us very clearly, that the boundary of Europe is this river on the other side of Georgia; they were very keen to make sure that Georgia is part of Europe. Now, the politicians have to decide where the borders of Europe are. It is good that the EMS exists; but you should think rather broadly about how it is evolving as Europe evolves, as the world evolves, as mathematics evolves. What should its function be? How should it relate to national societies? How should it relate to the AMS? How should it relate to the governmental bodies? It is an opportunity! It has a role to play! Sir Michael Francis Atiyah and Isadore M. Singer 29

Apart from mathematics...

Could you tell us in a few words about your main interests besides mathematics? SINGER I love to play tennis, and I try to do so 2–3 times a week. That refreshes me and I think that it has helped me work hard in mathematics all these years.

ATIYAH Well, I do not have his energy! I like to walk in the hills, the Scottish hills – I have retired partly to Scotland. In Cambridge, where I was before, the highest hill was about this (gesture) big. Of course you have got even bigger ones in Norway. I spent a lot of my time outdoors and I like to plant trees, I like nature. I believe that if you do mathematics, you need a good relaxation which is not intellectual – being outside in the open air, climbing a mountain, working in your garden. But you actually do mathematics meanwhile. While you go for a long walk in the hills or you work in your garden – the ideas can still carry on. My wife complains, because when I walk she knows I am thinking of mathematics.

SINGER I can assure you, tennis does not allow that!

Thank you very much on behalf of the Norwegian, the Danish, and the Europe- an Mathematical Societies!

Abel Prize 2004 laureates Sir Michael Atiyah (left) and Isadore Singer. (Photo: Anne Lise Flavik)

Abel Prize 2005: Peter D. Lax

The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2005 to

Peter D. Lax Courant Institute of Mathematical Sciences, New York University

“for his groundbreaking contributions to the theory and application of partial differential equations and to the computation of their solutions.”

Ever since Newton, differential equations have been the basis for the scientific understanding of nature. Linear differential equations, in which cause and effect are directly propor- tional, are reasonably well understood. The equations that arise in such fields as aerody- namics, meteorology and elasticity are nonlinear and much more complex: their solutions can develop singularities. Think of the shock waves that appear when an airplane breaks the sound barrier. In the 1950s and 1960s, Lax laid the foundations for the modern theory of nonlinear equations of this type (hyperbolic systems). He constructed explicit solutions, identified classes of especially well-behaved systems, introduced an important notion of entropy, Abel Laureate Peter D. Lax. and, with Glimm, made a penetrating study (Photo : Anne Lise Flavik) of how solutions behave over a long period of time. In addition, he introduced the widely used Lax–Friedrichs and Lax–Wendroff numerical schemes for computing solutions. His work in this area was important for the further theoretical developments. It has also been extraordinarily fruitful for practical applications, from weather prediction to airplane design. Another important cornerstone of modern numerical analysis is the “Lax Equiv- alence Theorem”. Inspired by Richtmyer, Lax established with this theorem the conditions under which a numerical implementation gives a valid approximation to the solution of a differential equation. This result brought enormous clarity to the subject.

* 1.5.1926 31 32 Abel Prize 2005

A system of differential equations is called “integrable” if its solutions are completely characterized by some crucial quantities that do not change in time. A classical example is the spinning top or gyroscope, where these conserved quantities are energy and angular momentum. Integrable systems have been studied since the 19th century and are important in pure as well as applied mathematics. In the late 1960s a revolution occurred when Kruskal and co-workers discovered a new family of examples, which have “soliton” solutions: single-crested waves that maintain their shape as they travel. Lax became fascinated by these mysterious solutions and found a unifying concept for understanding them, rewriting the equations in terms of what are now called “Lax pairs”. This developed into an essential tool for the whole field, leading to new constructions of integrable systems and facilitating their study. Scattering theory is concerned with the change in a wave as it goes around an obstacle. This phenomenon occurs not only for fluids, but also, for instance, in atomic physics (Schrödinger equation). Together with Phillips, Lax developed a broad theory of scattering and described the long-term behaviour of solutions (specifically, the decay of energy). Their work also turned out to be important in fields of mathematics apparently very distant from differential equations, such as number theory. This is an unusual and very beautiful example of a framework built for applied mathematics leading to new insights within pure mathematics. Peter D. Lax has been described as the most versatile mathematician of his generation. The impressive list above by no means states all of his achievements. His use of geometric optics to study the propagation of singularities inaugurated the theory of Fourier Integral Operators. With Nirenberg, he derived the definitive Gårding-type estimates for systems of equations. Other celebrated results include the Lax–Milgram lemma and Lax’s version of the Phragmén–Lindelöf principle for elliptic equations. Peter D. Lax stands out in joining together pure and applied mathematics, combining a deep understanding of analysis with an extraordinary capacity to find unifying concepts. He has had a profound influence, not only by his research, but also by his writing, his lifelong commitment to education and his generosity to younger mathematicians.

Peter Lax receives the prize from Crown Prince Haakon of Norway. (Photo: Knut Falch) Peter D. Lax 33

On behalf of the Norwegian and Danish Mathematical Societies we would like to congratulate you on winning the Abel Prize for 2005. You came to the U.S. in 1941 as a 15-year old kid from Hungary. Only three years later, in 1944, you were drafted into the U.S army. Instead of being shipped overseas to the war front, you were sent to Los Alamos in 1945 to participate in the Manhattan project, building the first atomic bomb. It must have been awesome as a young man to come to Los Alamos taking part in such a momentous endeavour, and to meet so many legendary famous scientists: Fermi [1901–1954], Bethe [1906–2005], Szilard [1898–1964], Wigner, Teller [1908–2003], Feynman [1918–1988], to name some of the physicists, and von Neumann [1903–1957] and Ulam [1909–1984], to name some of the mathematicians. How did this experience shape your view of mathematics and influence your choice of research field within mathematics? In fact, I returned to Los Alamos after I got my Ph.D. in 1949 for a year’s stay and then spent many summers as a consultant. The first time I spent in Los Alamos, and especially the later exposure, shaped my mathematical thinking. First of all it was the experience of being part of a scientific team, not just of mathematicians, people with different outlooks, and the aim being not a theorem, but a product. One can not learn that from books, one must be a participant, and for that reason I urge my students to spend at least a summer as a visitor at Los Alamos. Los Alamos has a very active visitor’s program. Secondly, it was there – that was in the 50s – that I became imbued with the utter importance of computing for science and mathematics. Los Alamos, under the influence of von Neumann, was for a while in the 50s and the early 60s the undisputed leader in computational science.

Research contributions

May we come back to computers later? First some questions about some of your main research contributions to mathematics: You have made outstanding contributions to the theory of non-linear partial differential equations. For the theory and numerical solutions of hyperbolic systems of conservation laws your contribution has been decisive, not to mention your contribution to the understanding of the propagation of discontinuities, so called shocks. Could you describe in a few words how you were able to overcome the formidable obstacles and difficulties this area of mathematics presented? Well, when I started to work on it I was very much influenced by two papers. One was Eberhard Hopf’s [1902–1983] on the viscous limit of Burgers’ equation, and the other was the von Neumann–Richtmyer paper on artificial viscosity. And looking at these examples I was able to see what the general theory might look like.

The astonishing discovery by Kruskal [1925–2006] and Zabusky in the 1960’s of the role of solitons for solutions of the Korteweg–de Vries (KdV) equation, and the no less astonishing subsequent explanation given by several people 34 Abel Prize 2005 that the KdV equation is completely integrable, represented a revolutionary development within the theory of non-linear partial differential equations. You entered this field with an ingenious original point of view, introducing the so-called Lax-pair, which gave an understanding of how the inverse scattering transform applies to equations like the KdV, and also to other non-linear equations which are central in mathematical physics, like the sine-Gordon and the non-linear Schrödinger equation. Could you give us some thoughts on how important you think this theory is for mathematical physics and for applications, and how do you view the future of this field? Perhaps I start by pointing out that the astonishing phenomenon of the interaction of solitons was discovered by numerical calculations, as was predicted by von Neumann some years before, namely that calculations will reveal extremely interesting phenomena. Since I was a good friend of Kruskal I learned early about his discoveries, and that started me thinking. It was quite clear that there are infinitely many conserved quantities, and so I asked myself: How can you generate all at once an infinity of conserved quantities. I thought if you had a transformation that pre- served the spectrum of an operator then that would be such a transformation, and that turned out to be a very fruitful idea applicable quite widely. Now you ask how important is it? I think it is pretty important. After all, from the point of view of technology for the transmission of signals, signalling by solitons is very important and a promising future technology in trans-oceanic transmission. This was developed by Linn Mollenauer, a brilliant engineer at Bell Labs. It has not yet been put into practice, but it will some day. The interesting thing about it is that classical signal theory is entirely linear, and the main point of soliton signal transmission is that the equations are non-linear. That’s one aspect of the practical importance of it. As for the theoretic importance: the KdV equation is completely integrable, and then an astonishing number of other completely integrable systems were discovered. Completely integrable systems can really be solved in the sense that the general population uses the word solved. When a mathematician says he has solved the problem he means he knows the solution exists, that it’s unique, but very often not much more. Now the question is: Are completely integrable systems exceptions to the behavior of solutions of non-integrable systems, or is it that other systems have similar behaviour, only we are unable to analyse it? And here our guide might well be the Kolmogorov-Arnold-Moser theorem which says that a system near a completely integrable system behaves as if it were completely integrable. Now, what near means is one thing when you prove theorems, another when you do experiments. It’s another aspect of numerical experimentation revealing things. So I do think that studying completely integrable systems will give a clue to the behaviour of more general systems as well. Who could have guessed in 1965 that completely integrable systems would become so important?

The next question is about your seminal paper “Asymptotic solutions of oscil- lating initial value problems” from 1957. This paper is by many people consid- Peter D. Lax 35 ered to be the genesis of Fourier In- tegral Operators. What was the new viewpoint in the paper that proved to be so fruitful? It is a micro-local description of what is going on. It combines looking at the problem in the large and in the small. It combines both aspects and that gives it its strengths. The numerical implementation of the micro-local point of view is by wavelets and similar approaches, which are very powerful numerically.

May we touch upon your collaboration with Ralph Phillips – on and off over a span of more that 30 years – on scattering theory, applying it in a number of settings. Could you comment on this collaboration, and what do you consider to be the most important results you obtained? That was one of the great pleasures of my life! Ralph Phillips is one of the great analysts of our time and we formed a very close friendship. We had a new way of viewing the scattering process with incoming and outgoing subspaces. We were, so to say, carving a semi-group Peter Lax lecturing at Oslo University. out of the unitary group, whose infin- (Photo: Terje Bendiksby) itesimal generator contained almost all the information about the scattering process. So we applied that to classical scattering of sound waves and electromagnetic waves by potensials and obstacles. Follow- ing a very interesting discovery of Faddeev [1923–2017] and Pavlov [1936–2016], we studied the spectral theory of automorphic functions. We elaborated it further, and we had a brand new approach to Eisenstein series for instance, getting at spectral representation via translation representation. And we were even able to contemplate – following Faddeev and Pavlov – the Riemann hypothesis peeking around the corner.

That must have been exciting! Yes! Whether this approach will lead to the proof of the Riemann hypothesis, stating it, as one can, purely in terms of decaying signals by cutting out all standing waves, is unlikely. The Riemann hypothesis is a very elusive thing. You may remember in Peer Gynt there is a mystical character, the Boyg, which bars Peer Gynt’s way wherever he goes. The Riemann hypothesis resembles the Boyg! 36 Abel Prize 2005

Which particular areas or questions are you most interested in today? I have some ideas about the zero dispersion limit.

Pure and applied mathematics

May we raise a perhaps contentious issue with you: pure mathematics versus applied mathematics. Occasionally one can hear within the mathematical community statements that the theory of non-linear partial differential equations, though profound and often very important for applications, is fraught with ugly theorems and awkward arguments. In pure mathematics, on the other hand, beauty and aesthetics rule. The English mathematician G. H. Hardy [1877–1947] is an extreme example of such an attitude, but it can be encountered also today. How do you respond to this? Does it make you angry? I don’t get angry very easily. I got angry once at a dean we had, terrible son of a bitch, destructive liar, and I got very angry at the mob that occupied the Courant Institute and tried to burn down our computer. Scientific disagreements do not arouse my anger. But I think this oppinion is definitely wrong. I think Paul Halmos [1916–2006] once claimed that applied mathematics was, if not bad mathematics, at least ugly mathematics, but I think I can point to those citations of the Abel Com- mittee dwelling on the elegance of my works! Now about Hardy: When Hardy wrote Apology of a Mathematician he was at the end of his life, he was old, I think he had suffered a debilitating heart-attack, he was very depressed. So that should be taken into account. About the book itself: There was a very harsh criticism by the chemist Frederick Soddy [1877–1956], who was one of the codiscoverers of the isotopes – he shared the Nobel Prize with Rutherford [1871–1937]. He looked at the pride that Hardy took in the uselessness of his mathematics and wrote: “From such cloistral clowning the world sickens”. It was very harsh because Hardy was a very nice person. My friend Joe Keller [1923–2016], a most distinguished applied mathematician, was once asked to define applied mathematics and he came up with this: “Pure mathematics is a branch of applied mathematics”. Which is true if you think a bit about it. Mathematics originally, say after Newton [1643–1727], was designed to solve very concrete problems that arose in physics. Later on these subjects developed on their own and became branches of pure mathematics, but they all came from applied background. As von Neumann [1903–1957] pointed out, after a while these pure branches that develop on their own need invigoration by new empirical material, like some scientific questions, experimental facts and, in particular, some numerical evidence.

In the history of mathematics, Abel [1802–1829] and Galois [1811–1832] may have been the first great mathematicians that one may describe as “pure mathematicians”, not being interested in any “applied” mathematics as such. How- ever, Abel did solve an integral equation, later called “Abel’s integral equa- Peter D. Lax 37 tion”, and Abel gave an explicit solution, which incidentally may have been the first time in the history of mathematics that an integral equation had been formulated and solved. Interestingly, by a simple reformulation one can show that the Abel integral equation and its solution are equivalent to the Radon Transform, the mathematical foundation on which modern medical tomog- raphy is based. Examples of such totally unexpected practical applications of pure mathematical results and theorems abound in the history of mathematics – group theory that evolved from Galois’ work is another striking example. What are your thoughts on this phenomenon? Is it true that deep and important theories and theorems in mathematics will eventually find practical applications, for example in the physical sciences? Well, as you pointed out this has very often happened: Take for example Eugene Wigner’s use of group theory in quantum mechanics. And this has happened too often to be just a coincidence. Although, one might perhaps say that other theories and theorems which did not find applications were forgotten. It might be interesting for a historian of mathematics to look into that phenomenon. But I do believe that mathematics has a mysterious unity which really connects seemingly distinct parts, which is one of the glories of mathematics.

You have said that Los Alamos was the birthplace of computational dynamics, and I guess it is safe to say that the U.S. war effort in the 1940’s advanced and accelerated this development. In what way has the emergence of the high-speed computer altered the way mathematics is done? Which role will high-speed computers play within mathematics in the future? It has played several roles. One is what we saw in Kruskal’s and Zabusky’s discovery of solitons, which would not have been discovered without computational evidence. Likewise the Fermi–Pasta–Ulam phenomenon of recurrence was also a very striking thing which may or may not have been discovered without the computer. That is one aspect. But another is this: in the old days, to get numerical results you had to make enormously drastic simplifications if your computations were done by hand, or by simple computing machines. And the talent of what drastic simplifications to make was a special talent that did not appeal to most mathematicians. Today you are in an entirely different situation. You don’t have to put the problem on a Procrustean bed and mutilate it before you attack it numerically. And I think that has attracted a much larger group of people to numerical problems of applications – you could really use the full theory. It invigorated the subject of linear algebra, which as a research subject died in the 1920s. Suddenly the actual algorithms for carrying out these operations became important. It was full of surprises, like fast matrix multiplication. In the new edition of my linear algebra book I will add a chapter on the numerical calculation of the eigenvalues of symmetric matrices. You know it’s a truism that due to increased speed of computers, a problem that took a month 40 years ago can be done in minutes, if not seconds today. Most of the speed-up is attributed, at least by the general public, to increased speed of comput- 38 Abel Prize 2005 ers. But if you look at it, actually only half of the speed-up is due to this increased speed. The other half is due to clever algorithms, and it takes mathematicians to invent clever algorithms. So it is very important to get mathematicians involved, and they are involved now.

Could you give us personal examples of how questions and methods from applied points of view have triggered “pure” mathematical research and results? And conversely, are there examples where your theory of nonlinear partial differential equations, especially your explanation of how discontinuities propagate, have had commercial interests? In particular, concerning oil exploration, so important for Norway! Yes, oil exploration uses signals generated by detonations that are propagated through the earth and through the oil reservoir and are recorded at distant sta- tions. It’s a so-called inverse problem. If you know the distribution of the densities of materials and the associated waves’ speeds, then you can calculate how signals propagate. The inverse problem is that if you know how signals propagate, then you want to deduce from it the distribution of the materials. Since the signals are discontinuities, you need the theory of propagation of discontinuities. Otherwise it’s somewhat similar to the medical imaging problem, also an inverse problem. Here the signals do not go through the earth but through the human body, but there is a similarity in the problems. But there is no doubt that you have to understand the direct problem very well before you can tackle the inverse problem.

Hungarian mathematics

Now to some questions related to your personal history. The first one is about your interest in, and great aptitude for, solving problems of a type that you call “Mathematics Light” yourself. To mention just a few, already as a 17 year old boy you gave an elegant solution to a problem that was posed by Erdös [1903– 1996] and is related to a certain inequality for polynomials, which was earlier proved by Bernstein [1880–1968]. Much later in your career you studied the so-called Polya function which maps the unit interval continuously onto a right-angled triangle, and you discovered its amazing differentiability properties. Was problem solving specifically encouraged in your early mathematical education in your native Hungary, and what effect has this had on your career later on? Yes, problem solving was regarded as a royal road to stimulate talented youngsters, and I was very pleased to learn that here in Norway they have a successful high- school contest, where the winners were honoured this morning. But after a while one shouldn’t stick to problem solving, one should broaden out. I return ever once in a while to it, though. Back to the differentiability of the Polya function: I knew Polya [1887–1985] quite well having taken a summer course with him in ’46. The differentiability question Peter D. Lax 39 came about this way: I was teaching a course on real variables and I presented Polya’s example of an area-filling curve, and I gave as homework to the students to prove that it’s nowhere differentiable. Nobody did the homework, so then I sat down and I found out that the situation was more complicated. There was a tradition in Hungary to look for the simplest proof. You may be familiar with Erdös’ concept of The Book. That’sThe Book kept by the Lord of all theorems and the best proofs. The highest praise that Erdös had for a proof was that it was out of The Book. One can overdo that, but shortly after I had got my Ph.D., I learned about the Hahn–Banach theorem, and I thought that it could be used to prove the existence of Green’s function. It’s a very simple argument – I believe it’s the simplest – so it’s out of The Book. And I Prize winners Atiyah, Lax and Singer. think I have a proof of Brouwer’s Fixed (Photo: Ørn E. Borgen) Point Theorem, using calculus and just change of variables. It is probably the simplest proof and is again out of The Book. I think all this is part of the Hungarian tradition. But one must not overdo it.

There is an impressive list of great Hungarian physicists and mathematicians of Jewish background that had to flee to the US after the rise of fascism, Nazism and anti-Semitism in Europe. How do you explain this extraordinary culture of excellence in Hungary that produced people like de Hevesy [1885–1966], Szilard, Wigner, Teller, von Neumann, von Karman [1881–1963], Erdös, Szegö [1895–1985], Polya, yourself, to name some of the most prominent ones? There is a very interesting book written by John Lukacs with the title Budapest 1900: A Historical Portrait of a City and its Culture, and it chronicles the rise of the middle class, rise of commerce, rise of industry, rise of science, rise of literature. It was fuelled by many things: a long period of peace, the influx of mostly Jewish population from the East eager to rise, an intellectual tradition. You know in mathematics, Bolyai was a culture hero to Hungarians, and that’s why mathematics was particularly looked upon as a glorious profession.

But who nurtured this fantastic flourishing of talent, which is so remarkable? Perhaps much credit should be given to Julius König [1849–1913], whose name is probably not known to you. He was a student of Kronecker [1823–1891], I believe, 40 Abel Prize 2005 but he also learned Cantor’s [1845–1918] set theory and made some basic contribution to it. I think he was influential in nurturing mathematics. His son was a very distinguished mathematician, Denes König, really the father of modern graph theory. And then there arose extraordinary people. Leopold Fejér, for instance, had enormous influence. There were too many to fill positions in a small country like Hungary, so that’s why they had to go abroad. Part of it was also anti-Semitism. There is a charming story about the appointment of Leopold Fejér, who was the first Jew proposed for a professorship at Budapest University. There was opposi- tion to it. At that time there was a very distinguished theologian, Ignatius Fejér, in the Faculty of Theology. Fejér’s original name was Weiss. So one of the opponents, who knew full well that Fejér’s original name had been Weiss, said pointedly: “This professor Leopold Fejér that you are proposing, is he related to our distinguished colleague Father Ignatius Fejér?” And Eötvös, the great physicist who was pushing the appointment, replied without batting an eyelash: “Illegitimate son”. That put an end to it.

And he got the job? He got the job.

Scribbles that changed the course of human affairs

The mathematician Stanisław Ulam was involved with the Manhattan Project and is considered to be one of the fathers of the hydrogen bomb. He wrote in his autobiography Adventures of a Mathematician: “It is still an unending source of surprise for me to see how a few scribbles on a blackboard, or on a sheet of paper, could change the course of human affairs”. Do you share this feeling? And what are your feelings to what happened to Hiroshima and Nagasaki, to the victims of the explosions of the atomic bombs that brought an end to World War II? Well, let me answer the last question first. I was in the army, and all of us in the army expected to be sent to the Pacific to participate in the invasion of Japan. You remember the tremendous slaughter that the invasion of Normandy brought about. That would have been nothing compared to the invasion of the Japanese mainland. You remember the tremendous slaughter on Okinawa and Iwo Jima. The Japanese would have resisted to the last man. The atomic bomb put an end to all this and made an invasion unnecessary. I don’t believe reversionary historians who say: “Oh, Japan was already beaten, they would have surrendered anyway”. I don’t see any evidence for that. There is another point which I raised once with someone who had been involved with the atomic bomb project. Would the world have had the horror of nuclear war if it had not seen what one bomb could do? The world was inoculated against using nuclear weapon by its use. I am not saying that alone justifies it, and it certainly was not the justification for its use. But I think that is a historical fact. Peter D. Lax 41

Now about scribbles changing history: Sure, the special theory of relativity, or quantum mechanics, it would be unimaginable today without scribbles. Inciden- tally, Ulam was a very interesting mathematician. He was an idea man. Most mathematicians like to push their ideas through. He preferred throwing out ideas. His good friend Rota [1932–1999] even suggested that he did not have the technical ability or patience to work them out. But if so, then it’s an instance of Ulam turning a disability to tremendous advantage. I learned a lot from him.

It is amazing for us to learn that an 18 year old immigrant was allowed to participate in a top-secret and decisive weapon development during WWII. The war created an emergency. Many of the leaders of the Manhattan Project were foreigners, so being a foreigner was no bar.

Collaboration. Work Style

Your main workplace has been the Courant Institute of Mathematical Sciences in New York, which is part of New York University. You served as its director for an eight year period in the 70s. Can you describe what made this institute, which was created by the German refugee Richard Courant [1888–1972] in the 1930s, a very special place from the early days on, with a particular spirit and atmosphere? And is the Courant Institute today still a special place that differs from others? To answer your first question, certainly the personality of Courant was decisive. Courant saw mathematics very broadly, he was suspicious of specialisation. He wanted it drawn as broadly as possible, and that’s how it came about that applied topics and pure mathematics were pursued side by side, often by the same people. This made the Courant Institute unique at the time of its founding, as well as in the 40s, 50s and 60s. Since then there are other centres where applied mathematics is respected and pursued. I am happy to say that this original spirit is still present at the Courant Institute. We still have large areas of applied interest, meteorology and climatology under Andy Majda, solid state and material science under Robert Kohn and others, and fluid dynamics. But we also have differential geometry as well as some pure aspects of partial differential equations, even some algebra. I am very pleased how the Courant Institute is presently run. It’s now the third generation that’s running it, and the spirit that Courant instilled in it – kind of a family feeling – still prevails. I am happy to note that many Norwegian mathematicians received their training at the Courant Institute, and later rose to become leaders in their field.

You told us already about your collaboration with Ralph Phillips. Generally speaking, looking through your publication list and the theorems and methods you and your collaborators have given name to, it is apparent that you have had a vast collaboration with a lot of mathematicians. Is this sharing of ideas 42 Abel Prize 2005 a particularly successful, and maybe also joyful, way of making advances for you? Sure, sure. Mathematics is a social phenomenon after all. Collaboration is a psycho- logical and interesting phenomenon. A friend of mine, Vera John-Steiner, has written a book (“Creative Collaboration”) about it. Two halves of a solution are supplied by two different people, and something quite wonderful comes out of it.

Many mathematicians have a very particular work style when they work hard on certain problems. How would you characterise your own particular way of thinking, working, and writing? Is it rather playful or rather industrious? Or both? Phillips thought I was lazy. He was a product of the Depression which imposed a certain strict discipline on people. He thought I did not work hard enough, but I think I did!

Sometimes mathematical insights seem to rely on a sudden unexpected inspiration. Do you have examples of this sort from your own career? And what is the background for such sudden inspiration in your opinion? The question reminds me of a story about a German mathematician, Schottky [1851–1932], when he reached the age of 70 or 80. There was a celebration of the event, and in an interview like we are having, he was asked: “To what do you attribute your creativity and productivity”. The question threw him into great confusion. Finally he said: “But gentlemen, if one thinks of mathematics for 50 years, one must think of something!” It was different with Hilbert [1862–1943]. This is a story I heard from Courant. It was a similar occasion. At his 70th birthday he was asked what he attributed his great creativity and originality to. He had the answer immediately: “I attribute it to my very bad memory”. He really had to reconstruct everything, and then it became something else, something better. So maybe that is all I should say. I am between these two extremes. Incidentally, I have a very good memory.

Teaching

You have also been engaged in the teaching of calculus. For instance, you have written a calculus textbook with your wife Anneli as one of the co-authors. In this connection you have expressed strong opinions about how calculus should be exposed to beginning students. Could you elaborate on this? Our calculus book was enormously unsuccessful, in spite of containing many excellent ideas. Part of the reason was that certain materials were not presented in a fashion that students could absorb it. A calculus book has to be fine-tuned, and I didn’t have the patience for it. Anneli would have had it, but I bullied her too much, I am afraid. Sometimes I dream of redoing it because the ideas that were in there, and that I have had since, are still valid. Peter D. Lax 43

Of course, there has been a calculus reform movement and some good books have come out of it but I don’t think they are the answer. First of all, the books are too thick, often more than 1000 pages. It’s unfair to give such a book into the hands of an unsuspecting student who can barely carry it. And the reaction to it would be: “Oh, my God, I have to learn all that is in it?” Well, all that is not in it! Secondly, if you compare it to the old standards, Thomas, say, it’s not so different – the order of the topics and concepts, perhaps. In my calculus book, for instance, instead of continuity at a point, I advocated uniform continuity. This you can explain much easier than defining continuity at a point and then say the function is continuous at every point. You lose the students; there are too many quantifiers in that. But the mathematical communities are enormously conservative: “continuity has been defined pointwise, and so it should be!” Other things that I would emphasize: To be sure there are applications in these new books. But the applications should all stand out. In my book there were chap- ters devoted to the applications, that’s how it should be, they should be featured prominently. I have many other ideas as well. I still dream of redoing my calculus book, and I am looking for a good collaborator. I recently met someone who expressed admiration for the original book, so perhaps it could be realized, if I have the energy. I have other things to do as well, like the second edition of my linear algebra book, and revising some old lecture notes on hyperbolic equations. But even if I could find a collaborator on a calculus book, would it be accepted? Not clear. In 1873, Dedekind {1831–1916] posed the important question: “What are, and what should be, the real numbers”? Unfortunately, he gave the wrong answer as far as calculus students are concerned. The right answer is: infinitesimals. I don’t know how such a joke will go down?

Heading large institutions

You were several times the head of large organisations: director of the Courant Institute in 1972–1980, president of the American Mathematical Society in 1977–1980, leader of what was called the Lax Panel on the National Science Board in 1980–1986. Can you tell us about some of the most important decisions that had to be taken in these periods? The president of the Mathematical Society is a figurehead. His influence lies in appointing members of committees. Having a wide friendship and reasonable judgement are helpful. I was very much helped by the secretary of the Mathematical Society, Everett Pitcher [1912–2006]. As for being the director of the Courant Institute, I started my directorship at the worst possible time for New York University. They had just closed down their School of Engineering, and that meant that mathematicians from the engineering school were transferred to the Courant Institute. This was the time when the Com- puter Science Department was founded at Courant by Jack Schwartz [1930–2009]. There was a group of engineers that wanted to start activity in informatics, which 44 Abel Prize 2005

is the engineers’ word for the same thing. As a director I fought very hard to stop that. I think it would have been very bad for the university to have two computing departments – it certainly would have been very bad for our Computer Sci- ence Department. Other From left to right: Martin Raussen, Christian Skau and Peter things: Well, I was instru- Lax. (Photo: Knut Falch) mental in hiring Charlie Peskin at the recommendation of Alexander Chorin, I was very pleased with that. Likewise, hiring Sylvain Cappell at the recommendation of Bob Kohn. Both were enormous successes. What were my failures? Well, maybe when the Computer Science Department was founded I should have insisted on having a very high standard of hiring. We needed people to teach courses, but in hindsight I think we should have exercised more restraint in our hiring. We might have become the number one computer science department. Right now the quality has improved very much – we have a wonderful chairwoman, Margaret Wright. Being on the National Science Board was my most pleasant administrative experience. It’s a policy-making body for the National Science Foundation (NSF), so I found out what making policy means. Most of the time it just means nodding ‘yes’, and a few times saying ‘no’. But then there are sometimes windows of opportunity, and the Lax Panel was a response to such a thing. You see, I noticed through my own experience and those of my friends who are interested in large scale computing, in particular, Paul Garabedian [1927–2010], who complained that university computational scientists had no access to the supercomputers. At a certain point the government, which alone had enough money to purchase these supercomputers, stopped placing them at universities. Instead they went to national labs and industrial labs. Unless you happened to have a friend there with whom you collaborated, you had no access. That was very bad from the point of view of the advance of computational science, because the most talented people were at the universities. At that time accessing and computing at remote sites became possible thanks to ARPANET, which then became a model for the Internet. So the panel that I established made strong recommendation that the NSF establish computing centres, and that was followed up. My quote on our achievement was a paraphrase of Emerson [1803–1882]: “Nothing can resist the force of an idea that is ten years overdue”.

A lot of mathematical research in the US has been funded by contracts from DOD, DOE, the atomic energy commission, the NSA. Is this dependence of mutual benefit? Are there pitfalls? Peter D. Lax 45

I am afraid that our leaders are no longer aware of the subtle but close connection between scientific vigour and technological sophistication.

Personal interests

Would you tell us a bit about your interests and hobbies that are not directly related to mathematics? I love poetry. Hungarian poetry is particularly beautiful, but English poetry is perhaps even more beautiful. I love to play tennis. Now my knees are a bit wobbly and I can’t run anymore, but perhaps these can be replaced – I’m not there yet. My son and three grandsons are tennis enthusiasts so I can play doubles with them. I like to read. I have a knack for writing. Alas, these days I write obituaries – it’s better to write them than being written about.

You have also written Japanese haikus? You’re right. I got this idea from a nice article by Marshall Stone [1903–1989] – I forget exactly where it was – where he wrote that the mathematical language is enormously concentrated, it is like haikus. And I thought I would take it one step further and actually express a mathematical idea by a haiku.

Speed depends on size Balanced by dispersion Oh, solitary splendour.

Professor Lax, thank you very much for this interview on behalf of the Norwe- gian, the Danish, and the European Mathematical Societies! I thank you. 46 Abel Prize 2005

Peter D. Lax was received like a filmstar by all the children who had been waiting for him in Bergen. Abel Prize 2006: Lennart Carleson

The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2006 to

Lennart Carleson Royal Institute of Technology, Sweden

“for his profound and seminal contributions to harmonic analysis and the theory of smooth dynamical systems.”

In 1807, the versatile mathematician, engineer and Egyptologist Jean Baptiste Joseph Fourier made the revolutionary discovery that many phenomena, ranging from the typical profiles describing the propagation of heat through a metal bar to the vibrations of violin strings, can be viewed as sums of simple wave patterns called sines and cosines. Such sum- mations are now called Fourier series. Har- monic analysis is the branch of mathematics that studies these series and similar objects. For more than 150 years after Fourier’s discovery, no adequate formulation and justification was found of his claim that every function equals the sum of its Fourier series. In hindsight this loose statement should be interpreted as regarding every function for Abel Laureate Lennart Carleson. which “it is possible to draw the graph”, or (Photo: Anne Lise Flavik) more precisely, every continuous function. Despite contributions by several mathematicians, the problem remained open. In 1913 it was formalized by the Russian mathematician Lusin in the form of what became known as Lusin’s conjecture. A famous negative result of Kolmogorov in 1926, together with the lack of any progress, made experts believe that it would only be a matter of time before someone constructed a continuous function for which the sum of its Fourier series failed to give the function value anywhere. In 1966, to the surprise

* 18.3.1928

47 48 Abel Prize 2006 of the mathematical community, Carleson broke the decades-long impasse by proving Lusin’s conjecture that every square-integrable function, and thus in particular every continuous function, equals the sum of its Fourier series “almost everywhere”. The proof of this result is so difficult that for over thirty years it stood mostly isolated from the rest of harmonic analysis. It is only within the past decade that mathematicians have understood the general theory of operators into which this theorem fits and have started to use his powerful ideas in their own work. Carleson has made many other fundamental contributions to harmonic analysis, complex analysis, quasi-conformal mappings, and dynamical systems. Standing out among them is his solution of the famous corona problem, so called because it examines structures that become apparent “around” a disk when the disk itself is “obscured”, poetically analogous to the corona of the sun seen during an eclipse. In this work he introduced what has become known as Carleson measures, now a fundamental tool of both complex and harmonic analysis. The influence of Carleson’s original work in complex and harmonic analysis does not limit itself to this. For example, the Carleson-Sjölin theorem on Fourier multipli- ers has become a standard tool in the study of the “Kakeya problem”, the prototype of which is the “turning needle problem”: how can we turn a needle 180 degrees in a plane, while sweeping as little area as possible? Although the Kakeya problem originated as a toy, the description of the volume swept in the general case turns out to contain important and deep clues about the structure of Euclidean space. Dynamical systems are mathematical models that seek to describe the behaviour in time of large classes of phenomena, such as those observed in meteorology, financial markets, and many biological systems, from fluctuations in fish populations to epi- demiology. Even the simplest dynamical systems can be mathematically surprisingly complex. With Benedicks, Carleson studied the Hénon map, a dynamical system first proposed in 1976 by the astronomer Michel Hénon, a simple system exhibiting the intricacies of weather dynamics and turbulence. This system was generally believed to have a so-called strange attractor, drawn in beautiful detail by computer graphics tools, but poorly understood mathematically. In a great tour de force, Benedicks and Carleson provided the first proof of the existence of this strange attractor in 1991; this development opened the way to a systematic study of this class of dynamical systems. Carleson’s work has forever altered our view of analysis. Not only did he prove extremely hard theorems, but the methods he introduced to prove them have turned out to be as important as the theorems themselves. His unique style is characterized by geometric insight com- Lennart Carleson receives the prize from Queen Sonja of Norway. bined with amazing control of the branching (Photo: Knut Falch) complexities of the proofs. Lennart Carleson 49

The interview was conducted in Oslo prior to the Abel prize celebration and was later shown on Norwegian TV. The first two questions and their answers were originally phrased in the three Scandinavian languages: Norwegian, Danish and Swedish. They are reproduced here translated into English.

Introduction

On behalf of the Norwegian and Danish mathematical societies, we want to congratulate you on winning the Abel prize for 2006. This year we commemorate the 100th centenary of the death of the Norwe- gian dramatist and poet Henrik Ibsen [1828–1906]. He passed away on the 23rd of May just a stone’s throw away from this place. The longest poem he ever wrote is called “Balloon letter to a Swedish lady” and it contains a verse which reads as follows:

“aldri svulmer der en løftning av et regnestykkes drøftning ti mot skjønnhet hungrer tiden”

Translated into English this becomes:

“never arises elation from the analysis of an equation for our age craves beauty“

Without drawing too far-reaching conclusions, Ibsen seems to express a feeling shared by many people, i.e. that mathematics and beauty or art are opposed to each other, that they belong to different spheres. What are your comments to this view? I do not think that Ibsen was very well-oriented about beauty in mathematics, which you certainly can find and enjoy. And I would even maintain that the beauty of many mathematical arguments can be easier to comprehend than many modern paintings. But a lot of mathematics is void of beauty. Maybe particularly in modern mathematics, where problem areas have often gotten extremely complex and complicated, with the result that the solution can only be formulated on several hundreds of pages. And that can scarcely be called beautiful. But in classical mathematics you find many striking theorems and arguments that hit you as something really original. It is reasonable to use the term beauty for those.

Mathematicians all over Scandinavia are proud of counting one of their own among the very first recipients of the Abel Prize. How would you characterize and evaluate Scandinavian, and particularly Swedish, mathematics in an international perspective? 50 Abel Prize 2006

I think that Scandinavia does quite well in this respect. In Sweden, we have a fine new generation of young mathematicians. And I think it looks very much alike in the other Scandinavian countries. It is difficult to perceive a new Abel on the horizon, but that is probably too much to hope for.

Could you please characterize the unique contribution that the Finnish/Swed- ish school of Phragmen [1863–1937], Lindelöf [1870–1946], M. Riesz [1886– 1969], Carleman [1892–1949], R. Nevanlinna [1895–1980], Beurling [1905– 1986] and Ahlfors [1907–1996] brought to analysis in the first half of the 20th century, which was formative and decisive for your own contribution to hard analysis? In your list, you miss another Scandinavian mathematician: J. L. Jensen [1859–1925]. The importance of “Jensen’s inequality” can hardly be exaggerated. He and Lin- delöf started the Scandinavian school, building of course on Riemann’s [1826–1866] approach to complex analysis rather than that of Cauchy–Weierstrass; Nevanlinna and Carleman continued, followed by Ahlfors and Beurling, a remarkable concentration of talent in Scandinavia. My lecture tomorrow will give more details.

Mathematical achievements in context

Abel [1802–1829] first thought that he had solved the general quintic by radicals. Then he found a mistake and subsequently he proved that it was impossible to solve the quintic algebraically. The famous and notoriously difficult problem about the pointwise convergence almost everywhere of L2-functions, that Lusin [1883–1950] formulated in 1913 and actually goes back to Fourier [1768–1830] in 1807, was solved by you in the mid-1960s. We understand that the prehistory of that result was converse to that of Abel’s, in the sense that you first tried to disprove it. Could you comment on that story? Yes, of course. I met the problem already as a student when I bought Zygmund’s book on trigonometric series. Then I had the opportunity to meet Zygmund [1900– 1992]. He was at Harvard in ’50 or ’51. I was at that time working on Blaschke products and I said maybe one could use those to produce a counterexample. Zygmund was very positive and said “of course, you should do that”. I tried for some years and then I forgot about it before it again came back to me. Then, in the beginning of the ’60s, I suddenly realized that I knew exactly why there had to be a counterexample and how one should construct one. Somehow, the trigonometric system is the type of system where it is easiest to provide counterexamples. Then I could prove that my approach was impossible. I found out that this idea would never work; I mean that it couldn’t work. If there were a counterexample for the trigonometric system, it would be an exception to the rule. Then I decided that maybe no one had really tried to prove the converse. From then on it only took two years or so. But it is an interesting example of ‘to prove something hard, it is extremely important to be convinced of what is right and Lennart Carleson 51 what is wrong’. You could never do it by alternating between the one and the other because the conviction somehow has to be there.

Could we move to another problem, the so-called Corona problem that you solved in 1962? In this connection, you introduced the so-called Carleson measure, which was used extensively by other mathematicians afterwards. Could you try to explain why the notion of the Carleson measure is such a fruitful and useful notion? Well, I guess because it occurs in problems related to the general theory of BMO and H 1-spaces. I wish this class of measures had been given a more neutral name. In my original proof of the Corona problem, the measures were arc lengths on the special curves needed there. Beurling suggested that I should formulate the inequality for general measures. The proof was the same and quite awkward. Stein soon gave a natural and simple proof and only then the class deserved a special name.

We will move to another one of your achievements. Hardy [1877–1947] once said that mathematics is a young man's game. But you seem to be a counterexample; after you passed sixty years of age, you and Michael Benedicks managed to prove that the so called Hénon map has strange attractors exhibiting chaotic behaviour. The proof is extremely complicated. It’s a tour de force that took many years to do. With this as a background, what is your comment on mathematical creativity and age? I guess and hope that you don’t get more stupid when you get older. But I think your stamina is less, your perseverance weakens (keeping lots of facts in your mind at the same time). Probably this has to do with the circulation of the blood or something like that. So I find it now much harder to concentrate for a long period. And if you really want to solve complicated problems, you have to keep many facts available at the same time.

Mathematical problems

You seem to have focused exclusively on the most difficult and profound problems of mathematical analysis. As soon as you have solved any one of these, you leave the further exploration and elaboration to others, while you move on to other difficult and seemingly intractable problems. Is this a fair assessment of your mathematical career and of your mathematical urge? Yes, I think so. Problem solving is my game, rather than to develop theories. Cer- tainly the development of mathematical theories and systems is very important but it is of a very different character. I enjoy starting on something new, where the background is not so complicated. If you take the Hénon case, any schoolboy can understand the problem. The tools also are not really sophisticated in any way; we do not use a lot of theory. 52 Abel Prize 2006

The Fourier series problem of course used more machinery that you had to know. But that was somehow my background. In the circles of dynamical systems people, I always consider myself an amateur. I am not educated as an expert on dynamical systems.

Have there been mathematical problems in analysis that you have worked on seriously, but at which you have not been able to succeed? Or are there any particular problems in analysis that you especially would have liked to solve? Yes, definitely. There is one in dynamical systems, which is called the standard map. This is like the Hénon map but in the area preserving case. I spent several years working on it, collaborating with Spencer for example, but we never got anywhere. If you want to survive as a mathematician, you have to know when to give up also. And I am sure that there have been many other cases also. But I haven’t spent any time on the Riemann hypothesis… and it wouldn’t have worked either.

Characterization of great mathematicians

What are the most important features, besides having a good intellectual capacity of course, that characterize a great mathematician? I don’t think they are the same for everybody. They are not well defined really. If you want to solve problems, as in my case, the most important property is to be very, very stubborn. And also to select problems which are within reach. That needs some kind of intuition, I believe, which is a little closer to what we talked about initially, about beauty. You must somehow have a feeling for mathematics: What is right, what is wrong and what is feasible. But, of course, there are many other mathematicians who create theories and they combine results into new buildings and keep other people working. It is a different kind of a mathematician. I don’t think you should try to find a simple formula for people.

For several decades, you have worked hard on problems that were known to be exceptionally difficult. What drove you and what kept you going for years, with no success guaranteed? What drives a person to devote so much energy to an arcane subject that may only be appreciated by a handful of other mathematicians? Yes, that’s a big issue. Stubbornness is important; you don’t want to give up. But as I said before, you have to know when to give up also. If you want to succeed you have to be very persistent. And I think it’s a drive not to be beaten by stupid problems.

Your main research contribution has been within mathematical analysis. What about your interest in algebra and topology/geometry? Geometry is of course very much part of the analysis. But I have no feeling for algebra or topology, I would say. I have never tried to… I should have learned more! Lennart Carleson 53

Mathematics of the future

What do you consider to be the most challenging and exciting area of mathematics that will be explored in the 21st century? Do you have any thoughts on the future development of mathematics? Yes, of course I have had thoughts. Most of the influence comes from the outside. I think we are still lacking a good understanding of which kind of methods we should use in relation to computers and computer science. And also in relation to problems depending on a medium sized number of variables. We have the machinery for a small number of variables and we have probability for a large number of variables. But we don’t even know which questions to ask, much less which methods to use, Lennart Carleson thanks the Norwegian people when we have ten variables or twenty for the Abel prize. (Photo: Knut Falch) variables.

This leads to the next question. What is the significance of computers in mathematics? Is it mainly checking experimentally certain conjectures? Or is it com- pleting proofs by checking an enormous amount of special cases? What are your thoughts on computers in mathematics? There are a few instances that I have been involved with. I had a student, Warwick Tucker, who proved that the Lorenz attractor exists. The proof was based on explicit computations of orbits. And in that case you could get away with a finite number of orbits. This is very different from the Hénon map, where you could never succeed in that way. You could never decide whether a parameter was good or bad. But for the Lorenz attractor he actually proved it for the specific values that Lorenz had prescribed. Because it is uniformly expanding, there is room for small changes in the parameter. So this is an example of an actual proof by computer. Of course then you could insist on interval arithmetics. That’s the fine part of the game so to say, in order to make it rigorous for the people who have very formal requirements.

But what about computers used, for instance, for the four colour problem, checking all these cases? 54 Abel Prize 2006

Probably unavoidable, but that’s okay. I wouldn’t like to do it myself. But it’s the same with group structures, the classification of simple groups, I guess. We have to accept that.

The solution of the 350 year old Fermat conjecture, by Andrew Wiles in 1994, uses deep results from algebraic number theory. Do you think that this will be a trend in the future, that proofs of results which are simple to state will require a strong dose of theory and machinery? I don’t know. The striking part in the proof of the Fermat theorem is the connection between the number theory problem and the modular functions. And once you have been able to prove that, you have moved the problem away from what looked like an impossible question about integers, into an area where there exists machinery.

Career. Teachers.

Your CV shows that you started your university education already at the age of 17 and that you took your PhD at Uppsala University when you were 22 years old. Were you sort of a wunderkind? No, I didn’t feel like a wunderkind.

Can you elaborate about what aroused your mathematical interests? And when did you become aware that you had an exceptional mathematical talent? During high school I inherited some books on calculus from my sister. I read those but otherwise I didn’t really study mathematics in any systematic way. When I went to university it was natural for me to start with mathematics. Then it just kept going somehow. But I was not born a mathematician.

You already told us about your PhD advisor, Arne Beurling, an exceptional Swedish mathematician, who is probably not as well known as he deserves. Could you characterize him as a person and as a researcher in a few sentences? Did he have a lasting influence on your own work? Yes, definitely. He was the one who set me on track. We worked on the same type of problems but we had a different attitude towards mathematics. He was one of the few people about whom I would use the word genius. Mathematics was part of his personality somehow. He looked at mathematics as a piece of art. Ibsen would have profited from meeting him. He also considered his papers as pieces of art. They were not used for education and they were not used to guide future researches. But they were used as you would use a painting. He liked to hide how he found his ideas. If you would ask him how he found his result, he would say a wizard doesn’t explain his tricks. So that was a rather unusual education. But of course I learned a lot from him. As you said, he has never been really recognized in a way which he deserves. Lennart Carleson 55

Abel laureates Peter Lax (2005), Lennart Carleson (2006), and Jean-Pierre Serre (2003). (Photo: Scanpix)

Apart from Arne Beurling, which other mathematicians have played an important part in your development as a mathematician? I have learnt from many others, in particular from the people I collaborated with and in particular from Peter Jones. I feel a special debt to Michel Herman. His thesis, where he proved the global Arnold conjecture on diffeomorphisms of the circle, gave me a new aspect on analysis and was my introduction to dynamical systems.

You have concentrated your research efforts mainly on topics in hard analysis, with some spices from geometry and combinatorics. Is there a specific background for this choice of area? I don’t think so. There is a combinatorial part in all of the three problems we have discussed here. And all of them are based on stopping time arguments. You make some construction and then you stop the construction, and you start all over again.

This is what is called renormalization? Yes, renormalization. That was something I didn’t learn. Probability was not a part of the Uppsala school. And similarly for coverings, which is also part of the combinatorics.

Which mathematical area and what kind of mathematical problems are you currently the most interested in? Well, I like to think about complexity. I would like to prove that it’s harder to multiply than to add.

That seems to be notoriously difficult, I understand. Well, I am not so sure. It’s too hard for me so far.

You have a reputation as a particularly skilful advisor and mentor for young mathematicians; 26 mathematicians were granted a PhD under your super- 56 Abel Prize 2006 vision. Do you have particular secrets on how to encourage, to advise and to educate young promising mathematicians? The crucial point, I think, is to suggest an interesting topic for the thesis. This is quite hard since you have to be reasonably sure that the topic fits the student and that it leads to results. And you should do this without actually solving the problem! A good strategy is to have several layers of the problem. But then many students have their own ideas. I remember one student who wanted to work on orthogonal polynomials. I suggested that he could start by reading Szegö’s [1895–1985] book. “Oh, no!” he said, “I don’t want to have any preconceived ideas”.

Publishing mathematics

We would like to move to the organization of research. Let’s start with the journal Acta Mathematica. It is a world famous journal founded by Gösta Mit- tag-Leffler [1846–1927] back in 1882 in Stockholm as a one-man enterprise at that time. It rose very quickly to be one of the most important mathematical journals. You were its editor in chief for a long period of time. Is there a particular recipe for maintaining Acta as a top mathematical journal? Is very critical refereeing important? It is the initial period that is crucial, when you build up a reputation so that people find it attractive to have a paper published there. Then you have to be very serious in your refereeing and in your decisions. You have to reject a lot of papers. You have to accept being unpopular.

Scientific publication at large is about to undergo big changes. The number of scientific journals is exploding and many papers and research results are sometimes available on the internet many years before they are published in print. How will the organization of scientific publication develop in the future? Will printed journals survive? Will peer review as today survive for the next decades? I’ve been predicting the death of the system of mathematical journals within ten years for at least 25 years. And it dies slowly, but it will only die in the form we know it today. If I can have a wish for the future, I would wish that we had, say, 100 journals or so in mathematics, which would be very selective in what they publish and which wouldn’t accept anything that isn’t really finalized, somehow. In the current situation, people tend to publish half-baked results in order to get better promo- tions or to get a raise in their salary. The printing press was invented by Gutenberg [ca. 1400–1468] 500 years ago in order to let information spread from one person to many others. But we have completely different systems today which are much more efficient than going through the printing process and we haven’t really used that enough. I think that refereeing is exaggerated. Let people publish wrong results and let other people criticize. As long as it’s available on the net it won’t be any great prob- Lennart Carleson 57 lem. Moreover, referees aren’t very reliable; it doesn’t really work anyway. I am predicting a great change, but it’s extremely slow in coming. And in the meantime the printers make lots of money.

Research institutions

I’ve just returned from a nice stay at the Institute Mittag-Leffler, which is sit- uated in Djursholm, north of Stockholm; one of the leading research institutes of our times. This institute was, when you stepped in as its director in 1968, something that I would characterize as a sleeping beauty. But you turned it into something very much different, very active within a few years. By now around thirty mathematicians work together there at any given time but there is almost no permanent staff. What was the inspiration for the concept of the Institute Mittag-Leffler as it looks like today? And how was it possible to get the necessary funds for this institute? Finally, how would you judge the present activities of the institute? To answer the last question first, I have to be satisfied with the way it worked out and the way it continues also. I just hope that it can stay on the same course. In the ’60s, there was a period when the Swedish government (and maybe also other governments) was willing to invest in science. There was a discussion about people moving to the United States. Hörmander had already moved and the question was whether I was going to move as well. In this situation, you could make a bargain with them. So we got some money, which was of course the important part. But there was a rather amusing connection with the Acta, which is not so well known. From Mittag-Leffler’s days, there was almost no money in the funds of the academy for the Mittag-Leffler Institute. But we were able to accumulate rather large sums of money by selling old volumes of the Acta. Mittag-Leffler had printed large stocks of the old Acta journals which he never sold at the time. They were stored in the basement of the institute. During the ’50s and early ’60s one could sell the complete set of volumes. I don’t remember what a set could be sold for, maybe 1000 dollars or so. He had printed several hundred extra copies, and there were several hundred new universities. If you multiply these figures together you get a large amount of money. And that is still the foundation of the economy of the institute.

A bit later, you became the president of the International Mathematical Un- ion, an organization that promotes international cooperation within mathematics. This happened during the cold war and I know that you were specifically concerned with integrating Chinese mathematics at the time. Could you share some of your memories from your presidency? Well, I considered my main concern to be the relation to the Soviet Union. The Chinese question had only started. I went to China and talked to people in Taiwan, and to people in mainland China. But it didn’t work out until the next presidential 58 Abel Prize 2006 period and it simply ripened. The main issue was always whether there was to be a comma in a certain place, or not, in the statutes. It was somehow much more serious with the Russians. You know, they threat- ened to withdraw from international cooperation altogether. The IMU committee and I considered that the relation between the West and the East was the most important issue of the International Mathematical Union. So that was exciting. Negotiations with Pontryagin [1908–1988] and Vinogradov [1891–1983] were kind of special.

Did these two express some anti-semitic views also? No, not officially. Well they did, of course, in private conversation. I remember Vinogradov being very upset about a certain Fields Medal being given to somebody, probably Jewish, and he didn’t like that. He said this is going to ruin the Fields Prize forever. Then I asked him if he knew who received the first Nobel Prize in literature. Do you? It was a French poet called Sully Prudhomme [1839–1907]; and that was during a period when Tolstoy [1828–1910], Ibsen and Strindberg [1849–1912] were available to get the prize. Well, the Nobel Prize survived.

Mathematics for our times

You wrote a book, “Matematik för vår tid” or “Mathematics for our times”, which was published in Sweden in 1968. In that book, you took part in the debate on so-called New Mathematics, but you also described concrete mathematical problems and their solutions. Among other things you talked about the separation between pure and applied mathematics. You described it as being harmful for mathematics and harmful for contact with other scientists. How do you see recent developments in this direction? What are the chances of cross-fertilization between mathematics on the one side and, say, physics, biology or computer science on the other side? Isn’t computer science somehow presently drifting away from mathematics? Yes, but I think we should blame ourselves; mathematics hasn’t really produced what we should, i.e. enough new tools. I think this is, as we talked about before, really one of the challenges. We still have lots of input from physics, statistical physics, string theory, and I don’t know what. I stand by my statement from the sixties. But that book was written mostly as a way to encourage the teachers to stay with established values. That was during the Bourbaki and New Math period and mathematics was really going to pieces, I think. The teachers were very worried and they had very little backing. And that was somehow the main reason for the book.

If you compare the sixties with today, mathematics at a relatively elevated level is taught to many more people and other parts of the subject are empha- sized. For example the use of computers is now at a much higher state then at Lennart Carleson 59 that time, where it almost didn’t exist. What are your main points of view concerning the curriculum of mathematics at, say, high school level and the early years of university? Are we at the right terms? Are we teaching in the right way? No, I don’t think so. From left to right: Lennart Carleson, Martin Raussen and Again, something pre- Christian Skau. (Photo: Terje Bendiksby) dictable happens very slowly. How do you incorporate the fact that you can do many computations with these hand-computers into mathematics teaching? But in the meantime, one has also expelled many things from the classroom which are related to the very basis of mathematics, for example proofs and definitions and logical thinking in general. I think it is dangerous to throw out all computational aspects; one needs to be able to do calculations in order to have any feeling for mathematics. You have to find a new balance somehow. I don’t think anybody has seriously gotten there. They talk a lot about didactics, but I’ve never understood that there is any progress here. There is a very strong feeling in school, certainly, that mathematics is a God-given subject. That it is once and for all fixed. And of course that gets boring.

Public awareness

Let us move to public awareness of mathematics: It seems very hard to explain your own mathematics to the man in the street; we experience that right now. In general pure mathematicians have a hard time when they try to justify their business. Today there is an emphasis on immediate relevance and it’s quite hard to explain what mathematicians do to the public, to people in politics, and even to our colleagues from other sciences. Do you have any particular hints on how mathematicians should convey what they are doing in a better way? Well, we should at least work on it; it’s important. But it is also very difficult. A comment which may sound kind of stupid is that physicists have been able to sell their terms much more effectively. I mean who knows what an electron is? And who knows what a quark is? But they have been able to sell these words. The first thing we should try to do is to sell the words so that people get used to the idea of a derivative, or an integral, or whatever. 60 Abel Prize 2006

As something mysterious and interesting, right? Yes, it should be something mysterious and interesting. And that could be one step in that direction, because once you start to talk about something you have a feeling about what it is. But we haven’t been able to really sell these terms. Which I think is too bad.

Thank you very much for this interview on behalf of the Norwegian, the Danish and the European Mathematical Societies! Abel Prize 2007: Srinivasa S. R. Varadhan

The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2007 to

Srinivasa S. R. Varadhan Courant Institute of Mathematical Sciences, New York

“for his fundamental contributions to probability theory and in particular for creating a unified theory of large deviations.”

Probability theory is the mathematical tool for analyzing situations governed by chance. The law of large numbers, discovered by Jacob Bernoulli in the eighteenth century, shows that the average outcome of a long sequence of coin tosses is usually close to the expected value. Yet the unexpected happens, and the question is: how? The theory of large deviations studies the occurrence of rare events. This subject has concrete applications to fields as diverse as physics, biology, economics, statistics, computer science, and engineering. The law of large numbers states that the probability of a deviation beyond a given level goes to zero. However, for practical applications, it is crucial to know how fast it Abel Laureate Srinivasa S. R. Varadhan. vanishes. For example, what capital reserves (Photo: Anne Lise Flavik) are needed to keep the probability of default of an insurance company below acceptable levels? In analyzing such actuarial “ruin problems”, Harald Cramér discovered in 1937 that standard approximations based on the Central Limit Theorem (as visualized by the bell curve) are actually mislead- ing. He then found the first precise estimates of large deviations for a sequence of independent random variables. It took 30 years before Varadhan discovered the

* 2.1.1940

61 62 Abel Prize 2007 underlying general principles and began to demonstrate their tremendous scope, far beyond the classical setting of independent trials. In his landmark paper “Asymptotic probabilities and differential equations” in 1966 and his surprising solution of the polaron problem of Euclidean quantum field theory in 1969, Varadhan began to shape a general theory of large deviations that was much more than a quantitative improvement of convergence rates. It addresses a fundamental question: what is the qualitative behaviour of a stochastic system if it deviates from the ergodic behaviour predicted by some law of large numbers or if it arises as a small perturbation of a deterministic system? The key to the answer is a powerful variational principle that describes the unexpected behaviour in terms of a new probabilistic model minimizing a suitable entropy distance to the initial probability measure. In a series of joint papers with Monroe D. Donsker exploring the hierarchy of large deviations in the context of Markov processes, Varadhan demonstrated the relevance and the power of this new approach. A striking application is their solution of a conjecture of Mark Kac concerning large time asymptotics of a tubular neighbourhood of the Brownian motion path, the so-called “Wiener sausage”. Varadhan’s theory of large deviations provides a unifying and efficient method for clarifying a rich variety of phenomena arising in complex stochastic systems, in fields as diverse as quantum field theory, statistical physics, population dynamics, econo- metrics and finance, and traffic engineering. It has also greatly expanded our ability to use computers to simulate and analyze the occurrence of rare events. Over the last four decades, the theory of large deviations has become a cornerstone of modern probability, both pure and applied.

Srinivasa S. R. Varadhan receives the Abel Prize from King Harald. (Photo: Terje Bendiksby) Srinivasa S. R. Varadhan 63

Varadhan has made key contributions in several other areas of probability. In joint work with Daniel W. Stroock, he developed a martingale method for characterizing diffusion processes, such as solutions of stochastic differential equations. This new approach turned out to be an extremely powerful way of constructing new Markov processes, for example infinite-dimensional diffusions arising in population genetics. Another major theme is the analysis of hydrodynamical limits describing the macroscopic behaviour of very large systems of interacting particles. A first breakthrough came in joint work with Maozheng Guo and George C. Papanicolaou on gradient models. Varadhan went even further by showing how to handle non-gradient models, greatly extending the scope of the theory. His ideas also had a strong influence on the analysis of random walks in a random environment. His name is now attached to the method of “viewing the environment from the travelling particle”, one of the few general tools in the field. Varadhan’s work has great conceptual strength and ageless beauty. His ideas have been hugely influential and will continue to stimulate further research for a long time.

Srinivasa S. R. Varadhan at the Abel banquet with Jens Stoltenberg, then Prime Minister of Norway (and instrumental, with his government, in setting up the Abel Prize). (Photo: Heiko Junge) 64 Abel Prize 2007

Professor Varadhan, first of all I would like to congratulate you for having been awarded the Abel Prize this year. By extension, our congratulations go to the field of probability and statistics since you are the first recipient from this area. Incidentally, last year at the International Congress of Mathematicians in Madrid, Fields medals were given to mathematicians with expertise in this area for the first time, as well. How come that it took so long time before probability and statistics were recognized so prestigiously, first at the International Congress of Mathema- ticians last year and with the Abel Prize this year? Is it pure coincidence that this happens two years in a row? Could you add some comments on the development of the relations between probability and statistics on the one hand and the rest of mathematics on the other hand? Probability became a branch of mathematics very recently, in the 1930s after Kol- mogorov [1903–1987] wrote his book. Until then it was not really considered as a proper branch of mathematics. In that sense it has taken some time for the mathematical community to feel comfortable with probability the way they are comfortable with number theory and geometry. Perhaps that is one of the reasons why it took a lot of time. In recent years probability has been used in many areas. Mathematical finance for example uses a lot of probability. These days, probability has a lot of exposure and connections with other branches of mathematics have come up. The most recent example has to do with conformal invariance for which the Fields medal was given last year. These connections have brought probability to the attention of the mathematics community, and the awards are perhaps a reflection of that.

Career

The next question is about your career. You were born in Chennai, the capital of Tamil Nadu, on the South-East coast of India, in 1940. You went to school there and then to the Presidency College at Madras University. We would like to ask you about these formative years: What was the first reason for your interest in mathematical problems? Did that happen under the influence of your father, who was a teacher of mathematics? Were there other people, or were there specific problems that made you first interested in mathematics? My father was in fact a teacher of science, not so much mathematics. In my early school days I was good in mathematics, which just meant that I could add, subtract and multiply without making mistakes. Anyway, I had no difficulty with mathematics. At high school I had an excellent mathematics teacher who asked some of his better students to come to his house during weekends, Saturday or Sunday, and give them extra problems to work on. We thought of these problems just as intellectual games that we played, it was not like an exam; it was more for enjoyment. That gave me the idea that mathematics is something that you can enjoy doing like playing chess or solving puzzles. That attitude made mathematics a much more friendly Srinivasa S. R. Varadhan 65 subject, not something to be afraid of, and that played a role in why I got interested in mathematics. After that I went to college for five years. I had excellent teachers throughout. By the time I graduated with a master degree in statistics, I had three years of solid grounding in pure mathematics. My background was strong when I graduated from College.

Was there a specific reason that you graduated in statistics rather than in other branches of mathematics? The option at that time was either to go into mathematics or into statistics. There was not that much difference between these two. If you went into mathematics, you studied pure and applied mathematics; if you went into statistics, you studied pure mathematics and statistics. You replaced applied mathematics with statistics; that was the only difference between the two programs. Looking back, part of the reason for going into statistics rather then mathematics, was the perception that if you went into statistics your job opportunities were better; you could be employed in the industry and so on. I f you went into mathematics, you would end up as a school teacher. There was that perception; I do not know how real it was.

With your degree in statistics it seemed quite natural that you continued at the Indian Statistical Institute at Kolkata. There you found yourself quite soon in a group of bright students that, seemingly without too much influence from their teachers, started to study new areas of fundamental mathematics and then applied those to problems coming from probability theory; with a lot of success: You were able to extend certain limit theorems for stochastic processes to higher dimensional spaces; problems that other mathematicians from outside India had been working on for several years without so much success. Could you tell us a bit about this development and whom you collaborated with? The Indian system at that time was very like much the British system: If you decided to study for a doctoral degree, there were no courses; you were supposed to do research and to produce a thesis. You could ask your advisor questions and he would answer you, but there was no formal guidance as is the case in the USA for example. When I went there I had the idea that I would be looking for a job within some industry. I was told that I should work on statistical quality control, so I spent close to 6 or 8 months studying statistical quality control; in the end, that left me totally unsatisfied. Then I met Varadarajan, Parthasarathy and Ranga Rao, who worked in probability from a totally mathematical point of view. They convinced me that I was not spending my time usefully, and that I better learn some mathematics if I wanted to do anything at all. I got interested, and I think in the second year I was there, we said to ourselves: let us work on a problem. We picked a problem concerning probability distributions on groups. That got us started; we eventually solved the problem and in the process also learned the tools that were needed for it. 66 Abel Prize 2007

It was a lot of fun: the three of us constantly exchanged ideas starting at seven o’clock in the morning. We were all bachelors, living in the same dormitory. The work day lasted from 7 am to 9 pm; it was a terrific time to learn. In fact, the second paper we wrote had Abel in its title, because it has something to do with locally compact abelian groups.

From what you tell us, it seems that your work can serve as an example for the fact that the combination of motivations and insights from real world problems on the one hand and of fundamental abstract mathematical tools on the other hand has shown to be extremely fruitful. This brings me to a question about the distinction between pure and applied mathematics that some people focus on. Is it meaningful at all – in your own case and maybe in general? I think that distinction, in my case at least, is not really present. I usually look at mathematics in the following way: There is a specific problem that needs to be solved. The problem is a mathematical problem, but the origin of the problem could be physics, statistics or could be another application, an economic application perhaps. But the model is there, and it is clear what mathematical problem you have to solve. But of course, if the problem came from physics or some application, there is an intuition that helps you to reason what the possible answer could be. The challenge is how to translate this intuition into rigorous mathematics. That requires tools, and sometimes the tools may not be around and you may have to invent these tools and that is where the challenge and the excitement of doing mathematics is, as far as I am concerned. That is the reason why I have been doing it.

India and the 3rd world

May we come back to your Indian background? You are the first Abel Prize recipient with an education from a 3rd world country. In 1963, you left Kolk- ata and went to the Courant Institute of Mathematical Sciences in New York, where you still are working. We wonder whether you still strongly feel your In- dian background – in your mathematical training, your life style, your religion and philosophy? For 23 years, I grew up in India, and I think that part of your life always stays with you. I am still very much an Indian in the way I live. I prefer Indian food to anything else, and I have some religious feelings about Hinduism and I am a practising Hindu. So my religious beliefs are based on my real life, and my lifestyle is very much Indian. But when you are living in the United States you learn to adjust a little bit, you perhaps have a combination of the two that you are comfortable with. My training in India has been mainly in classical analysis. No matter what you do, even if you do the most abstract mathematics, you use it as a tool. At crucial points, I think you need to go back to your classical roots and do some tough estimates here and there; I think the classical training definitely helps there. The abstract mathematical tools then help you to put some results in perspective. You Srinivasa S. R. Varadhan 67 can see what the larger impact of what you have done is. To assess that, modern training gives you some help.

The best known Indian mathematician of the past, at least here in the West, is certainly Srinivasa Ra- manujan [1887–1920]. He is known for his very untraditional methods in obtaining deep results, and his Terry Lyons, Ofer Zeitouni, Srinivasa Varadhan and George Papanicolaou at the University of note books are still studied by a Oslo in connection with the Abel Lectures. lot of mathematicians around the (Photo: Terje Bendiksby) world. He is certainly also known for his tragically fate and his untimely death. Has he played a specific role in your life as a role model? Is that still true for many Indian mathematicians? I think the name of Ramanujan has been familiar to most Indians today. Maybe, when I was growing up, it was more familiar to people from the South than from the North, because he came from the southern part of India, but we definitely knew of him as a great mathematician. At that age, I did not really know the details of his work. Even now, I have only a vague idea of what it is about. People still do not seem to know how exactly he arrived at those results. He seemed to have a mental process that led him to these things, which he has not fully explained in his work. In spite of spending years with Hardy [1877–1947], the West was not able to penetrate the barrier and understand how his mind worked. I do not think we can do anything about it now.

Mathematical institutions

You spent the last years of your life in India at the Indian Statistical Institute (ISI) at Kolkata. There is another well-known research institute in India, the Tata Institute. We know that there has been some competition between these two institutions although they are specialising in different fields. Can you comment on this competition, the ongoing relations between the two institutes and their respective strengths? I do not know when the competition started. The Indian Statistical Institute was founded by Mahalanobis [1893–1972] in 1931; the Tata Institute was founded by Bhabha [1909–1966] in 1945. They were both great friends of Jawaharlal Nehru [1889–1964], the prime minister at the time, he encouraged them both. Maybe, there are some rivalries at that level, the institutional level. The mathematics division of the Indian Statistical Institute had Dr. C. R. Rao, who was my advisor, as its scientific director, and the mathematics division of the Tata Institute was headed by Dr. Chandrasekharan [1920–2007]; he was the moving force behind the mathematics school of Tata Institute. Maybe, there is some competition there. 68 Abel Prize 2007

I know many of the faculty of the Tata Institute; in fact many of them were from the same region in the South and they went to the same university, the same college, perhaps even to the same high school. So the relationships between the two faculties have always been friendly. It is true, the emphasis is very different. At Tata, they have concentrated on number theory and algebraic geometry and certain parts of abstract mathematics. The Indian Statistical Institute on the other hand has concentrated more on probability and statistics. Although there has been some overlap, it is really not that much.

We have heard that you still entertain close relations to India and to Chennai and its Mathematical Institute, in particular. And in general, you are interested in the academic development of 3rd World countries, in particular through the Third World Academy of Sciences. Please tell us about your connections and your activities there? I go to Chennai maybe once a year now. Earlier it used to be twice a year, when my parents were alive. I use to go and spend a month or two in Chennai, and I visit the two mathematical institutions in Chennai: There is the Chennai Mathematics Institute, and there is also the Institute of Mathematical Sciences in Chennai. I have visited both of them at different times; I have close contacts with their leadership and their faculty. In earlier times, I visited the Bangalore centre of the Tata Institute: The Tata Institute in Mumbai has a Centre for Applicable Mathematics in Bangalore. I spent some time visiting them, and we have had students from there coming to the Cou- rant Institute to take their degrees and so on. To the extent possible, I try to go back and keep in touch. Nowadays, with e-mail, they can ask me for advice, and I try to help out as much I can. The next couple of years, I have some plans to spend part of my sabbatical in Chennai lecturing at Chennai Mathematics Institute.

You are already the second Abel Prize winner working at the Courant Institute of Mathematical Sciences in New York, after Peter Lax. At least in the world of applied mathematics, the Courant Institute seems to play a very special role. Could you explain how this worked out? What makes the Courant Institute to such a special place? We are back to the 1930s, when the Courant Institute was started. There was no applied mathematics in the United States. Richard Courant [1888–1972] came and he started this mathematics institute with the emphasis on applied mathematics. His view of applied mathematics was broad enough so that it included pure mathematics. I mean, he did not see the distinction between pure and applied mathematics. He needed to apply mathematics, and he developed the tools, he needed to do it. And from that point of view, I think analysis played an important role. The Courant Institute has always been very strong in applied mathematics and analysis. And in the 1960s, there was a grant from the Sloan foundation to develop probability and statistics at the Courant Institute. They started it, and probability Srinivasa S. R. Varadhan 69 was successful, I think. Statistics did not quite work out, so we still do not have really much statistics at the Courant Institute. We have a lot of probability, analysis, and applied mathematics, and in recent years some differential geometry as well. In these areas we are very strong. The Courant Institute has always been successful in hiring the best faculty. The emphasis has mostly been on analysis and applied mathematics. Perhaps that reflects why we have had two Abel prize winners out of the first five.

Mathematical research: Process and results

You have given deep and seminal contributions to the area of mathematics which is called probability theory. What attracted you to probability theory in the first place? When I joined my undergraduate program in statistics, probability was part of statistics; so you had to learn some probability. I realised that I had some intuition for probability in the sense that I could sense what one was trying to do, more than just calculating some number. I cannot explain it, I just had some feeling for it. That helped a lot; that motivated me to go deeper into it.

Modern probability theory, as you mentioned earlier, started with Kolmogor- ov in the 1930s. You had an interesting encounter with Kolmogorov: He wrote from Moscow about your doctoral thesis at the Indian Statistical Institute, that you finished when you were 22 years: “This is not the work of a student, but of a mature master”. Did you ever meet Kolmogorov? Did you have any mathematical interaction with him later? Yes, I have met him; he came to India in 1962. I had just submitted my thesis, and he was one of the examiners of the thesis, but he was going to take the thesis back to Moscow and then to write a report; so the report was not coming at that time. He spent a month in India, and some of us graduate students spent most of our time travelling with him all over India. There was a period where we would meet him every day. There were some reports about it mentioned in the Indian press recently, which were not quite accurate. But there is one incident that I remember very well. I was giving a lecture on my thesis work with Kolmogorov in the audience. The lecture was supposed to last for an hour, but in my enthusiasm it lasted an hour and a half. He was not protest- ing, but some members in the audience were getting restless. When the lecture ended, he got up to make some comments and people started leaving the lecture hall before he could say something, and he got very angry. He threw the chalk down with great force and stormed out of the room. My immediate reaction was: There goes my PhD! A group of students ran after him to where he was staying, and I apol- ogized for taking too much time. He said: No no; in Russia, our seminars last three hours. I am not angry at you, but those people in the audience, when Kolmogorov stands up to speak, they should wait and listen. 70 Abel Prize 2007

That is a nice story! Among your many research contributions, the one which is associated with so-called large deviations must rank as one of the most important. Can you tell us first what large deviations are and why the study of these is so important; and what are the applications? The subject of large deviations goes back to the early thirties. It in fact started in Scan- dinavia, with actuaries working for the insurance industry. The pioneer who started that subject was named Esscher1. He was interested in a situation where too many claims could be made against the insurance company, he was worried about the total claim amount exceeding the reserve fund set aside for paying these claims, and he wanted to calculate the probability of this. Those days the standard model was that each individual claim is a random variable, you assume some distribution for it, and the total claim is then the sum of a large number of independent random variables. And what you are really interested in is the probability that the sum of a large number of independent random variables exceeds a certain amount. You are interested in estimating the tail probabilities of sums of independent random variables. People knew the central limit theorem at the time, which tells you that the distribution of sums of independent random variables has a Gaussian approximation. If you do the Gaussian approximation, the answer you get is not correct. It is not correct in the sense that the Gaussian approximation is still valid, but the error is measured in terms of difference. Both these numbers are very small, therefore the difference between them is small, so the central limit theorem is valid. But you are interested in how small it is, you are interested in the ratio of these two things, not just the difference of these small numbers. The idea is: how do you shift your focus so that you can look at the ratio rather then just at the difference. Esscher came up with this idea, that is called Esscher’s tilt; it is a little technical. It is a way of changing the measure that you use in a very special manner. And from this point of view, what was originally a tail event, now becomes a central event. So you can estimate it much more accurately and then go from this estimate to what you want, usually by a factor which is much more manageable. This way of estimation is very successful in calculating the exact asymptotics of these tail probabilities. That is the origin of large deviations. What you are really interested in is estimating the probabilities of certain events. It does not matter how they occur; they arise in some way. These are events with very small probability, but you would like to have some idea of how small it is. You would like to measure it in logarithmic scale, “e to the minus how big”. That is the sense in which it is used and formulated these days.

Large deviations have lots of applications, not the least in finance; is that correct? I think in finance or other areas, what the theory actually tells you is not just what the probability is, but it also tells you if an event with such a small probability

1 F. Esscher, On the probability function in the collective theory of risk. Skandinavisk Actuariet- idskrift 15 (1932), 175–195. Srinivasa S. R. Varadhan 71 occurred, how it occurred. You can trace back the history of it and explain how it occurred and what else would have occurred. So you are concerned of analysing entire circumstances. In Esscher’s method, there is the tilt that produced it; then that tilt could have produced other things, too; they would all happen if this event happened; it gives you more information than just how small the probability is. This has become useful in mathematical finance because you write an option which means: if something happens at a certain time, then you promise to pay somebody something. But what you pay may depend on not just what happened at that time, it may depend on the history. So you would like to know if something happened at this time, what was the history that produced it? Large deviation theory is able to predict this.

Together with Donsker you reduced the general large deviation principle to a powerful variational principle. Specifically, you introduced the so-called Don- sker–Varadhan rate function and studied its behaviour. Could you elaborate a little how you proceeded, and what type of rate functions you could handle and analyse? If you go back to the Esscher theory of large deviations for sums of random variables, that requires the calculation of the moment generating function. Since they are independent random variables, the moment generating functions are products of the individual ones; if they are all the same, you get just the n-th power of one moment generating function. What really controls the large deviation is the logarithm of the moment generating function. The logarithm of the n-th power is just a multiple of the logarithm of the original moment generating function, which now controls your large deviation. On the other hand, if your random variables are not independent, but dependent like in a Markov chain or something like that, then there is no longer just one moment generating function. It is important to know how the moment generating function of the sum grows; it does not grow like a product but it grows in some way. This is related by the Feynman–Kac formula to the principal eigenvalue of the generator of the Markov process involved. There is a connection between the rate function and the so-called principal eigenvalue. This is what our theory used considerably. The rate function is constructed as the Legendre transform or the convex conjugate of the logarithm of the principal eigenvalue.

Before we leave the subject of the large deviation principle, could you please comment on the so-called Varadhan integral lemma which is used in many areas. Why is that? I do not think Varadhan’s lemma is used that much, probably large deviation theory is used more. The reason why I called it a lemma is that I did not want to call it a theorem. It is really a very simple thing that tells you that if probabilities behave in a certain way, then certain integrals behave in a certain way. The proof just requires approximating the integral by a sum and doing the most elementary estimate. What is important there is just a point of view and not so much the actual estimates in the work involved; this is quite minimal. 72 Abel Prize 2007

But it pops up apparently in many different areas; is that correct? The basic idea in this is very simple: if you take two positive numbers a and b and raise them to a very high power and you look at the sum, the sum is just like the power of the larger one; the smaller one is insignificant, you can replace the logarithm of the sum by just a maximum. The logarithm of the sum of the exponential behaves just like the maximum. That is the idea, when you have just a finite number of exponentials, then in some sense integrating is not different from summation if you have the right estimates. That was how I looked at it, and I think this arises in many different contexts. One can use the idea in many different places, but the idea itself is not very complicated.

That is often the case with important results in mathematics. They go back to a simple idea, but to come up with that idea, that is essential! You realized that Mark Kac’s old formula for the first eigenvalue of the Schrödinger operator can be interpreted in terms of large deviations of a certain Brownian motion. Could you tell us how you came to this realization? It was in 1973, I just came back from India after a sabbatical, and I was in Donsker’s office. We always spent a lot of time talking about various problems. He wanted to look at the largest eigenvalue which controls the asymptotic behaviour of a Kac integral: I think people knew at that time that if you take the logarithm of the expec- tation of a Kac type exponential function, its asymptotic growth rate is the first eigenvalue. The first eigenvalue is given by a variational formula; that is classical. We knew that if we do large deviations and calculate asymptotically the integrals, you get a variational formula, too. So, he wanted to know if the two variational formulas have anything to do with each other: Is there a large deviation interpretation for this variational formula? I was visiting Duke University, I remember, some time later that fall, and I was waiting in the library at Duke University for my talk which was to start in half an hour or so. Then it suddenly occurred to me what the solution to this problem was: It is very simple, in the Rayleigh–Ritz variational formula; there are two objects that compete. One is the integral of the potential multiplied by the square of a function; the other one is the Dirichlet form of the function. If you replace the square of the function and call it a new function, then the Dirichlet form becomes the Dirichlet form of the square root of that function. It is as simple as that. And then the large deviation rate function is nothing but the Dirichlet form of the square root of the density. Once you interpret it that way, it is clear what the formula is; and once you know what the formula is, it is not that difficult to prove it.

This brings me naturally to the next question: If you occasionally had a sudden flash of insight, where you in an instant saw the solution to a problem that you had struggled with, as the one you described right now: Do these flashes depend on hard and sustained preparatory thinking about the problem in question? Yes, they do: What happens is, once you have a problem you want to solve, you have some idea of how to approach it. You try to work it out, and if you can solve it the Srinivasa S. R. Varadhan 73 way you thought you could, it is done, and it is not interesting. You have done it, but it does not give you a thrill at all. On the other hand, if it is a problem, in which everything falls in to place, except for one thing you cannot do; if only you could do that one thing, then you would have the whole building, but this foundation is missing. So you struggle and struggle with it, sometimes for months, sometimes for years and sometimes for a life-time! And eventually, suddenly one day you see how to fix that small piece. And then the whole structure is complete. That was the missing piece. Then that is a real revelation, and you enjoy a satisfaction which you cannot describe.

How long does the euphoria last when you have this experience? It lasts until you write it up and submit it for publication; and then you go on to the next problem!

Your cooperation with Daniel Stroock on the theory of diffusions led to several landmark papers. The semigroup approach by Kolmogorov and Feller [1906– 1970] had serious restrictions, we understand, and Paul Levy [1886–1971] suggested that a diffusion process should be represented as a stochastic differential equation. Îto also had some very important contribution. Could you explain how you and Stroock managed to turn this into a martingale problem? I have to step back a little bit: Mark Kac [1914–1984] used to be at Rockefeller University. Between New York University and Rockefeller University, we used to have a joint seminar; we would meet one week here and one week there and we would drive back and forth. I remember once going to Rockefeller University for a seminar and then coming back in a taxi to NYU. Somebody mentioned a result of Ciesielski, a Polish probabilist who was visiting Marc Kac at that time: You can look at the fundamental solution of a heat equation, for the whole space, and look at the fundamental solution with Dirichlet boundary data in a region. The fundamental solution for the Dirichlet boundary data is smaller, by the maximum principle, than the other one. If you look at the ratio of the two fundamental solutions, then it is always less than or equal to one. The question is: As t, the time variable in the fundamental solution, goes to zero, when does this ratio go to 1 for all points x and y in the region? The answer turns out to be: if and only if the region is convex! Of course, there are some technical aspects, about sets of capacity zero and so on. Intuitively, the reason it is happening is that the Brownian path, if it goes from x to y, in time t, as time t goes to zero, it would have to go in a straight line. Because its mean value remains the same as that of the Brownian bridge, which is always linear, and thus a line connecting the two points. The variance goes to zero, if you do not give it much time. That means it follows a straight line. That suggests that, if your space were not flat but curved, then it should probably go along the geodesics. One would expect therefore that the fundamental solution of the heat equation with variable coefficients should look like e to the minus the square of the geodesic distance divided by 2 t; just like the heat equation does with the Euclidean distance. 74 Abel Prize 2007

This occurred to me on the taxi ride back. That became the paper on the behaviour of the fundamental solution for small time. In fact, I think that was the paper that the PDE people at Courant liked, and that gave me a job. At that time, I was still a postdoc. Anyway, at that point, the actual proof of it used only certain martingale properties of this process. It did not really use so much PDE, it just used certain mar- tingales. Stroock was a graduate student at Rockefeller University at that time; we used to talk a lot. I remember, that spring, before he finished, we would discuss it. We thought: If it is true that we could prove this by just the martingale properties, then those martingale properties perhaps are enough to define it. Then we looked at it and asked ourselves: Can you define all diffusion processes by just martingale properties? It looked like it unified different points of view: Kolmogorov and Feller through the PDE have one point of view, stochastic differential people have another point of view, semigroup theory has still another point of view. But the martingale point of view unifies them. It is clear that it is much more useful; and it turned out, after investigation, that the martingale formulation is sort of the weakest formulation one can have; that is why everything implies it. Being the weakest formulation, it became clear that the hardest thing would be to prove uniqueness. Then we were able to show that whenever any of the other methods work, you could prove uniqueness for this. We wanted to extend it and prove uniqueness for a class which had not been done before, and that eluded us for nearly one and a half year until one day the idea came, and we saw how to do it and everything fell into place.

That was another flash of inspiration? That was another flash; that meant that we could do a lot of things for the next four to five years that kept us busy.

Before we leave your mathematical research, we would like to ask you about your contribution to the theory of hydrodynamic limits that is describing the macroscopic behaviour of very large systems of interacting particles. Your work in this area has been described as viewing the environment from the travelling particle. Could you describe what this means? I will try to explain it. The subject of hydrodynamic scaling as it is called, or hydrodynamic limits is a subject that did not really start in probability. It started from classical mechanics, Hamiltonian equations, and it is the problem of deriving Euler equations of fluid flow directly as a consequence of Hamiltonian motion. After all, we can think of a fluid as a lot of individual particles and the particles interact, ignoring quantum effects, according to Newtonian rules. We should be able to describe how every particle should move. But this requires solving a 10 to the 68 -dimensional ODE, and only then you are in good shape. Instead we replace this huge system of ODEs by PDEs, a small system of nonlinear PDEs, and these nonlinear PDEs describe the motion of conserved quantities. Srinivasa S. R. Varadhan 75

If there are no conserved quantities, then things reach equilibrium very fast, and nothing really moves. But if there are conserved quantities, then they change very slowly locally, and so you have to speed up time to a different scale. Then you can observe change of these things. Mass is conserved, that means, density is one of the variables; momenta are conserved, so fluid velocity is one of the variables; the energy is conserved, so temperature becomes one of the variables. For these conserved quantities, you obtain PDEs. When you solve your partial differential equations, you get a solution that is supposed to describe the macroscopic properties of particles in that location. And given these parameters, there is a unique equilibrium for these fixed values of the parameters which are the average values. In a Hamiltonian scheme, that would be a fixed surface with prescribed energy and momentum etc. On that surface the motion is supposed to be ergodic, so that there is a single invariant measure. This invariant measure describes how locally the particles are behaving over time. That is only described in statistical terms; you cannot really pin down which particle is where; and even if you could, you do not really care. This program, although it seems reasonable in a physical sense, it has not been carried out in a mathematical sense. The closest thing that one has come to is the result by Oscar Lanford [1940–2013] who has shown, that for a very small time scale, you can start from the Hamiltonian system and derive the Boltzmann equations. Then to go from Boltzmann to Euler requires certain scales to be large, it is not clear if the earlier results work in this regime. The mathematical level of these things is not where it should be. On the other hand, if you put a little noise in your system, so that you look at not a deterministic Hamiltonian set of equations, but stochastic differential equations, with particles that move and jump randomly, then life becomes much easier. The problem is the ergodic theory. The ergodic theory of dynamical systems is very hard. But the ergodic theory of Markov processes is a lot easier. With a little bit of noise, it is much easier to keep these things in equilibrium. And then you can go through this program and actually prove mathematical results. Now coming to the history: We were at a conference in Marseille at Luminy, which is the Oberwolfach of the French Mathematical Society. My colleague George Papanicolaou, who I think should be here in Oslo later today, and I, we were taking a walk down to the calanques. And on the way back, he was describing this problem. He was interested in interacting particles, Brownian motion interacting under some potential. He wanted to prove the hydrodynamic scaling limit. I thought the solution should be easy; it seemed natural somehow. When I came back and looked at it, I got stuck regardless how much I tried. There were two critical steps, I figured out, needed to be done; one step I could do, the second step I could not do. For the time being, I just left it at that. Then, a year later, we had a visitor at the Courant Institute, Josef Fritz from Hungary. He gave a talk on hydrodynamic limits; he had a slightly different model. By using different techniques, he could prove the theorem for that model. Then I realized that the second step on which I got stuck in the original model, I could do 76 Abel Prize 2007 it easily in this model. So we wrote a paper with George Papanicolaou and one of his students Guo; that was my first paper on hydrodynamic limits. This work was more for a field than for an actual particle system which was what got me interested in the subject. When you look at particles, you can ask two different questions. You can ask what is happening to the whole system of particles, you do not identify them; you just think of it as a cloud of particles. Then you can develop how the density of particles changes over time. But it does not tell you which particle moves where. Imagine particles have two different colours. Now you have two different densities, one for each colour. You have the equation of motion for the sum of the two densities, but you do not have an equation of motion for each one separately. Because to do each one separately, you would have to tag the particles and to keep track of them! It becomes important to keep track of the motion of a single particle in the sea of particles. A way to analyse it that I found useful was to make the particle that you want to tag the centre of the Universe. You change your coordinate scheme along with that particle. Then this particle does not move at all; it stays where it is, and the entire Universe revolves around it. So you have a Markov process in the space of universes. This is of course an infinite dimensional Markov process, but if you can analyse it and prove ergodic theorems for it, then you can translate back and see how the tagged particle would move; because in some sense how much the Uni- verse moves around it or it moves around the Universe is sort of the same thing. I found this method to be very useful, and this is the system looked from the point of view of the moving particle.

Work style

Very interesting! A different question: Can you describe your work style? Do you think in geometric pictures or rather in formulas? Or is there an analytic way of thinking? I like to think physically in some sense. I like to build my intuition as a physicist would do: What is really happening, understanding the mechanism which produces it, and then I try to translate it into analysis. I do not like to think formally, starting with an equation and manipulating and then see what happens. That is the way I like to work: I let my intuition guide me to the type of analysis that needs to be done.

Your work in mathematics has been described by a fellow mathematician of yours as “Like a Bach fugue, precise yet beautiful”. Can you describe the feeling of beauty and aesthetics that you experience when you do mathematics? I think the quotation you are referring to can be traced back to the review of my book with Stroock by David Williams. I think mathematics is a beautiful subject because it explains complicated behaviour by simple means. I think of mathematics as simplifying, giving a simple explanation for much complex behaviour. It helps you Srinivasa S. R. Varadhan 77 to understand why things behave in a certain manner. The underlying reasons why things happen are usually quite simple. Finding this simple explanation of complex behaviour, that appeals most to me in mathematics. I find beauty in the simplicity through mathematics. From left to right: Christian Skau, Martin Raussen and Srinivasa Varadhan. (Photo: Håkon Mosvold Larsen) Public awareness

May we now touch upon the public awareness of mathematics: There appears to be a paradox: Mathematics is everywhere in our life, as you have already witnessed from your perspective: in technical gadgets, in descriptions and calculations of what happens on the financial markets, and so on. But this is not very visible for the public. It seems to be quite difficult for the mathematical community to convince the man in the street and the politicians of its importance. Another aspect is that it is not easy nowadays to enrol new bright students in mathematics. As to graduate students, in the United States more than half of the PhD students come from overseas. Do you have any suggestions what the mathematical community could do to enhance its image among the public, and how we might succeed to enrol more students into this interesting and beautiful subject? Tough questions! People are still trying to find the answer. I do not think it can be done by one group alone. For a lot of reasons, probably because of the nature of their work, most mathematicians are very introverted by nature. In order to convince the public, you need a kind of personality that goes out and preaches. Most research mathematicians take it as an intrusion on their time to do research. It is very difficult to be successful, although there are a few examples. The question then becomes: How do you educate politicians and other powerful circles that can do something about it about the importance of education? I think that happened once before when the Russians sent the Sputnik in 1957, I do not know how long it will take to convince people today. But I think it is possible to make an effort and to convince people that mathematics is important to society. And I think that signs are there, because one of the powerful forces of the society today are the financial interests, and the financial interests are beginning to realize that mathematics is important for them. There will perhaps be pressure from their side to improve mathematics education and the general level of mathematics in the country; and that might in the long run prove beneficial; at least we hope so. 78 Abel Prize 2007

In connection with the Abel Prize, there are also other competitions and prizes, like the Niels Henrik Abel competition and the Kapp Abel for pupils, the Holm- boe prize to a mathematics teacher, and furthermore the Ramanujan prize for an outstanding 3rd world mathematician. How do you judge these activities? I think these are very useful. They raise the awareness of the public. Hopefully, all of this together will have very beneficial effect in the not too distant future. I think it is wonderful what Norway is doing.

Private interests

In our very last question, we would like to leave mathematics behind and ask you about your interests and other aspects of life that you are particularly fond of. What would that be? I like to travel. I like the pleasure and experience of visiting new places, seeing new things and having new experiences. In our profession, you get the opportunity to travel, and I always take advantage of it. I like music, both classical Indian and a little bit of classical Western music. I like to go to concerts if I have time; I like the theatre, and New York is a wonderful place for theatre. I like to go to movies. I like reading Tamil literature, which I enjoy. Not many people in the world are familiar with Tamil as a language. It is a language which is 2000 years old, almost as old as Sanskrit. It is perhaps the only language which today is not very different from the way it was 2000 years ago. So, I can take a book of poetry written 2000 years ago, and I will still be able to read it. To the extent I can, I do that.

At the end, we would like to thank you very much for this interesting interview. These thanks come also on behalf of the Norwegian, the Danish and the Euro- pean Mathematical Society. Thank you very much. Thank you very much. I have enjoyed this interview, too. Abel Prize 2008: John Griggs Thompson and Jacques Tits

The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2008 to

John Griggs Thompson and Jacques Tits University of Florida Collège de France

“for their profound achievements in algebra and in particular for shaping modern group theory.”

Abel Laureate John Griggs Thompson. Abel Laureate Jacques Tits. (Photo: Anne Lise Flavik) (Photo: Anne Lise Flavik)

* 13.10.1932 * 12. 8.1930

79 80 Abel Prize 2008

Modern algebra grew out of two ancient traditions in mathematics, the art of solving equations, and the use of symmetry as for example in the patterns of the tiles of the Alhambra. The two came together in late eighteenth century, when it was first conceived that the key to understanding even the simplest equations lies in the symmetries of their solutions. This vision was brilliantly realised by two young mathematicians, Niels Henrik Abel and Evariste Galois, in the early nineteenth century. Eventually it led to the notion of a group, the most powerful way to capture the idea of symmetry. In the twentieth century, the group theoretical approach was a crucial ingredient in the development of modern physics, from the understanding of crystal- line symmetries to the formulation of models for fundamental particles and forces. In mathematics, the idea of a group proved enormously fertile. Groups have striking properties that unite many phenomena in different areas. The most important groups are finite groups, arising for example in the study of permutations, and linear groups, which are made up of symmetries that preserve an underlying geometry. The work of the two laureates has been complementary: John Thompson concentrated on finite groups, while Jacques Tits worked predominantly with linear groups. Thompson revolutionised the theory of finite groups by proving extraordinarily deep theorems that laid the foundation for the complete classification of finite simple groups, one of the greatest achievements of twentieth century mathematics. Simple groups are atoms from which all finite groups are built. In a major breakthrough, Feit and Thompson proved that every non-elementary simple group has an even number of elements. Later Thompson extended this result to establish a classification of an important kind of finite simple group called an N-group. At this point, the classifica-

John Griggs Thompson (left) and Jacques Tits receive the Abel Prize from King Harald. (Photo: Heiko Junge) John Griggs Thompson and Jacques Tits 81 tion project came within reach and was carried to completion by others. Its almost incredible conclusion is that all finite simple groups belong to certain standard families, except for 26 sporadic groups. Thompson and his students played a major role in understanding the fascinating properties of these sporadic groups, including the largest, the so-called Monster. Tits created a new and highly influential vision of groups as geometric objects. He introduced what is now known as a Tits building, which encodes in geometric terms the algebraic structure of linear groups. The theory of buildings is a central unifying principle with an amazing range of applications, for example to the classification of algebraic and Lie groups as well as finite simple groups, to Kac–Moody groups (used by theoretical physicists), to combinatorial geometry (used in computer science), and to the study of rigidity phenomena in negatively curved spaces. Tits’s geometric approach was essential in the study and realisation of the sporadic groups, including the Monster. He also established the celebrated “Tits alternative”: every finitely generated linear group is either virtually solvable or contains a copy of the free group on two generators. This result has inspired numerous variations and applications. The achievements of John Thompson and of Jacques Tits are of extraordinary depth and influence. They complement each other and together form the backbone of modern group theory.

Jacques Tits and John Griggs Thompson at the Abel monument in Oslo. (Photo: Scanpix) 82 Abel Prize 2008

Early experiences

On behalf of the Norwegian, Danish and European Mathematical Societies we want to congratulate you for having been selected as Abel Prize winners for 2008. In our first question we would like to ask you when you first got interested in mathematics. Were there any mathematical results or theorems that made a special impression on you in your childhood or early youth? Did you make any mathematical discoveries during that time that you still remember? TITS: I learned the rudiments of arithmetic very early; I was able to count as a small child – less than four years, I believe. At the age of thirteen, I was reading mathematical books that I found in my father’s library and shortly after I started tutoring youngsters five years older than me who were preparing for the entrance examination at the École Polytechnique in Brussels. That is my first recollection. At that time I was interested in analysis but later on I became a geometer. Concerning my work in those early years, I certainly cannot talk about great discoveries but I think that some of the results I obtained then are not without interest. My starting subject in mathematical research has been the study of strictly triple transitive groups; that was the subject essentially given to me by my professor (Paul Libois). The problem was this: we knew axiomatic projective geometry in dimension greater than one. For the one-dimensional case, nobody had given an axiomatic definition. The one-dimensional case corresponds to PSL(2). My teacher gave me the problem of formulating axiomatics for these groups. The idea was to take triple transitivity as the first axiom. So I started with this kind of problem: giving axiomatics of projective geometry based on triple transitivity. Of course, I was then led naturally to consider quadruple and quintuple transitivity. That is how I rediscovered all the Mathieu groups, except, strangely enough, the biggest one, the quintuple transitive. I had to rediscover that one in the literature!

So you didn’t know about the Mathieu groups when you did this work? TITS: No, I didn’t.

How old were you at that time? TITS: 18 years old, I suppose. In fact, I first found all strictly quadruple transitive groups. They were actually known by Camille Jordan [1838–1922]. But I didn’t know the work of Camille Jordan at the time. I rediscovered that.

You must have been much younger than your fellow students at the time. Was it a problem to adjust in an environment where you were the youngest by far? TITS: I am very grateful to my fellow students and also to my family. Because I was what is sometimes called a little genius. I was much quicker than all the others. But nobody picked up on that; they just let it go. My father was a little bit afraid that I would go too fast. My mother knew that this was exceptional but she never boasted about it. In fact, a female neighbour said to my mother: “If I had a son like that, I would go around and boast about it”. My mother found that silly. I was not at all put on a pedestal. John Griggs Thompson and Jacques Tits 83

Hardy [1877–1947] once said that mathematics is a young man’s game. Do you agree? TITS: I think that it is true to a certain extent. But there are people who do very deep things at a later age. After all, Chevalley’s [1909–1984] most important work was done when he was more than 40 years old and perhaps even later. It is not an absolute rule. People like to state such rules. I don’t like them really.

THOMPSON: Well, it is true that you don’t have any childhood geniuses in politics. But in chess, music and mathematics, there is room for childhood exceptionalism to come forth. This is certainly very obvious in the case of music and chess and to some extent in mathematics. That might sort of skew the books in a certain direction. As far as Hardy’s remark is concerned, I don’t know what he was feeling about himself at the time he made that remark. It could be a way for a person to say: “I am checking out now; I have reached the age where I don’t want to carry on”. I don’t know what the sociologists and psychologists say; I leave it to them. I have seen mathematicians reach the age of 50 and still be incredibly lively. I don’t see it as a hard and fast rule. But then Tits and I are really in no position to talk given our age.

John von Neumann [1903–1957] said, exaggerating a little, that whatever you do in mathematics beyond 30 is not worth anything, at least not compared to what you had done before 30. But when he himself reached the age of 30, he pushed this limit. Experience comes in, etc… THOMPSON: But he was a prodigy. So he knows the childhood side of it.

TITS: We all have known very young and bright mathematicians. The point is that to find deep mathematics, it is not necessary to have all the techniques. They can find results that are deep without having all of those techniques at hand.

What about your memories on early mathematical experiences, Professor Thompson? THOMPSON: I don’t have any particularly strong memories. I have an older brother, three years older than me, who was very good at mathematics. He was instrumental, I guess, in interesting me in very elementary things. He was obviously more advanced than I was. We also played cards in our family. I liked the combinatorics in card play. At that time, I was 10 or 12 years old, I also liked playing chess. I never got any good at it but I liked it. When my brother went to the university, he learned about calculus and he tried to explain it to me. I found it totally incompre- hensible but it intrigued me, though. I did get books out of the library myself. But I didn’t make too much progress without him. Early group theory

You have received this year’s Abel Prize for your achievements in group theory. Can we start with a short historical introduction to the subject? We would like 84 Abel Prize 2008 to ask you to tell us how the notion of a group came up and how it was developed during the 19th century. In fact, Norwegian mathematicians played quite an important role in that game, didn’t they? TITS: Well, when you talk about groups it is natural to talk about Galois [1811– 1832]. I think Abel [1802–1829] did not use groups in his theory – do you know?

THOMPSON: At least implicitly. I think the equation of the fifth degree comes in there. It was a great eye opener. I myself looked at a very well-known paper of Lagrange [1756–1813], I think around 1770 – before the French revolution. He examined equations and he also said something about equations of degree five. He was definitely getting close to the notion of a group. I don’t know about the actual formal definition. I guess we have to attribute it to Galois. Anyway, it was certainly he that came up with the notion of a normal subgroup – I am pretty sure that was Galois’ idea. He came up with the idea of a normal subgroup which is really basic.

TITS: But the theorem on the equation of degree five was discovered first by Abel, I think. Of course Galois had a technique which helped with many equations of different types that Abel did not have. Galois was really basically an algebraist, whereas Abel was also an analyst. When we now talk about Abelian functions – these ideas go back to Abel.

Can you explain why simple groups are so important for the classification of finite groups in general? That realization came about, we guess, with Camille Jordan and his decomposition theorem. Is that correct? TITS: You see, I think that one of the dreams of these people was always to describe all groups. And if you want to describe all groups you decompose them. The factors are then simple. I think that was one of the aims of what they were doing. But of course they didn’t go that far. It is only very recently that one could find all finite simple groups, a solution to the problem to which John Thompson contributed in a major way.

What about Sylow [1832–1918] and Lie [1842–1899] in the beginning of group theory? THOMPSON: Those are two other Norwegians.

TITS: Lie played an important role in my career. In fact, practically from the beginning, the main subject of my work has centred around the so-called exceptional Lie groups. So the work of Lie is fundamental in what I have done.

Could you comment on the work of Frobenius [1849–1917] and Burnside [1852–1927]? THOMPSON: Of course. After the last half of the 19th century Frobenius among other things put the theory of group characters on a solid basis. He proved the orthogonality relations and talked about the transfer map. Burnside eventually got John Griggs Thompson and Jacques Tits 85 on the wagon there. And eventually he proved his p aq b-theorem, the two prime theorem, using character theory, namely that groups of these orders are solvable. That was a very nice step forward, I feel. It showed the power of character theory, which Frobenius had already done. Frobenius also studied the character theory of the symmetric groups and multiply transitive permutation groups. I don’t know how much he thought of the Mathieu groups. But they were pretty curious objects that had been discovered before character theory. For a while there was quite a bit of interest in multiply transitive permutation groups – quite complicated combinatorial arguments. Burnside and Frobenius were very much in the thick of things at that stage.

TITS: When I was a young mathematician, I was very ignorant of the literature. For instance, I rediscovered a lot of results that were known about multiply transitive groups, in particular on the strictly 4-fold and 5-fold transitive groups. Fortunately, I did this with other methods than the ones that were used before. So these results were in fact new in a certain sense.

Was it a disappointment to discover that these results had been discovered earlier? TITS: Not too much.

Burnside was also interesting because he posed problems and conjectures that you and others worked on later, right? THOMPSON: Right. Well, I sort of got started on working on the Frobenius conjecture, which was still open. I think it was Reinhold Baer [1902–1979] or maybe Marshall Hall [1910–1990] who told me about the Frobenius conjecture: the Frobe- nius kernel of the Frobenius group was conjectured to be nilpotent. I liked that conjecture for the following reason: if you take the group of proper motions of the Euclidean plane, it is a geometric fact that every proper motion is either a translation or a rotation. I hope kids are still learning that. It is a curious phenomenon. And the translations form a normal subgroup. So that is something you could actually trace back to antiquity. No doubt Frobenius knew that. So when he proved his theorem about the existence of the normal complement, that was a link back to very old things to be traced in geometry, I feel. That was one of the appeals. And then the attempt to use Sylow’s theorems and a bit of character theory, whatever really, to deal with that problem. That is how I first got really gripped by pure mathematics.

Mathieu [1835–1890] discovered the first sporadic simple groups, the Mathieu groups, in the 1860s and 1870s. Why do you think we had to wait 100 years before the next sporadic group was found by Janko, after your paper with Feit [1930–2004]? Why did it take so long a time? THOMPSON: Part of the answer would be the flow of history. The attention of the mathematical community was drawn in other directions. I wouldn’t say that group theory, certainly not finite group theory, was really at the centre of mathematical development in the 19th century. For one thing, Riemann [1826–1866] came along; 86 Abel Prize 2008 topology gained and exerted tremendous influence and, as Jacques has mentioned, analysis was very deep and attracted highly gifted mathematicians. It is true, as you mentioned earlier, that Frobenius was there and Burnside; so group theory wasn’t completely in the shadows. But there wasn’t a lot going on. Now, of course, there is a tremendous amount going on, both within pure and applied mathematics. There are many things that can attract people, really. So why there was this gap between these groups that Mathieu found and then the rather rapid development in the last half of the 20th century of the simple groups, including the sporadic groups – I have to leave that to the historians. But I don’t find it all that mysterious, really. You know, mathematics is a very big subject.

The Feit–Thompson theorem

The renowned Feit–Thompson theorem – finite groups of odd order are solvable – that you proved in the early 1960s, that was originally a conjecture by Burnside, right? THOMPSON: Burnside had something about it, yes. And he actually looked at some particular integers and proved that groups of that order were solvable. So he made a start.

When you and Feit started on this project, were there any particular results preceding your attack on the Burnside conjecture that made you optimistic about being able to prove it? THOMPSON: Sure. A wonderful result of Michio Suzuki [1926–1998], the so-called CA-theorem. Absolutely basic! Suzuki came to adulthood just at the end of the Sec- ond World War. He was raised in Japan. Fortunately, he came to the University of Illinois. I think it was in 1952 that he published this paper on the CA-groups of odd order and proved they were solvable by using exceptional character theory. It is not a very long paper. But it is incredibly ingenious, it seems to me. I still really like that paper. I asked him later how he came about it, and he said he thought about it for two years, working quite hard. He finally got it there. That was the opening wedge for Feit and me, really. The wedge got wider and wider.

TITS: Could you tell me what a CA-group is?

THOMPSON: A CA-group is a group in which the centralizer of every non-identity element is Abelian. So we can see Abel coming in again. Abelian centralizer – that is what the A means.

The proof that was eventually written down by Feit and you was 255 pages long, and it took one full issue of the Pacific Journal to publish. THOMPSON: It was long, yes. John Griggs Thompson and Jacques Tits 87

It is such a long proof and there were so many threads to connect. Were you nervous that there was a gap in this proof? THOMPSON: I guess so, right. It sort of unfolded in what seemed to us a fairly natural way: part group theory, part character theory and this funny little number-theoretic thing at the end. It all seemed to fit together. But we could have made a mistake, really. It has been looked at by a few people since then. I don’t lose sleep about it.

It seems that, in particular in finite group theory, there did not exist that many connections to other fields of mathematics like analysis, at least at the time. This required you to develop tools more or less from scratch, using ingenious arguments. Is that one of the reasons why the proofs are so long? THOMPSON: That might be. It could also be that proofs can become shorter. I don’t know whether that will be the case. I certainly can’t see that the existing proofs will become tremendously shorter in my lifetime. These are delicate things that need to be explored.

TITS: You see, there are results that are intrinsically difficult. I would say that this is the case for the Feit-Thompson result. I personally don’t believe that the proof will be reduced to scratch.

THOMPSON: I don’t know whether it will or not. I don’t think mathematics has reached the end of its tether, really.

TITS: It may of course happen that one can go around these very fine proofs, like John’s proof, using big machinery like functional analysis. That one suddenly gets a big machine which crushes the result. That is not completely impossible. But the question is whether it is worth the investment.

The theory of buildings

Professor Tits, you already mentioned Lie groups as a point of departure. Simple Lie groups had already been classified to a large extent at the end of the 19th century, first by Killing [1847–1923] and then by Élie Cartan [1869– 1951], giving rise to a series of matrix groups and the five exceptional simple Lie groups. For that purpose, the theory of Lie algebras had to be developed. When you started to work on linear algebraic groups, there were not many tools available. Chevalley had done some pioneering work but the picture first became clear when you put it in the framework of buildings, this time associat- ing geometric objects to groups. Could you explain to us the idea of buildings – consisting of apartments, chambers, all of these suggestive words – how it was conceived, what it achieved and why it has proven to be so fruitful? 88 Abel Prize 2008

TITS: First of all, I should say that the terminology like buildings, apartments and so on is not mine. I discovered these things but it was Bourbaki who gave them these names. They wrote about my work and found that my terminology was a shambles. They put it in some order and this is how the notions like apartments and so on arose. I studied these objects because I wanted to understand these exceptional Lie groups geometrically. In fact, I came to mathematics through projective geometry; what I knew about was projective geometry. In projective geometry you have points, lines and so on. When I started studying exceptional groups I sort of looked for objects of the same sort. For instance, I discovered – or somebody else discovered, actually – that the group E6 is the collineation group of the octonion projective plane. And a little bit later, I found some automatic way of proving such results, starting from the group to reconstruct the projective plane. I could use this procedure to give geometric interpretations of the other exceptional groups, e.g. E6 , E7 , E8. That was really my starting point. Then I tried to make an abstract construction of these geometries. In this construction I used terms like skeletons, for instance, and what became apartments were called skeletons at the time. In fact, in one of the volumes of Bourbaki, many of the exercises are based on my early work.

An additional question about buildings. This concept has been so fruitful and made connections to many areas of mathematics that maybe you didn’t think of at the time, like rigidity theory for instance? TITS: For me it was really the geometric interpretations of these mysterious groups, the exceptional groups, that triggered everything. Other people have then used these buildings for their own work. For instance, some analysts have used them. But in the beginning I didn’t know about these applications.

You asked some minutes ago about CA-groups. Maybe we can ask you about BN-pairs: what are they and how do they come in when you construct buildings? TITS: Again, you see, I had an axiomatic approach towards these groups. The BN-pairs were an axiomatic way to prove some general theorems about simple algebraic groups. A BN-pair is a pair of two groups B and N with some simple properties. I noticed that these properties were sufficient to prove, I wouldn’t say deep but far-reaching results – for instance, proving the simplicity property. If you have a group with a BN-pair you have simple subgroups free of charge. The notion of BN-pairs arises naturally in the study of split simple Lie groups. Such groups have a distinguished conjugacy class of subgroups, namely the Borel subgroups. These are the Bs of a distinguished class of BN-pairs.

The classification of finite simple groups

We want to ask you, Professor Thompson, about the classification project – the attempt to classify all finite simple groups. Again, the paper by Feit and you in John Griggs Thompson and Jacques Tits 89

1962 developed some techniques. Is it fair to say that without that paper the project would not have been doable or even realistic? THOMPSON: That I can’t say.

TITS: I would say yes.

THOMPSON: Maybe, but the history has bifurcations so we don’t know what could have happened.

The classification theorem for finite simple groups was probably the most mon- umental collaborative effort done in mathematics, and it was pursued over a long period of time. Many people have been involved; the final proof had 10,000 pages, at least originally. A group of people, originally led by Goren- stein [1923–1992], are still working on making the proof more accessible. We had an interview here five years ago with the first Abel Prize recipient Jean-Pierre Serre. At that time, he told us that there had been a gap in the proof, which was only about to be filled in at the time of the interview with him. Before, it would have been premature to say that one actually had the proof. The quasi-thin case was left. How is the situation today? Can we really trust that this theorem finally has been proved? THOMPSON: At least that quasi-thin paper has been published now. It is quite a massive work itself (by Michael Aschbacher and Stephen Smith) and quite long, well over 1000 pages. Several of the sporadic simple groups come up. They characterize them because they are needed in quasi-thin groups. I forget which ones come up but the Rudvalis group certainly is among them. It is excruciatingly detailed. It seems to me that they did an honest piece of work. Whether one can really believe these things is hard to say. It is such a long proof that there might be some basic mistakes. But I sort of see the sweep of it, really. It makes sense to me now. In some way it rounded itself off. I can sort of see why there are probably no more sporadic simple groups, but not really conceptually. There is no conceptual reason that is really satisfactory. But that’s the way the world seems to be put together. So we carry on. I hope people will look at these papers and see what the arguments are and see how they fit together. Gradually this massive piece of work will take its place in the accepted canon of mathematical theorems.

TITS: There are two types of group theorists. Those who are like St. Thomas, they don’t believe because they have not seen every detail of the proof. I am not like them and I believe in the final result although I don’t know anything about it. The people who work on or who have worked on the classification theorem may of course have forgotten some little detail somewhere. But I don’t believe these details will be very important. And I am pretty sure that the final result is correct. 90 Abel Prize 2008

May we ask about the groups that are associated with your names? You have a group that’s called the Thompson group among the sporadic simple groups. How did it pop up? How were you involved in finding it? THOMPSON: That is, in fact, a spin-off from the Monster Group. The so-called Thompson group is essentially the centralizer of an element of order three in the Monster. Conway and Norton and several others were beavering away – this was before Griess constructed the Monster – working on the internal structure where this group came up, along with the Harada–Norton group and the Baby Monster. We were all working trying to get the characters. The Monster itself was too big. I don’t think it can be done by hand. Livingstone got the character table, the ordinary complex irreducible characters of the Monster. But I think he made very heavy use of a computing machine. And I don’t think that has been eliminated. That’s how the figure 196883 came about, the degree of the smallest faithful complex representation of the Monster Group. It is just too big to be done by hand. But we can do these smaller subgroups.

The Tits group was found by hand, wasn’t it? And what is it all about? TITS: Yes, it was really sort of a triviality. One expects that there would be a group there except that one must take a subgroup of index two so that it becomes simple. And that is what I know about this.

Professor Tits, there is a startling connection between the Monster Group, the biggest of these sporadic groups, and elliptic function theory or elliptic curves via the j-function. Are there some connections with other exceptional groups, for instance in geometry? TITS: I am not a specialist regarding these connections between the Monster Group, for instance, and modular functions. I don’t really know about these things, I am ashamed to say. I think it is not only the Monster that is related to modular forms but also several other sporadic groups. But the case of the Monster is especially satisfactory because the relations are very simple in that case. Somehow, smaller groups give more complicated results. In the case of the Monster, things sort of round up perfectly.

The inverse Galois problem

May we ask you, Professor Thompson, about your work on the inverse Galois problem? Can you explain first of all what the problem is all about? And what is the status right now? THOMPSON: The inverse Galois problem probably goes back already to Galois. He associated a group to an equation, particularly to equations in one variable with integer coefficients. He then associated to this equation a well-defined group now called the Galois group, which is a finite group. It captures quite a bit of the nature of the roots, the zeros, of this equation. Once one has the notion of a field, the field John Griggs Thompson and Jacques Tits 91 generated by the roots of an equation has certain automorphisms and these automorphisms give us Galois groups. The inverse problem is: start with a given finite group. Is there always an equation, a polynomial with one indeterminate with integer coefficients, whose Galois group is that particular group? As far as I know it is completely open whether or not this is true. And as a test case if you start with a given finite simple group, does it occur in this way? Is there an equation waiting for it? If there is one equation there would be infinitely many of them. So we wouldn’t know how to choose a standard canonical equation associated to this group. Even in the case of simple groups, the inverse problem of Galois Theory is not solved. For most general finite groups, I leave it to the algebraic geometers or whoever else has good ideas whether this problem is amenable. A lot of us have worked on it and played around with it, but I think we have just been nibbling at the surface. For example the Monster is a Galois group over the rationals. You can’t say that about all sporadic groups. The reason that the Monster is a Galois group over the rationals comes from character theory. It is just given to you.

TITS: This is very surprising; you have this big object and the experts can tell you that it is a Galois group. In fact, I would like to see an equation.

Is there anything known about an equation? THOMPSON: It would have to be of degree of at least 1020. I found it impressive, when looking a little bit at the j-function literature before the days of computers, that people like Fricke [1861–1930] and others could do these calculations. If you look at the coefficients of the j-functions, they grow very rapidly into the tens and hundreds of millions. They had been computed in Fricke’s book. It is really pleasant to see these numbers out there before computers were around – numbers of size 123 million. And the numbers had to be done by hand, really. And they got it right. TITS: It is really fantastic what they have done.

Could there be results in these old papers by Fricke and others that people are not aware of? THOMPSON: No. People have gone through them; they have combed through them.

TITS: Specialists do study these papers.

The E8-story

There is another collaborative effort that has been done recently, the so-called E8-story: a group of mathematicians has worked out the representations of E8. In fact, they calculated the complete character table for E8. The result was pub- licized last year in several American newspapers under the heading “A calculation the size of Manhattan” or something like that. 92 Abel Prize 2008

THOMPSON: It was a little bit garbled maybe. I did see the article.

Can you explain why we should all be interested in such a result, be it as a group theorist or as a general mathematician or even as a man on the street? THOMPSON: It is interesting in many ways. It may be that physicists have something to do with the newspapers. Physicists – they are absolutely fearless as a group. Any mathematical thing they can make use of they will gobble right up and put in a context that they can make use of, which is good. In that sense mathematics is a handmaiden for other things. And the physicists have definitely gotten interested in exceptional Lie groups. And E8 is out there, really. It is one of the great things.

Is there any reason to believe that some of these exceptional groups or sporadic groups tell us something very important – in mathematics or in nature? THOMPSON: I am not a physicist. But I know physicists are thinking about such things, really.

TITS: It is perhaps naive to say this but I feel that mathematical structures that are so beautiful like the Monster must have something to do with nature.

Mathematical work

Are there any particular results that you are most proud of? THOMPSON: Well, of course one of the high points of my mathematical life was the long working relationship I had with Walter Feit. We enjoyed being together and enjoyed the work that we did, and, of course, the fusion of ideas. I feel lucky to have had that contact and proud that I was in the game there.

TITS: I had a very fruitful contact for much of my career with François Bruhat [1929–2007] and it was very pleasant to work together. It was really working together like you did it, I suppose, with Walter Feit.

Was not Armand Borel [1923–2003] also very important for your work? TITS: Yes, I also had much collaboration with Borel. But in a sense that was different. But that was different in the following sense: When I worked with Borel, I had, very often, the impression that we both had found the same thing. We just put the results together in order not to duplicate. We wrote our papers practically on results that we had both found separately. Whereas with Bruhat, it was really joint work, complementary work.

Have either of you had the lightning flash experience described by Poincaré [1854–1912] – seeing all of a sudden the solution to a problem you have struggled with for a long time? John Griggs Thompson and Jacques Tits 93

TITS: I think this happens pretty often in mathematical research, that one suddenly finds that something is working. But I cannot recall a specific instance. I know that it has happened to me and it has happened to John, certainly. So certainly some of the ideas one has worked out, but it sort of disappears in a fog.

THOMPSON: I think my wife will vouch for the fact that when I wake in the morning I am ready to get out there and get moving right away. So my own naïve thinking is that while I am asleep there are still things going on. And you wake up and say: “Let’s get out there and do it”. And that is a wonderful feeling.

You have both worked as professors of mathematics in several countries. Could you comment on the different working environments in these places and people you worked with and had the best cooperation with? TITS: I think the country which has the best way of working with young people is Russia. Of course, the French have a long tradition and they have very good, very young people but I think Russian mathematics is in a sense more lively than French mathematics. French mathematics is too immediately precise. I would say that these are the two countries where the future of mathematics is the clearest. But of course Germany has had such a history of mathematics that they will continue. And nowadays, the United States have in a sense become the centre of mathematics because they have so much money – that they can…

… buy the best researchers? TITS: That’s too negative a way of putting it. Certainly many young people go the United States because they cannot earn enough money in their own country. And of course the catastrophe that happened in Europe in the 1930s with Nazism. A lot of people went to the United States.

What about you, Professor Thompson? You were in England for a long time. How was that experience compared to work at an American university? THOMPSON: Well, I am more or less used to holding my own role. People didn’t harass me very much any place. I have very nice memories of all the places I have visited, mainly in the United States. But I have visited several other countries, too, for shorter periods, including Russia, Germany and France. Mathematically, I feel pretty much comfortable everywhere I am. I just carry on. I have not really been involved in higher educational decision making. So in that sense I am not really qualified to judge what is going on at an international basis.

Thoughts on the development of mathematics

You have lived in a period with a rapid development of mathematics (in particular in your own areas) including your own contributions. Some time ago Lennart Carleson , who received the Abel Prize two years ago, said in an inter- 94 Abel Prize 2008 view that the 20th century had possibly been the Golden Age of Mathematics and that it would be difficult to imagine a development as rapid as we have witnessed it. What do you think? Have we already had the Golden Age of Mathematics or will development continue even faster? TITS: I think it will continue at its natural speed, which is fast – faster than it used to be.

THOMPSON: I remember reading a quote attributed to Laplace. He said that mathematics might become so deep, that we have to dig down so deep, that we will not be able to get down there in the future. That’s a rather scary image, really. It is true that prerequisites are substantial but people are ingenious. Pedagogical techniques might change. Foundations of what people learn might alter. But mathematics is a dynamic thing. I hope it doesn’t stop.

TITS: I am confident that it continues to grow.

Traditionally, mathematics has been mainly linked to physics. Lots of motivations come from there and many of the applications are towards physics. In recent years, biology, for example with the Human Genome Project, economics with its financial mathematics, computer science and computing have been around as well. How do you judge these new relations? Will they become as important as physics for mathematicians in the future? TITS: I would say that mathematics coming from physics is of high quality. Some of the best results we have in mathematics have been discovered by physicists. I am less sure about sociology and human science. I think biology is a very important subject but I don’t know whether it has suggested very deep problems in mathematics. But perhaps I am wrong. For instance, I know of Gromov, who is a first class mathematician and who is interested in biology now. I think that this is a case where mathematics, really highbrow mathematics, goes along with biology. What I said before about sociology, for instance, is not true for biology. Some biologists are also very good mathematicians.

THOMPSON: I accept that there are very clever people across the intellectual world. If they need mathematics they come up with mathematics. Either they tell mathematicians about it or they cook it up themselves.

Thoughts on the teaching of mathematics

How should mathematics be taught to young people? How would you encourage young people to get interested in mathematics? THOMPSON: I always give a plug for Gamow’s book One, Two, Three … Infinity and Courant and Robbins’ What is Mathematics? and some of the expository work John Griggs Thompson and Jacques Tits 95 that you can get from the libraries. It is a wonderful thing to stimulate curiosity. If we had recipes, they would be out there by now. Some children are excited and others are just not responsive, really. You have the same phenomenon in music. Some children are very responsive to music; others just From left to right: Jacques Tits, John G. Thompson, Christian don’t respond. We don’t Skau and Martin Raussen. (Photo: Kyrre Lien) know why.

TITS: I don’t know what to say. I have had little contact with very young people. I have had very good students but always advanced students. I am sure it must be fascinating to see how young people think about these things. But I have not had the experience.

Jean-Pierre Serre once said in an interview that one should not encourage young people to do mathematics. Instead, one should discourage them. But the ones that, after this discouragement, are still eager to do mathematics, you should really take care of them. THOMPSON: That’s a bit punitive. But I can see the point. You try to hold them back and if they strain at the leash then eventually you let them go. There is something to it. But I don’t think Serre would actually lock up his library and not let the kids look at it.

Maybe he wants to stress that research mathematics is not for everyone. THOMPSON: Could be, yeah.

TITS: But I would say that, though mathematics is for everyone, not everyone can do it with success. Certainly it is not good to encourage young people who have no gift to try to do something because that will result in sort of a disaster.

Personal interests

In our final question we would like to ask you both about your private interests besides mathematics. What are you doing in you spare time? What else are you interested in? TITS: I am especially interested in music and, actually, also history. My wife is an historian; therefore I am always very interested in history. 96 Abel Prize 2008

What type of music? Which composers? TITS: Oh, rather ancient composers.

And in history, is that old or modern history? TITS: Certainly not contemporary history but modern and medieval history. All related to my wife’s speciality.

THOMPSON: I would mention some of the same interests. I like music. I still play the piano a bit. I like to read. I like biographies and history – general reading, both contemporary and older authors. My wife is a scholar. I am interested in her scholarly achievements. Nineteenth century Russian literature; this was a time of tremendous achievements. Very interesting things! I also follow the growth of my grandchildren.

TITS: I should also say that I am very interested in languages, Russian for instance.

Do you speak Russian? TITS: I don’t speak Russian but I have been able to read some Tolstoy in Russian. I have forgotten a little. I have read quite a lot. I have learned some Chinese. In the course of years I used to spend one hour every Sunday morning studying Chinese. But I started a little bit too old so I forgot what I learned.

Are there any particular authors that you like? TITS: I would say all good authors.

THOMPSON: I guess we are both readers. Endless.

Let us finally thank you very much for this pleasant interview on behalf of the Norwegian, the Danish and the European Mathematical Societies. Thank you very much. THOMPSON: Thank you.

TITS: Thank you for the interview. You gave us many interesting topics to talk about! Abel Prize 2009: Mikhail Gromov

The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2009 to

Mikhail Leonidovich Gromov Permanent Professor, Institut des Hautes Études Scientifiques, France,

“for his revolutionary contributions to geometry.”

Geometry is one of the oldest fields of mathematics; it has engaged the attention of great mathematicians through the centuries, but has undergone revolutionary change during the last 50 years. Mikhail Gromov has led some of the most important developments, producing profoundly original general ideas, which have resulted in new perspectives on geometry and other areas of mathematics. Riemannian geometry developed from the study of curved surfaces and their higher dimensional analogues and has found applications, for instance, in the theory of general relativity. Gromov played a decisive role in the creation of modern global Riemannian geometry. His solutions of important problems in global geometry relied on new general Abel Laureate Mikhail Gromov. concepts, such as the convergence of Rieman- (Photo: Knut Falch) nian manifolds and a compactness principle, which now bear his name. Gromov is one of the founders of the field of global symplectic geometry. Hol- omorphic curves were known to be an important tool in the geometry of complex manifolds. However, the environment of integrable complex structures was too rigid. In a famous paper in 1985, he extended the concept of holomorphic curves to J-holomorphic curves on symplectic manifolds. This led to the theory of Gromov–Witten

* 23.12.1943

97 98 Abel Prize 2009 invariants, which is now an extremely active subject linked to modern quantum field theory. It also led to the creation of symplectic topology and gradually penetrated and transformed many other areas of mathematics. Gromov’s work on groups of polynomial growth introduced ideas that forever changed the way in which a discrete infinite group is viewed. Gromov discovered the geometry of discrete groups and solved several outstanding problems. His geometrical approach rendered complicated combinatorial arguments much more natural and powerful. Mikhail Gromov is always in pursuit of new questions and is constantly thinking of new ideas for solutions of old problems. He has produced deep and original work throughout his career and remains remarkably creative. The work of Gromov will continue to be a source of inspiration for many future mathematical discoveries.

Mikhail Gromov is received in audience by King Harald and Queen Sonja. (Photo: Erlend Aas) Mikhail Gromov 99

A Russian education

First of all, we would like to congratulate you warmly for having been selected as the 2009 Abel Prize winner. We would like to start with some questions about your early years and your early career. You were born towards the end of World War II in a small town called Boksitogorsk, 245 km east of St. Petersburg (at that time Leningrad). My mother was a medical doctor in the fighting army – and to give birth at that time, she had to move a little away from the frontline.

Could you tell us about your background, your early education and who or what made you interested in mathematics? My first encounter with mathematics besides school was a book my mother bought me called “Numbers and Figures” by Rademacher [1892–1969] and Toeplitz [1881– 1940], which had a big influence on me. I could not understand most of what I was reading but I was excited all the same. I still retain that excitement by all the mysteries that you cannot understand but which make you curious.

Did you know you would go into mathematics while at high school? In my middle and later years at high school I was more interested in chemistry than in mathematics. But then I was hooked. There were some very good books in Russia on mathematical problems for youngsters. I was going through them and I immersed myself in all this for a year. In my last year of high school I was attending a so-called mathematics circle, something for youngsters at the university, run by two people, Vasia Malozemov and Serezha Maslov (Maslov became a logician; coinci- dentally, he was the one who suggested Hilbert’s tenth problem to Matiasevic). They were running an extremely good group for young children that I attended. This was in St. Petersburg in 1959, the year before I started at university, and it was the major reason for my decision to study mathematics.

You started studying mathematics at Leningrad University. Please tell us about the environment there, how you were brought up mathematically and about the teachers that were important for you. I think it was a pleasant environment despite the political surroundings, which were rather unpleasant. There was an extremely high spirit in the mathematical community and among professors. I remember my first teachers, including Professor Isidor Pavlovich Natanson [1906–1964], and also I attended a class run by Boris Mikhailovich Makarov. You could see the high intensity of these people and their devotion to science. That had a very strong impact on me, as well as the interactions with the senior students. Let me mention one, the young algebraist Tolia Yakov- lev who projected this image of absolute dedication to mathematics. On the other hand, there was a general trend in Leningrad of relating mathematics to science. This was influenced, I think, by Kolmogorov [1903–1987] and Gelfand [1903–2009] from Moscow. Kolmogorov made fundamental contributions to hydrodynamics 100 Abel Prize 2009 and Gelfand was working in biology and also in physics. Basically, there was an idea of the universality of knowledge, with mathematics being kind of the focus of intellectual ideas and developments. And that, of course, shaped everybody who was there, myself included. And I learned very much of the Moscow style of mathematics from Dima Kazhdan, with whom we were meeting from time to time.

Can you remember when and how you became aware of your exceptional mathematical talent? I do not think I am exceptional. Accidentally, things happened and I have qualities that you can appreciate. I guess I never thought in those terms.

At least towards the end of your studies, your academic teacher was Vladimir Rokhlin [1919–1984]. Do you still sense his influence in the way you do mathematics today? You see, Rokhlin himself was educated in Moscow and the Moscow mathematical way of thinking was very different from that in Leningrad. They had a different kind of school that was much more oriented towards Western mathematics. Leningrad was more closed and focused on classical problems; Moscow was more open to new developments. And that is what he brought to Leningrad. Another person with the same attitude was Boris Venkov, an algebraic geometer. From him and from Rokhlin, I got a much broader view and perception of mathematics than what I could have got from the traditional school in Leningrad. On the other hand, the traditional school was also very strong; for instance, the geometry school of Aleksandr Danilovich Alexandrov [1912–1999]. There were people like Zalgaller and Burago from whom I learned most of my geometry. Burago was my first teacher in geometry.

You were very successful at Leningrad University at the beginning of the 1970s. Still, you left Leningrad and the Soviet Union shortly after in 1974. What was the background for your desire to leave? This is very simple. I always say, if someone tells you you should not do something, you try to do exactly that. You know what happened when God prohibited Eve eating the apple. This is human nature. It was said that you cannot leave the country; it is just impossible, it is wrong, it is horrible. It is like in scientific work: if it is impossible, you try to do it anyway.

It was probably not that easy to get out of the Soviet Union at that time? For me it was relatively easy. I was very lucky. But in general it was difficult and risky. I had to apply, I waited for several months and then I got permission.

Russian mathematics

Jacques Tits, one of the Abel prize winners last year, praised Russian mathematical education and Russian schools for the strong personalities and the Mikhail Gromov 101 strong ties between motivations, applications and the mathematical appara- tus, as well as the lively seminars and discussions sometimes lasting for many hours. What is your perception: what is special about the Russian mathematical style and school? Like I said, it was somewhat different in Leningrad compared to Moscow. What Tits was probably referring to was Gelfand’s seminars in Moscow. I attended this seminar in Moscow only once when I was invited to give a talk, so my recollection might not be typical. But when I came, it took about two hours before the seminar could start because Gelfand was discussing various matters with the audience. Another seminar was run by Piatetsky-Shapiro [1929–2009] and that was very rigorous. When something was presented on the blackboard and the audience asked questions, then Shapiro would express his attitude, which was very strong and a bit aggressive: on what students should know and should not know, the idea that they should learn this and this and that… Extremely powerful indications of his personality!

Do you still feel that there is a specific Russian mathematical background that you build your work upon? Yes, definitely. There was a very strong romantic attitude towards science and mathematics: the idea that the subject is remarkable and that it is worth dedicating your life to. I do not know whether that is also true in other countries because I was not elsewhere at that time of my education. But that is an attitude that I and many other mathematicians coming from Russia have inherited.

Is there still a big difference between Russian mathematics and, say, Western mathematics in our days? Or is this difference about to disappear, due to the fact that so many Russians are working in the West? This I cannot tell given there are so many Russians working in the West. I do not know much about mathematical life in Russia nowadays; certainly, things have changed tremendously. In my time in Russia, this intensity was partly a reaction to the outside world. Academic life was a peaceful garden of beauty where you could leave a rather ugly political world outside. When this all changed, this sharp concentration went down. It might be so. I don’t know. This is only a conjecture.

Do you still have a lot of contact with Russian mathematicians? Do you go there once in a while? I have been there twice since I left the country. You still feel the intensity of life there but things go down, partially because so many gifted people are leaving. They are drawn to larger centres where they can learn more.

Can you tell us about other Russian mathematicians that have influenced you, like Linnik? Yes. Yuri Linnik [1915–1972] was a great scientist, professor and academician in Leningrad. He was running educational seminars in algebraic geometry one year. 102 Abel Prize 2009

A remarkable thing was that he always admitted his complete ignorance. He never pretended to know more than he did, rather the contrary. And secondly, there was always a complete equality between him and his students. I remember one time I was supposed to give a talk there but I overslept and arrived one hour late. But he was just laughing at it – not annoyed at all. And that, I think, exhibits some of his spirit in mathematics – the atmosphere of how we were all in the same boat, regardless of who you were.

How would you compare him with Rokhlin as a person? Rokhlin was a more closed person as he had gone through a very complicated life. He was a prisoner in the Second World War. He was Jewish but he somehow managed to conceal it. He had an extremely strong personality. After he was liberated, he was sent to a prison in Russia, a labour camp, because it was considered that he hadn’t finished his military service. Being a prisoner of war didn’t count as military service! After some work he came to Moscow. It was difficult to say what he thought. He was very closed and tried to keep high standards on everything but he was not so relaxed and open as Linnik was. It was at first unclear what it was but then you realised that he was shaped by those horrible experiences.

Was Linnik also Jewish? I think Linnik was half Jewish but he did not participate in the war. He had a different kind of life. He was better positioned in his career as a member of the Academy and so on. Rokhlin was always discriminated against by the authorities, for reasons I don’t know. I heard some rumours that he was getting into conflict with some officials in Moscow. For some time he was a secretary for Pontryagin [1908–1988] because Pontry- agin was blind and, as an academician, needed a secretary. Rokhlin had this position until he had defended his second thesis. Then he was kicked out of Moscow because he was over-qualified. A. D. Alexandrov, then the rector at Leningrad University, made a great effort to bring him to Leningrad in 1960. That had a very strong influence on the development of mathematics in Leningrad. The whole school of topology grew out of his ideas. Rokhlin was a very good teacher and organiser.

Is it true that Pontryagin was anti-Semitic? I believe he became anti-Semitic after his second marriage He was blind and it is unclear how independent his perception of the world was. In his later years he became anti-Semitic and he also wrote pamphlets that sounded absolutely silly. It is unclear what or who influenced him to get those ideas.

History of Geometry

You are the first Abel laureate to receive the prize explicitly for your “revolutionary contribution to geometry”. From Euclid`s time geometry was, so to Mikhail Gromov 103 say, the “face” of mathematics and a paradigm of how to write and to teach mathematics. Since the work of Gauss [1777–1855], Bolyai [1802–1860] and Lobachevsky [1792–1856] from the beginning of the nineteenth century, geometry has expanded enormously. Can you give us your thoughts on some of the highlights since then within geometry? I can only give a partial answer and my personal point of view. It is very difficult to find out about how people thought about the subject in ancient times. Seen from today, geometry as a mathematical subject was triggered from observations you make in the world; Euclid gave a certain shape of how to organise observations and made an axiomatic approach to mathematics and what followed from those. It happened that it worked very badly beyond the point that it was designed for. In particular, there was a problem with the parallel postulate and people tried to prove it. There was a mixture: on one hand they believed that the way you see the world was the only way for you to see it and they tried to justify that axiomatically. But it did not work. Eventually, mathematicians realised that they had to break out of the naïve way of thinking about axioms. The axioms happened to be very useful but only useful in a limited way. Eventually, you had to deny them. This is how they served. From this point on, mathematics started to move in different directions. In particular, Abel was one of the people who turned mathematics from just observing and formalizing what you see to formalizing what you cannot see directly – what you can only see in a very opaque way. Modern mathematics was shaped in the beginning of the nineteenth century. Then it became more and more structural. Mathematics not only deals with what you see with your eye but what you see in the structure of things. At a more fundamental level, I would say. If you formulate the problem in modern language, the mathematicians at the time faced trying to understand the limitations of Euclidean geometry; it is completely obvious. But it took centuries to develop this language. This work was started by Lobachevsky, Bolyai and Gauss, and in a different domain by Abel [1802–1829] and Galois [1811–1832].

The laureate’s research in geometry

It is said that you revolutionized Riemannian geometry in the late 1970s. Could you explain to us what your novel and original idea consisted of, the idea that turned out to be so groundbreaking? I cannot explain that since I never thought of them as groundbreaking or original. This happens to any mathematician. When you do something new, you don’t realise it is something new. You believe everybody knows it, that it is kind of immediate and that other people just have not expressed it. This happens in fact with many mathematical proofs; the ideas are almost never spoken out. Some believe they are obvious and others are not aware of them. People come from different backgrounds and perceive different things… 104 Abel Prize 2009

A hallmark of your work has been described as the softening of geometry, whereby equations are replaced by inequalities or approximate or asymptotic equations. Examples include the “coarse viewpoint” on Riemannian geometry, which considers all Riemannian structures at once. This is very original. No- body had thought about that before. Is not that true? That is probably true. But again, I am not certain whether somebody else had had this idea before. For me it was clear from the very beginning and I actually never articulated it for a long time believing everybody knew it. I believe that some people knew about it but they never had an occasion to say it aloud. In the end, I formulated it because I gave a course in France.

First of all, you had this new perspective. The basic ideas are perhaps very simple but you were the first to get any deep results in that direction. Well, there were predecessors. This trend in Riemannian geometry started with the work of Jeff Cheeger. Earlier, up to some point, people were thinking about manifolds in very abstract terms. There were many indices and you could not take the subject into your hand. I think that one of the first works in which Rieman- nian geometry was turned into something simple was by John Nash [1928–2015]. Actually, he had a tremendous influence on me. He was just taking manifolds in his hands and putting them in space, just playing with them. From this I first learned about this very concrete geometry. Simple things, but you had to project it to very high dimensions. And then there was the work by Jeff Cheeger, formally a very different subject but with the same attitude, realising that things got quite simple when formalized, if that was done properly. So I was just following in the steps of these people.

This means that you read Nash’s work and were impressed by it very early? Yes, I read it very carefully. And I still believe I am the only person who read his papers from the beginning to the end. By judging what people have written about it afterwards, I do not think they have read it.

Why not? At first, I looked at one of Nash’s papers and thought it was just nonsense. But professor Rokhlin said: “No, no. You must read it”. I still thought it was nonsense; it could not be true. But then I read it and it was incredible. It could not be true but it was true. There were three papers; the two more difficult ones, on embeddings, they looked nonsensical. Then you look at the way it is done and you also think that it looks nonsensical. After understanding the idea you try to do it better; many people tried to do it in a better way. But when you look at how they were doing it, and also what I tried, and then come back to Nash, you have to admit that he had done it in a better way. He had a tremendous analytic power combined with geometric intuition. This was a fantastic discovery for me: how the world may be different from what you think! Mikhail Gromov 105

John Nash received the Nobel Prize in economics and he was also the person behind the “Beautiful Mind” movie. Many people think he should have gotten the Fields Medal for his efforts. Do you subscribe to this idea? Yes. When you think about this guy and his achievements in science, forgetting about medals, the discoveries he made were fantastic. He was a person thinking in a most unusual way. At least, his work in geometry was contrary to what everybody would expect, concerning the results, the techniques and the ideas he used. He did various matters in an extremely simple way, so that everybody could see it but nobody would believe it could work. He also had a tremendous power of imple- menting it, with a dramatic analytic power. What he has done in geometry is, from my point of view, incomparably greater than what he has done in economics, by many orders of magnitude. It was an incredible change in attitude of how you think about manifolds. You can take them in your bare hands and what you do may be much more powerful than what you can do by traditional means.

So you admit that he had an important influence upon you and your work. Yes, absolutely. All over, his work and the work of Smale, which was explained to me by Sergei Novikov at a summer school in the early 60s, have had the most important influence on me.

You introduced the h-principle, where “h” stands for homotopy, in order to study a class of partial differential equations that arises in differential geometry rather than in physical science; it has proved to be a very powerful tool. Could you explain the h-principle and your ideas behind introducing the concept? This was exactly motivated by the work of Smale and Nash. And I realised then that they dealt more or less with the same topic – which had not been clear at all. In particular, if you use Nash’s techniques you immediately get all the results of immersion theory. You do not have to go deep. The first lemma in Nash proves all immersion theorems in topology! I was thinking about this for several years, trying to understand the mechanism behind it. I realised there was a simple general mechanism, which was rather formal but incorporated the ideas of Nash and Smale by combining them. This applies to a wide class of equations because you interpolate between rather remote topics and then you cover a very large ground.

You proved a celebrated theorem, precursors of which were theorems by Mil- nor-Wolf and Tits. It tells us that if a finitely generated group has polynomial growth then it contains a nilpotent subgroup of finite index. A particularly remarkable aspect of your proof is that you actually use Hilbert’s Fifth Problem, which was proved by Gleason [1921–2008], Montgomery [1909–1992] and Zippin [1905–1995]. And this is the first time, apparently, that this result has been used in a significant way. Can you explain and expand on this? I thought previously about applying this theorem in Riemannian geometry, though in a different context, inspired by Margulis’ 1967 paper on 3-dimensional Anosov flows and by his 1970 rendition of Mostow’s rigidity theorem, where Margulis intro- 106 Abel Prize 2009 duced and exploited quasi-isometries. I wanted to prove something that happened to be wrong. I tried to apply a version of the Shub-Franks construction in topological dynamics. It didn’t work either. Also, there was a paper by Hirsch concerning exactly this question about polynomial growth – a special case of this problem – where he tried to apply the classification of topological groups; and again it didn’t work. So I believed it couldn’t be applied. It was kind of clear to us that it was close but it didn’t seem to work. But when I was formalizing the idea of limits of manifolds, I tried to think in those terms and then I saw that it might work. This was kind of a surprise to me.

It must have been a very nice experience when you realised that this would work out? Well, it was not really a sudden insight. I realised what was needed was just a slight change in conceptions. Then it is not difficult to do it. The proof is extremely simple in a way. You take an obvious concept of a limit and then, by the power of analysis, you can go to the limit many times, which creates structures that you have not seen before. You think you have not done anything but, amazingly, you have achieved something. That was a surprise to me.

You introduced the idea of looking at a group from infinity, which is an apt description of looking at the limit of a sequence of metric spaces associated to the group in the so-called Gromov-Hausdorff metric. You have used this technique with impressive effect. Please give us some comments. After proving the theorem about polynomial growth using the limit and looking from infinity, there was a paper by Van den Dries and Wilkie giving a much better presentation of this using ultrafilters. Then I took it up again and I realised it applied to a much wider class of situations where the limits do not exist but you still have the ultralimits, and it gives you a very good view on many mathematical objects including groups. But it is still not tremendously powerful. In the context of groups, I was influenced by a survey of the small cancellation theory by Paul Schupp in the book Word Problems (1973) where he said – and I think this was a very honest and very useful remark – that “people don’t understand what small cancellation groups are”. And I felt very comfortable because I didn’t understand it either. I started thinking about what they could be and then I came up with this concept of hyperbolicity. This was rather pleasing to me but there were some technical points I could not handle for some time, such as the rough version of the Cartan-Hadamard theorem, before I could write an article about it.

When did you introduce the concept of a hyperbolic group? My first input on the geometry of groups came from Dima Kazhdan who explained to me in the middle of the 60s the topological proof of the Kurosh subgroup theorem. Later on I read, in the same 1971 issue of Inventiones, the paper by Griffiths on complex hyperbolicity and the paper by Klingenberg [1924–2010] on manifolds of hyperbolic type. The latter contained the idea of rough hyperbolicity, albeit the Mikhail Gromov 107 main theorem in this paper was incorrect. And, as I said, I had read the paper by Schupp. I presented the first definition of hyperbolicity during the ’78 meeting at Stony Brook under the name of Is(2)-groups as they satisfy the linear isoperimetric inequality in dimension two. The article appeared three years later. Also, I recall, I spoke about it at the Arbeitstagung in 1977. I tried for about ten years to prove that every hyperbolic group is realisable by a space of negative curvature, which I couldn’t do, and this is still unknown. Then Steve Gersten convinced me to write what I already knew about it and I wrote that but I was very dissatisfied because I couldn’t decide if you needed the theory of such groups. If they were “geometric”, the way I said, we would not need hyperbolicity theory and we would have much better theorems.

You said that almost all groups are hyperbolic? Right. That was actually the point. When I realised that we could see hyperbolicity in certain generic constructions better without an appeal to curvature then I accepted it as a worthwhile notion. In my first article I suggested a rather technical definition and terminology. I believed it was a preliminary concept. But then I realised eventually that it probably was the right concept, regardless of whether the geometrization theorem I was trying to prove was true or not. Also, I was encouraged by talking to Ilia Rips in the early 80s who, by that time, had developed hyperbolic group theory in a combinatorial framework, well beyond what I knew at the time, by the ongoing development of Thurston’s 3D-theory and by Cannon’s solution of Thurston’s rationality conjecture.

We move to a different area, symplectic geometry, that you have also made a revolutionary contribution to. You introduced methods from complex analysis, notably pseudo-holomorphic curves. Could you expand on this and explain how you got the idea for this novel approach? And also on the Gromov-Witten invariant, which is relevant for string theory and which came up in this connection. Yes, I remember very vividly this amazing discovery I made there. I was reading a book by Pogorelov [1909–2002] about rigidity of convex surfaces. He was using the so-called quasi-analytic functions developed by Bers [1914–1993] and Vekua [1907–1977]. He talked about some differential equations and said that the solutions were quasi-analytic functions. I couldn’t understand what the two had in common. I was looking in his books and in articles of these people but I couldn’t understand a single word; and I still don’t. I was extremely unhappy about this but then I thought about it in geometric terms. And then you immediately see there is an almost complex structure there and the solutions are just holomorphic curves for this almost complex structure. It is nothing special because any elliptic system in two variables has this property. It has the same principal symbol as the Cauchy-Riemann equation. The theorem he was using is obvious once you say it this way. You didn’t have to use any theory; it is obvious because complex numbers have a forced orientation. That’s all you use! 108 Abel Prize 2009

You say obvious but not many mathematicians were aware of this? Yes, exactly. They were proving theorems but they never looked at this. If you look at this in certain terms, it becomes obvious because you have experience with algebraic geometry. Once you know algebraic geometry you observe it as the same. We have this big science of complex analysis and algebraic geometry with a well-established theory; you know what these things are and you see there is no difference. You use only some part of this but in higher generality. Then, I must admit that for some time I was trying to use it to recapture Don- aldson theory but I couldn’t do it because there were some technical points that did not work. Actually, it was similar to the obstruction of being Kähler in dimension four. I spoke with Pierre Deligne and asked him whether there was an example of a complex surface that was not Kählerian and that would have certain unpleasant properties. He said, yes, and showed me such examples. I turned then to the symplectic case and I realised that it worked very well. And once again, things were very simple, once you knew where to go. It was so simple that I had difficulty believing it could work because there was only one precedent, due to Donaldson. It was Don- aldson’s theory that said that such mathematics can give you that kind of conclusion. It had never happened before Donaldson and that was very encouraging. Otherwise I probably wouldn’t have believed it would work if not for Donaldson’s discovery. Besides, I was prepared by Arnold’s conjectures, which I learned from Dima Fuks in the late 60s, by the symplectic rigidity ideas of Yasha Eliashberg developed by him in the 70s and by the Conley–Zehnder theorem.

Could you say something about the proof by Perelman and Hamilton of the Poincaré conjecture? Did they use some of your results? No. If at all, then just some very simple things. That is a completely different mathematics. There are interactions with the geometry I know but they are minor. It is essentially a quite different sort of mathematics, which I understand only superfi- cially, I must admit. But I must say that it is a domain that is basically unexplored compared to what we know about Cauchy-Riemann equations in a generalized sense, or Yang-Mills, Donaldson or Seiberg-Witten equations. Here, it is one theorem and it is still somewhat isolated. There is no broader knowledge around it and we have to wait and see what comes. We certainly expect great developments from this yet to come.

Do you have any interaction with Alain Connes? Oh yes, certainly. We have interacted quite a bit though we think in very different ways. He understands one half and I understand the other half, with only a tiny intersection of the two parts; amazingly, the outcome turns out to be valid sometimes. I have had two joint papers with him and Moscovici, proving particular cases of the Novikov conjecture.

You came up with an example of some expanders on some groups and thus produced a counter-example to the Baum-Connes conjecture. Mikhail Gromov 109

This counter-example is due to Higson, Lafforgues and Skandalis, where they used the construction of random groups.

Is there one particular theorem or result you are the most proud of? Yes. It is my introduction of pseudo-holomorphic curves, unquestionably. Everything else was just understanding what was already known and to make it look like a new kind of discovery.

You are very modest!

Mathematical biology

We have been told that you have been interested in questions and problems in mathematical biology recently. Can you describe your involvement and how your mathematical and geometric insights can be useful for problems in biology? I can explain how I got involved in that. Back in Russia, everybody was excited by ideas of René Thom [1923–2002] on applying mathematics to biology. My later motivation started from a mathematical angle, from hyperbolic groups. I realised that hyperbolic Markov partitions were vaguely similar to what happens in the process of cell-division. So I looked in the literature and spoke to people and I learned that there were so-called Lindenmayer systems. Many biologists think that they represent a very good way of describing the growth of plants by patterns of substitution and cell division. Then, at the base of that, we had a meeting at the IHES in Bures on pattern for- mation, in particular in biology. I got interested and I wanted to learn more about biology. Soon, I realised that there had been a huge development in molecular biology in the 80s, after the discoveries of genetic engineering and of PCR (polymerase chain reaction). It was really mathematical procedures applied to living cells. Mathematicians could invent PCR. It didn’t happen but mathematicians could have invented PCR. It was one of the major discoveries of the century. It changed molecular biology completely. I started to learn about these mathematical procedures and to realise that it led to fantastic mathematical questions. But it was hard to say exactly what it is; I just cannot formulate it. Of course there are very particular domains like sequencing and there are specific algorithms used there. But this is not new mathematics; it is old mathematics applied to this domain. I believe there is mathematics out there still unknown to us that is yet to be discovered. It will serve as a general framework, just like differential equations give a framework for classical mechanics. It will be rather abstract and formal but it should embed our basic knowledge of biology and maybe accumulate results that we still do not know. I still think about this but I do not know the answer.

Would you please explain the term PCR? 110 Abel Prize 2009

It means polymerase chain reaction and you can see it as follows. You come to a planet that is populated by rats and they all look the same. In your lab, you also have rats that are very similar. They look absolutely identical but they are of a different species. Now, one of the female rats escapes. One year later you want to decide whether it has survived or not. There are billions of those rats, so you cannot check all of them, so what do you do? Here is the idea. You throw in several billions of male rats and if the escaped rat is still there then you will find a certain population of your rats. Then you wait a little bit and the number of them will grow into billions. You take a sample and check if it contains your rat. This is how a polymerase chain reaction works but instead of rats you use DNA. There are billions of different DNAs of various kinds and if you want to know if a particular one of them is out there then there is a way to do that with a given mole- cule that amplifies exponentially. If one had been out there, you would have billions of them after several cycles. This incredible idea is very simple and powerful. One fundamental thing happening in biology is amplification; it is specific for biology. Mathematics should be useful for biologists. We cannot make it yet but I believe it can be done. It will have impact on problems in genetic engineering and identifying gene functions but it has not been developed yet. It will be very different from other kinds of mathematics.

Interaction between mathematics and science

Is it your impression that biologists recognise and appreciate your work and the work of other mathematicians? I have not done anything. I just communicated with biologists. But I think many of them were quite satisfied talking to me, as well as to other mathematicians. Not because we know something but because we ask many questions. Sometimes they cannot answer but that makes them think. That is about it but this is not so little in my opinion. In this way, mathematicians can be useful by being very good listeners. It happens very rarely that something is done by mathematicians in science. One of the most remarkable examples happened here in Norway in the middle of the nineteenth century. In collaboration, the mathematician Guldberg [1836–1902] and the chemist Waage [1833–1900] invented chemical kinetics. I do not know of any other situation since then where mathematicians have contributed to experimental science at this level. This shows that it is possible but it happened through a very close collaboration and in a special situation. I think something like that may happen in biology sometime but it cannot come so easily.

You came across Guldberg and Waage in connection with your interest in chemistry? Yes. This is kind of the fundamental equation in chemistry and also in molecular biology, always on the background of things. Mathematicians can have their word Mikhail Gromov 111 but it is not so easy. You cannot program it. You have to be involved. Sometimes, very rarely, something unexpected happens, with a very strong impact!

To our amazement, we realised that one of the Abel lectures in connection with the prize, the science lecture, was given on computer graphics. It is said that computer graphics or computer vision, and shape analysis in particular, ben- efits from your invention: the Gromov-Hausdorff distance. Can you explain where this notion comes in and how it is used? When you have to compare images, the question is how you compare them. Amaz- ingly – for a geometer it looks unbelievable – the early work on computer vision was based on matching images with another, taking differences in intensity – which is certainly completely contrary to what your eyes do! Actually, the idea of how eyes operate with images goes back to Poincaré [1854– 1912]. In his famous book called Science and Hypothesis he thinks, in particular, about how the human mind can construct Euclidean geometry from the experience we have. He gives an almost mathematical proof that it would be impossible if your eyes could not move. So, what you actually reconstruct, the way your brain records visual information, is based on the movement of your eyes and not so much on what you see. Roughly, the eye does this. It does not add images. It moves images. And it has to move them in the right category, which is roughly the category that appears in Riemannian geometry, with Hausdorff convergence or whatever, using small distortions and matchings of that. For a mathematician who has read Poincaré, this is obvious. But for the people in computer science, following different traditions from linear analysis, it was not obvious at all. And then, apparently, they brought these ideas from geometry to their domain… Actually, several times I attended lectures by G. Sapiro since I became interested in vision. He is someone who has thought for a long time about how you analyse images.

It seems that there is not enough interaction between science and mathematics. Absolutely, I completely agree. To say “not enough” is an understatement. It is close to zero. The communities have become very segregated due to technical reasons and far too little communication. A happy exception is the Courant Institute. We still have many people interacting and it happens that mathematicians fall in love with science. To see these young people at Courant is extremely encouraging because you don’t see this kind of applied mathematicians anywhere else. But they are well aware of the body of pure mathematics where they can borrow ideas and then apply them. Typically, applied mathematicians are separated from the pure ones. They, kind of, don’t quite like each other. That’s absurd. This has to be changed because we have the same goals. We just understand the world from different sides.

Do you have any ideas of how to improve this situation? No. But I think in any subject where you have this kind of problem, the only suggestion is that you have to start by studying the problem. I don’t know enough 112 Abel Prize 2009 about this; I just have isolated examples. We have to look at where it works, where it doesn’t work and just try to organise things in a new way. But it has to be done very gently because you cannot force mathematicians to do what they don’t like. The obvious way to do it is to design good combined educations in mathematics and science. Actually, there is a very good initiative by François Taddei in Paris who organ- ises classes with lectures on biology for non-biologists – for young people in mathematics and physics. He is extremely influential and full of enthusiasm. I attended some of those classes and it was fantastic. He was teaching biology at École Normale for mathematicians and physicists and he manages to make those ideas accessible for everybody. That is what I think should be done at the first stage. We have to have this special kind of education that is not in any curriculum; you cannot formalize it. Only people who have enough enthusiasm and knowledge can project this knowledge to young people. An institutionalized system is much harder to design, and it is very dangerous to make it in any way canonical, because it may just misfire. Forcing mathematics on non-mathematicians only makes them unhappy.

We have already talked about your affiliation to the Courant Institute in New York, but for a much longer time you have been affiliated with the Institut des Hautes Études Scientifique (IHES) at Bures-sur-Yvette, close to Paris. Can you explain the role of this institution for your research – and for your daily life, as well? It is a remarkable place. I knew about it before I came there; it was a legendary place because of Grothendieck [1928–2014]. He was kind of a god in mathematics. I had met Dennis Sullivan already at Stony Brook but then met him again at IHES, where I learned a lot of mathematics talking to him. I think he was instrumental bringing me there because he liked what I was doing and we interacted a lot. Dennis interacted with many people. He had a fantastic ability of getting involved in any idea – absorbing and helping to develop an idea. Another great man there was René Thom, but he was already into philosophy apart from doing mathematics. Pierre Deligne was also there. From Pierre I learned some stuff rather punctually; on several occasions, I got fantastic answers when I asked him questions. He would take an idea from your mind and turn it in another direction. Basically, the whole atmosphere created at this institution was very particular. You are almost completely free of anything except for doing research and talking to people – a remarkable place. I think my best memories go back to when I was there as a first year visitor. Then I was really free. When I became a part of it there were some obligations. Not much, but still. It is ideal for visitors to come for half a year and just relax but being there permanently was also not so bad.

Did you get your best results when you were at Bures? Yes. When I was between 35 and 39, I would say. That’s when I was the most productive. Mikhail Gromov 113

Computers for mathematicians and for mathematics

It is clear that the use of computers has changed the everyday life of mathematicians a lot. Everybody uses computers to communicate and editing is done with computer tools by almost everybody. But other people use computers also as essential research equipment. What are your own experiences? Do you use computers? No, unfortunately not. I am not adept with computers. I can only write my articles on a computer and even that I learned rather recently. I do believe that some mathematics, particular related to biology, will be inseparable from computers. It will be different mathematics when you, indeed, have to combine your thinking with computer experiments. We have to learn how to manipulate large amounts of data without truly understanding everything about it, only having general guidelines. This is, of course, what is happening but it is not happening fast enough. In biology, time is the major factor because we want to discover cures or at least learn about human diseases. And the faster we do it, the better it is. Mathematicians are usually timeless. You are never in a hurry. But here you are in a hurry and mathematicians can accelerate the process. And there, computers are absolutely a part of that. In this way, I believe computers are playing and will play a crucial role.

And that will change the way mathematics is done in the long run, say within the next fifty years? I think that within 50 years there will be a radical change in computers. Program- ming develops very fast and I also believe mathematicians may contribute to the development in a tremendous way. If this happens, we will have very different computers in 50 years. Actually, nobody has been able to predict the development of computers. Just look at how Isaac Asimov [1920–1992] imagined robots and computers thirty years ago when he was projecting into the 21st century how they looked like in the 70s. We probably cannot imagine what will happen within 50 years. The only thing one can say is that they will be very different from now; technology moves at a very fast speed.

What do you think about quantum computing? Well, I am not an expert to say anything about that. You have to ask physicists but they have very different opinions about it. My impression is that the experimental physicists believe we can do it and theorists say: “No, no, we cannot do it”. That is the overall impression I have but I cannot say for myself because I don’t understand either of the aspects of it.

Mathematical work style

You have been described as a mathematician who introduces a profoundly original viewpoint to any subject you work on. Do you have an underlying phi- 114 Abel Prize 2009 losophy of how one should do mathematics and, specifically, how one should go about attacking problems? The only thing I can say is that you have to work hard and that’s what we do. You work and work, and think and think. There is no other recipe for that. The only general thing I can say is that when you have a problem then – as mathematicians in the past have known – one has to keep the balance between how much you think yourself and how much you learn from others. Everybody has to find the right balance according to his or her abilities. That is different for different people so you cannot give any general advice.

Are you a problem solver more than a theory builder? Would you describe yourself in any of those terms? It depends upon the mood you are in. Sometimes you only want to solve one problem. Of course, with age, you become more and more theoretical. Partly because you get wiser but you can also say it is because you get weaker. I suppose it depends on how you look at it.

Concerning your mathematical work style, do you think about mathematics all the time? Yes, except when I have some problems of a personal nature; if there is something else that disturbs me then I cannot think. But if everything is ok and, at least, if there is nothing else to do at the moment, I immerse myself in mathematics, or other subjects, like biology, but in a mathematical way, so to say.

How many hours per day do you work with mathematics? Not as much as I used to. When I was young I could go on all day, sometimes from nine in the morning to eleven at night. Nothing could distract me. Of course, now I cannot do that any longer. I can only do five, six hours a day without getting tired.

When you were younger, you had more energy but now you are a lot wiser, right? You can say you become more experienced and wiser when you get older. But you also lose your mental powers and you become weaker. You certainly just have to accept that. Whether you become wiser is questionable. But it is obvious that you become weaker.

John von Neumann once said that you do the most important things in mathematics before you are thirty. When he himself turned thirty he added that you get wiser as you get older. Do you think that the best mathematics is done before you are thirty? I can say about myself that I think my best work was done when I was between thirty and forty years old. When I started, I didn’t have any perspective and was just doing whatever was coming first. As I was learning more, I kept changing my attitude all the time. Now, if I had to start anew, I would do something completely different, wrongly or rightly, I cannot judge. Mikhail Gromov 115

On the other hand, I must say that everything I think about now, I had already thought of forty years ago. Ideas were germinating in me for a long time. Well, some people probably create radically new work late in life but basically you develop certain feelings very early. Like your abilities to talk, right? You learn to talk when you are three years old but it doesn’t mean you say the same things when you are thirty as when you are three. That’s how it works.

We are surprised that you are so modest by playing down your own achievements. Maybe your ideas are naïve, as you yourself say; but to get results from these ideas, that requires some ingenuity, doesn’t it? It is not that I am terribly modest. I don’t think I am a complete idiot. Typically when you do mathematics you don’t think about yourself. A friend of mine was complaining that anytime he had a good idea he became so excited about how smart he was that he could not work afterwards. So naturally, I try not to think about it.

Having worked so hard as you say, have you ever suffered from depression because you have overexerted yourself? No. Sometimes some outside unhappy things have distracted my work. Of course, sometimes you get very tired and you are glad that someone interrupts your work but other times you cannot stop. You work and work, like an alcoholic, so then it is good to get some rest.

Abel and the Abel Prize

You once complained that the mathematical community only has digested a minor part of your work, rather the technical details than the underlying big ideas and vistas. Do you think that being awarded the Abel Prize may change that situation? First about this complaint: it was kind of a half-joke. There were some pieces of work where there happened to be ideas that could not be developed, unlike more successful ones, and I was unhappy about that. It depends on how you look at it; either the ideas were no good or people were not paying attention. You just never know. I wished something I was saying could be developed further but this was not happening. And that was my complaint, or rather the motivation for my complaint. It has nothing to do with the Abel Prize.

What do you think about prizes in general and, in particular, about the Abel Prize? Objectively, I don’t think we need these prizes for mathematicians who have already achieved much. We need more to encourage young people at all levels and we must put more effort into that. On the other hand, it is very pleasant to receive this prize. I enjoy it, and it may have some overall positive effect on the perception of the mathematical community in the eyes of the general public. That may be just 116 Abel Prize 2009 self-justification because I like it, of course, for appreciation of my work by my friends and by receiving this prize. But as the general scientific concern, the far more serious issue is projecting a much greater effort in getting funds for educating and motivating young people to embrace mathematics. What I have seen here in Oslo, at the high school I visited earlier today – with these young people – I was tremendously impressed. I want to see this kind of event everywhere in the world. Of course, mathematicians are not so ascetic that they don’t like prizes but in the long run it is not prizes that shape our future.

Coming back to Abel, do you admire him as a mathematician? Yes, absolutely. As I said, he was one of the major figures, if not the major figure, in changing the course of mathematics from what could be visualized and immediately experienced to the next level, a level of deeper and more fundamental structures.

There is a posthumous paper by Abel where he writes about the theory of equations, which later became Galois theory, and in the introduction he says something very interesting. He says something like: “A problem that seems insurmountable is just seemingly so because we have not asked the right question. You should always ask the right question and then you can solve the problem”. Absolutely. He changed the perspective on how we ask questions. I do not know enough about the history of mathematics but it is obvious that the work of Abel and his way of thinking about spaces and functions has changed mathematics. I do not know enough history to say exactly when this happened but the concept of underlying symmetries of structures comes very much from his work. We still follow that development. It is not exhausted yet. This continued with Galois theory and in the development of Lie group theory, due to Lie, and, in modern times, it was done at a higher level, in particular by Grothendieck [1842–1899]. This will continue and we have to go through all that to see where it brings us before we go on to the next stage. It is the basis of all we do now in mathematics.

Future of mathematics

After this excursion into the history of mathematics, may we speculate a little about the future of mathematics? You once compared the whole building of mathematics with a tree, Hilbert’s tree, with a metric structure expressing closeness or nearness between different areas and results. We know from Kurt Gödel [1906–1978] that there are parts of that tree we will never reach. On the other hand, we have a grasp of a certain part of the tree but we don’t know how big this part is. Do you think we know a reasonable part of Hilbert’s tree? Is human mind built for grasping bigger parts of it or will there stay areas left uncharted forever? Actually, I am thinking about that now. I don’t know the answer but I have a programme of how we can approach it. It is a rather long discussion. There are certain Mikhail Gromov 117 basic operations by which we can perceive the structure. We can list some of them and apparently they bring you to certain parts of this tree. They are not axioms. They are quite different from axioms. But eventually you cannot study the outcome with your hands and you have to use computers. With computers you come to some conclusions without knowing the intermediate steps. The computational size will be too huge for you. You have to formalize this approach to arrive at certain schemes of computations. This is what I think about now but I don’t know the answer. There are indirect indications that it is possible but those are of a non-mathematical nature, rather biological.

If you try to look into the future, 50 or 100 years from now… 50 and 100 is very different. We know more or less about the next 50 years. We shall continue in the way we go. But in 50 years from now, the Earth will run out of the basic resources and we cannot predict what will happen after that. We will run out of water, air, soil, rare metals, not to mention oil. Everything will essentially come to an end within 50 years. What will happen after that? I am scared. It may be okay if we find solutions but if we don’t then everything may come to an end very quickly! Mathematics may help to solve the problem but if we are not successful, there will not be any mathematics left, I am afraid!

Are you pessimistic? I don’t know. It depends on what we do. If we continue to move blindly into the future, there will be a disaster within 100 years and it will start to be very critical in 50 years already. Well, 50 is just an estimate. It may be 40 or it may be 70 but the problem will definitely come. If we are ready for the problems and manage to solve them, it will be fantastic. I think there is potential to solve them but this potential should be used and this potential is education. It will not be solved by God. People must have ideas and they must prepare now. In two generations people must be educated. Teachers must be educated now, and then the teachers will educate a new generation. Then there will be sufficiently many people who will be able to face the difficulties. I am sure this will give a result. If not, it will be a disaster. It is an exponential process. If we run along an exponential process, it will explode. That is a very simple computation. For example, there will be no soil. The soil is being exhausted everywhere in the world. It is not being said often enough. Not to mention water. It is not an insurmountable problem but it requires solutions on a scale we have never faced before, both socially and intellectually.

Education systems for the future

Education is apparently a key factor. You have earlier expressed your distress realising that the minds of gifted youths are not developed effectively enough. Any ideas about how education should change to get better adapted to very different minds? 118 Abel Prize 2009

Again I think you have to study it. There are no absolutes. Look at the number of people like Abel who were born 200 years ago. Now there are no more Abels. On the other hand, the number of educated people has grown tremendously. It means that they have not been educated properly because where are those people like Abel? It means that they have been destroyed. The education destroys these potential geniuses – we do not have them! This means that education does not serve this particular function. The crucial point is that you have to treat everybody in a different way. That is not happening today. We don’t have more great people now than we had 100, 200 or 500 years ago, starting from the renaissance, in spite of a much larger population. This is probably due to education. This is maybe not the most serious problem with education. Many people believe in very strange things and accordingly make very strange decisions. As you know, in the UK, in some of the universities, there are faculties of homeopathy that are supported by the government. They are tremendously successful in terms of numbers of students. And anybody can learn that nonsense. It is very unfortunate.

You point out that we don’t have anybody of Abel’s stature today, or at least very few of them. Is that because we, in our educational system, are not clever enough to take care of those that are exceptionally gifted because they may have strange ideas, remote from mainstream? The question of education is not obvious. There are some experiments on animals which indicate that the way you teach an animal is not the way you think it happens. The learning mechanism of the brain is very different from how we think it works: like in physics, there are hidden mechanisms. We superimpose our view from everyday experience, which may be completely distorted. Because of that, we can distort the potentially exceptional abilities of some children. There are two opposite goals education is supposed to achieve: firstly, to teach people to conform to the society they live in; on the other hand, to give them freedom to develop in the best possible way. These are opposite purposes and they are always in collision with each other. This creates the result that some people get suppressed in the process of adapting them to society. You cannot avoid this kind of collision of goals but we have to find a balance between the two and that is not easy, on all levels of education. There are very interesting experiments performed with Chimpanzee and Bon- obo apes and under which conditions they learn, or even how you teach a parrot to talk. How do you do that? The major factor is that it should not see the teacher. You put a mirror between you and the parrot and then you speak behind the mirror. The parrot then sees a bird – it talks to a bird. But if it sees you, it will learn very badly. That is not an obvious thing. The very presence of a teacher, an authority, moves students in a particular direction and not at all the direction the teacher wants them to move. With all this accumulated evidence, you cannot make any simple decision. If you say “do this and this”, you are wrong for sure. Solutions are not obvious; they can only come after analysing deeply what is actually known and by studying Mikhail Gromov 119 the possibilities. I think the answers will be unexpected. What children can learn and what they cannot learn, we don’t know because we don’t know how to conduct experiments to be ethical and instructive at the same time. It is a very non-trivial issue, which has not been studied much. With animals we have results but not very much with people. From left to right: Martin Raussen, Christian Skau and Let us come back to mathe- Mikhail Gromov. (Photo: Heiko Junge) matics and to mathematics education. It seems that many people stop dealing with mathematics as soon as they have left high school. But as mathematicians we know that mathematics is everywhere, though often hidden: as the workhorse in science and technology but also as a pillar in human culture, emphasising rigour and organised thinking. Do you have any ideas on how we can make this double role perceived and appreciated by society and how to make decision makers realise that mathematics needs support? It is a very difficult question because we have to project mathematical ideas to people who work very far from mathematics – to people that make decisions in society. The way we think is very different from the way they operate. I don’t know but I think that within our mathematical society we can make some steps towards education, like creating good mathematical sources for children. Today we have the Internet so we should try to make Internet presentations. Actually, in France there are some people trying to organise extra-curricula activities for younger children on a small scale. We should try to do something like that on a big scale: big centres of stimulating creativity in all directions. I would not only focus on mathematics but on science and art and whatever can promote creative activity in young people. When this develops, we may have some influence but not before that. Being inside our ivory tower, what can we say? We are inside this ivory tower and we are very comfortable there. But we cannot really say much because we don’t see the world well enough either. We have to go out but that is not so easy.

You mentioned that you first got interested in mathematics after reading the book “Numbers and Figures” by Rademacher and Toeplitz. We could also mention the book “What is mathematics?” by Courant [1888–1972] and Robbins [1915–2001]. Should we encourage pupils in high school who show an interest in mathematics to read books like that? 120 Abel Prize 2009

Yes. We have to produce more such books. Already there are some well-written books, by Martin Gardner [1914–2010], by Yakov Perelman [1882–1942] (Math- ematics can be fun), by Yaglom [1921–1988] and co-authors – very remarkable books. Other mathematicians can contribute by writing such books and combine this with the possibilities of the Internet, in particular visualization. It is relatively simple to write just one page of interesting mathematics. This should be done so that many different subjects in mathematics become easily available. As a community we should go out and create such structures on the Internet. That is relatively easy. The next level is more complicated; writing a book is not easy. Within the community we should try to encourage people to do that. It is a very honourable kind of activity. All too often mathematicians say: “Just vulgarization, not serious”. But that is not true; it is very difficult to write books with a wide appeal and very few mathematicians are actually able to do that. You have to know things very well and understand them very deeply to present them in the most evident way.

This could be a way to get more young people to take up mathematics? You will attract more young people. Moreover, the political figures will sense it on a much larger scale because it will have a much wider appeal than what we do internally.

Poetry

You have mentioned that you like poetry. What kind of poetry do you like? Of course, most of what I know is Russian poetry – the so-called Silver Age of Rus- sian Poetry at the turn of the twentieth century. There were some poets but you, probably, do not know them. They are untranslatable, I guess. People in the West know Akhmatova [1889–1966] but she was not the greatest poet. The three great poets were Tsvetaeva [1892–1941] (also a woman), Blok [1880–1921] and Mandel- stam [1891–1938].

What about Pushkin? You see, with Pushkin [1799–1837], the problem is as follows. He was taught at school and that has a tremendously negative impact. But 40 years later I rediscovered Pushkin and found him fantastic – when I had forgotten what I had learned in school.

What about modern poetry and English poetry? I have read some English poetry. I know some pieces but I don’t know it on a larger scale. It is difficult. Even with modern Russian poetry, e.g. Brodsky [1940–1996], I find it difficult to absorb a new style. To absorb a poet is non-trivial. For Eng- lish poetry, there are a few particular pieces that I learned and appreciate. Some of them are easy to deal with; some have Russian translations. A remarkable one is Mikhail Gromov 121

Edgar Allan Poe [1809–1849]. He is kind of simple in a way. But many other English poets are more remote from Russian style. I know a little bit of French poetry, like François Villon [1431–1463]; I can appreciate him in French. But modern poetry is very difficult for me.

To finish the interview, we would like to thank you very much on behalf of the Norwegian, the Danish and the European Mathematical Societies.

Mikhail Gromov during the Abel prize ceremony. (Photo: Erlend Aas)

Abel Prize 2010: John Tate

The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2010 to

John Torrence Tate University of Texas at Austin, USA

“for his vast and lasting impact on the theory of numbers.”

Beyond the simple arithmetic of 1, 2, 3,… lies a complex and intricate world that has chal- lenged some of the finest minds throughout history. This world stretches from the mysteries of the prime numbers to the way we store, transmit, and secure information in modern computers. It is called the theory of numbers. Over the past century it has grown into one of the most elaborate and sophisticated branches of mathematics, interacting profoundly with other areas such as algebraic geometry and the theory of automorphic forms. John Tate is a prime architect of this development. Tate’s 1950 thesis on Fourier analysis in number fields paved the way for the modern theory of automorphic forms and their Abel Laureate John T. Tate. L-functions. He revolutionized global class (Photo: Berit Roald) field theory with Emil Artin, using novel techniques of group cohomology. With Jon- athan Lubin, he recast local class field theory by the ingenious use of formal groups. Tate’s invention of rigid analytic spaces spawned the whole field of rigid analytic geometry. He found a p-adic analogue of Hodge theory, now called Hodge-Tate theory, which has blossomed into another central technique of modern algebraic number theory.

* 13.3.1925

123 124 Abel Prize 2010

A wealth of further essential mathematical ideas and constructions were initiated by Tate, including Tate cohomology, the Tate duality theorem, Barsotti–Tate groups, the Tate motive, the Tate module, Tate’s algorithm for elliptic curves, the Néron–Tate height on Mordell–Weil groups of abelian varieties, Mumford–Tate groups, the Tate isogeny theorem and the Honda–Tate theorem for abelian varieties over finite fields, Serre–Tate deformation theory, Tate–Shafarevich groups, and the Sato–Tate conjecture concerning families of elliptic curves. The list goes on and on. Many of the major lines of research in algebraic number theory and arithmetic geometry are only possible because of the incisive contribution and illuminating insight of John Tate. He has truly left a conspicuous imprint on modern mathematics.

Abel Laureate John Tate with King Harald. (Photo: Terje Pedersen) John Tate 125

Education

Professor Tate, you have been selected as this year’s Abel Prize Laureate for your decisive and lasting impact on number theory. Before we start to ask you questions we would like to congratulate you warmly on this achievement. You were born in 1925 in Minneapolis, USA. Your father was a professor of physics at the University of Minnesota. We guess he had some influence on your attraction to the natural sciences and mathematics. Is that correct? It certainly is. He never pushed me in any way, but on a few occasions he simply explained something to me. I remember once he told me how one could estimate the height of a bridge over a river with a stopwatch, by dropping a rock, explaining that the height in feet is approximately 16 times the square of the number of seconds it takes until the rock hits the water. Another time he explained Cartesian coordinates, and how one could graph an equation and, in particular, how the solution to two simultaneous linear equations is the point where two lines meet. Very rarely, but beautifully, he just explained something to me. He did not have to explain negative numbers – I learned about them from the temperature in the Minnesota winters. But I have always, in any case, been interested in puzzles and trying to find the answers to questions. My father had several puzzle books. I liked reading them and trying to solve the puzzles. I enjoyed thinking about them, even though I did not often find a solution.

Are there other people that have had an influence on your choice of fields of interest during your youth? No. I think my interest is more innate. My father certainly helped, but I think I would have done something like physics or mathematics anyway.

You started to study physics at Harvard University. This was probably during the Second World War? I was in my last year of secondary school in December 1941 when Pearl Harbour was bombed. Because of the war Harvard began holding classes in the summer, and I started there the following June. A year later I volunteered for a Naval Officer Training Program in order to avoid being drafted into the army. Some of us were later sent to MIT to learn meteorology, but by the time we finished that training and Midshipman School it was VE day.1 Our campaign in the Pacific had been so successful that more meteorologists were not needed, and I was sent to do mine- sweeping research. I was in the Navy for three years and never aboard a ship! It was frustrating.

1 Victory in Europe day: 8 May 1945. 126 Abel Prize 2010

Study conditions in those times must have been quite different from conditions today. Did you have classes regularly? Yes, for the first year, except that it was accelerated. But then in the Navy I had specific classes to attend, along with a few others of my choice I could manage to squeeze in. It was a good program, but it was not the normal one. It was not the normal college social life either, with parties and such. We had to be in bed or in a study hall by ten and were roused at 6:30 AM by a recording of reveille, to start the day with calisthenics and running.

Then you graduated in 1946 and went to Princeton? Yes, that’s true. Harvard had a very generous policy of giving credit for military activities that might qualify – for instance some of my navy training. This and the wartime acceleration enabled me to finish the work for my undergraduate degree in 1945. On my discharge in 1946, I went straight from the Navy to graduate school in Princeton.

When you went to Princeton University, it was still with the intention of becoming a physicist? That’s correct. Although my degree from Harvard was in mathematics, I entered Princeton graduate school in physics. It was rather silly and I have told the story many times: I had read the book Men of Mathematics by Eric Temple Bell [1883– 1960]. That book was about the lives of the greatest mathematicians in history, people like Abel [1802–1829]. I knew I wasn’t in their league and I thought that unless I was, I wouldn’t really be able to do much in mathematics. I didn’t realize that a less talented person could still contribute effectively. Since my father was a physicist that field seemed more human and accessible to me, and I thought that was a safer way to go, where I might contribute more. But after one term it became obvious that my interest was really in mathematics. A deeper interest, which should have been clear anyway, but I just was too afraid and thought I never would be able to do much research if I went into mathematics.

Were you particularly interested in number theory from the very beginning? Yes. Since I was a teenager I had an interest in number theory. Fortunately, I came across a good number theory book by L. E. Dickson [1874–1954] so I knew a little number theory. Also I had been reading Bell’s histories of people like Gauss [1777–1855]. I liked number theory. It’s natural, in a way, because many wonderful problems and theorems in number theory can be explained to any interested high- school student. Number theory is easier to get into in that sense. But of course it depends on one’s intuition and taste also.

Many important questions are easy to explain, but answers are often very tough to find. Yes. In number theory that is certainly true, but finding good questions is also an important part of the game. John Tate 127

Teachers and fellows

When you started your career at Princeton you very quickly met Emil Artin who became your supervisor. Emil Artin [1898–1962] was born in Austria and became a professor in mathematics at the University of Hamburg. He had to leave Germany in 1937 and came to the US. Can you tell us more about his background? Why did he leave his chair and how did he adjust when he came to the States? His wife was half Jewish and he eventually lost his position in Germany. The family left in ’37, but at that time there weren’t so many open jobs in the US. He took a position at the University of Notre Dame in spite of unpleasant memories of discipline at a Catholic school he had attended in his youth. After a year or two he accepted an offer from Indiana University, and stayed there until 1946. He and his wife enjoyed Bloomington, Indiana, very much. He told me it wasn’t even clear that he would have accepted Princeton’s offer in 1946 except that President H. B. Wells [1902–2000] of Indiana University, an educational visionary, was on a world tour, and somehow Indiana didn’t respond very well to Princeton’s offer. Artin went to Princeton the same year I did.

Artin apparently had a very special personality. First of all, he was an eminent number theorist, but also a very intriguing person – a special character. Could you please tell us a bit more about him? I think he would have made a great actor. His lectures were polished; he would finish at the right moment and march off the scene. A very lively individual with many interests: music, astronomy, chemistry, history … He loved to teach. I had a feeling that he loved to teach anybody anything. Being his student was a wonderful experience; I couldn’t have had a better start to my mathematical career. It was a remarkable accident. My favourite theorem, which I had first learned from Bell’s book, was Gauss’ law of quadratic reciprocity and there, entirely by chance, I found myself at the same university as the man who had discovered the ultimate law of reciprocity. It was just amazing.

What a coincidence! Yes, it really was.

You wrote your thesis with Artin, and we will certainly come back to it. After that you organised a seminar together with Artin on class field theory. Could you comment on this seminar? What was the framework and how did it develop? During his first two years in Princeton Artin gave seminars in algebraic number theory, followed by class field theory. I did not attend the former, but one of the first things I heard about Artin concerned an incident in it. A young British student, Douglas Northcott [1916–2005], who had been captured when the Japanese trapped the British army in Singapore, and barely survived in the Japanese prison camp, was in Princeton on a Commonwealth Fellowship after the war. Though his 128 Abel Prize 2010 thesis was in analysis under G. H. Hardy [1877–1947], he attended Artin’s seminar, and when one of the first speakers mentioned the characteristic of a field, Northcott raised his hand and asked what that meant. His question begot laughter from several students, whereupon Artin delivered a short lecture on the fact that one could be a fine mathematician without knowing what the characteristic of a field was. And indeed, it turned out that Northcott was the most gifted student in that seminar. But I’m not answering your question. I attended the second year, in which class field theory was treated, with Chevalley’s [1909–1984] non-analytic proof of the second inequality, but not much cohomology. This was the seminar at the end of which Wang [1915–1993] discovered that both published proofs of Grunwald’s [1909–1989] theorem, and in fact the theorem itself, were not correct at the prime 2. At about that time, Gerhard Hochschild [1915–2010] and Tadasi Nakayama [1912–1964] were introducing cohomological methods in class field theory, and used them to prove the main theorems, including the existence of the global fundamental class which André Weil [1906–1998] had recently discovered. In 1951–52 Artin and I ran another seminar giving a complete treatment of class field theory incorporating these new ideas. That is the seminar you are asking about. Serge Lang [1927–2005] took notes, and thanks to his efforts they were eventually published, first as informal mimeographed notes, and in 1968 commercially, under the title Class Field Theory. A new edition (2008) is available from AMS-Chelsea.

Serge Lang was also a student of Emil Artin and became a famous number theorist. He is probably best known as the author of many textbooks; almost every graduate student in mathematics has read a textbook by Serge Lang. He is also quite known for his intense temper, and he got into a lot of arguments with people. What can you tell us about Serge Lang? What are your impressions? He was indeed a memorable person. The memories of Lang in the May 2006 issue of the Notices of the AMS, written by about twenty of his many friends, give a good picture of him. He started Princeton graduate school in philosophy, a year after I started in physics, but he too soon switched to math. He was a bit younger than I and had served a year and a half in the US army in Europe after the war, where he had a clerical position in which he learned to type at incredible speed, an ability which served him well in his later book writing. He had many interests and talents. I think his undergraduate degree from Caltech was in physics. He knew a lot of history and he played the piano brilliantly. He didn’t have the volatile personality you refer to until he got his degree. It seemed to me that he changed. It was almost a discontinuity; as soon as he got his PhD he became more authoritative and asserted himself more. It has been noted that there are many mathematical notions linked to my name. I think that’s largely due to Lang’s drive to make information accessible. He wrote voluminously. I didn’t write easily and didn’t get around to publishing; I was always interested in thinking about the next problem. To promote access, Serge published some of my stuff and, in reference, called things “Tate this” and “Tate that” in a way I would not have done had I been the author. John Tate 129

Throughout his life, Serge addressed great energy to disseminating information; to sharing where he felt it was important. We remained friends over the years.

Research contributions

This brings us to the next topic: your PhD thesis from 1950, when you were 25 years old. It has been extensively cited in the literature under the sobriquet “Tate’s thesis”. Several mathematicians have described your thesis as unsur- passable in conciseness and lucidity, and as representing a watershed in the study of number fields. Could you tell us what was so novel and fruitful in your thesis? Well, first of all, it was not a new result, except perhaps for some local aspects. The big global theorem had been proved around 1920 by the great German mathematician Erich Hecke [1887–1947], namely the fact that all L-functions of number fields, abelian L-functions, generalizations of Dirichlet’s L-functions, have an analytic continuation throughout the plane with a functional equation of the expected type. In the course of proving it Hecke saw that his proof even applied to a new kind of L-function, the so-called L-functions with Grössencharacter. Artin suggested to me that one might prove Hecke’s theorem using abstract harmonic analysis on what is now called the adele ring, treating all places of the field equally, instead of using classical Fourier analysis at the archimedean places, and finite Fourier analysis with congruences at the p-adic places as Hecke had done. I think I did a good job – it might even have been lucid and concise! – but in a way it was just a wonderful exercise to carry out this idea. And it was also in the air. So often there is a time in mathematics for something to be done. My thesis is an example. Iwasawa would have done it had I not.

What do you think of the fact that, after your thesis, all places of number fields are treated on an equal footing in analytic number theory, whereas the situation is very different in the classical study of zeta functions; in fact, gamma factors are very different to non-Archimedean local factors. Of course there is a big difference between archimedean and non-archimedean places, in particular as regards the local factors, but that is no reason to discrimi- nate. Treating them equally, using adeles and ideles, is the simplest way to proceed, bringing the local – global relationship into clear focus.

The title of your thesis was “Fourier analysis in number fields and Hecke’s zeta-functions”. Atle Selberg [1917–2007] said in an interview five years ago that he preferred – and was most inspired by – Erich Hecke’s approach to algebraic number theory, modular forms and L-functions. Do you share that sentiment? Hecke and Artin were both at Hamburg University for a long time before Artin left. I think Artin came to number theory more from an algebraic side, whereas Hecke and Selberg came more from an analytic side. Their basic intuition was more ana- 130 Abel Prize 2010 lytic and Artin’s was more algebraic. Mine was also more algebraic, so the more I learned of Hecke’s work, the more I appreciated it, but somehow I did not instinc- tively follow him, especially as to modular forms. I didn’t know much about them when I was young. I have told the story before, but it is ironic that being at the same university, Artin had discovered a new type of L-series and Hecke, in trying to figure out what kind of modular forms of weight one there were, said they should correspond to some kind of L-function. The L-functions Hecke sought were among those that Artin had defined, but they never made contact – it took almost 40 years until this connection was guessed and ten more before it was proved, by Langlands. Hecke was older than Artin by about ten years, but I think the main reason they did not make contact was their difference in mathematical taste. Moral: be open to all approaches to a subject.

You mentioned that Serge Lang had named several concepts after you, but there are lots of further concepts and conjectures bearing your name. Just to mention a few: Tate module, Tate curve, Tate cohomology group, Tate conjecture, Shafarevich–Tate group, Sato–Tate conjecture, etc. Good definitions and fruitful concepts, as well as good problems, are perhaps as important as theorems in mathematics. You excel in all these categories. Did all or most of these concepts grow out of your thesis? No, I wouldn’t say that. In fact, I would say that almost none of them grew out of my thesis. Some of them, like the Tate curve grew out of my interest in p-adic fields which were also very central in my thesis, but they didn’t grow out of my thesis. They came from different directions. The Tate cohomology came from my understanding the cohomology of class field theory in the seminar that we discussed. The Shafarevich–Tate group came from applying that cohomology to elliptic curves and abelian varieties. In general, my conjectures came from an optimistic outlook, gen- eralizing from special cases. Although concepts, definitions and conjectures are certainly important, the bottom line is to prove a theorem. But you do have to know what to prove, or what to try to prove.

In the introduction to your delightful book Rational points on elliptic curves that you co-authored with your earlier PhD student Joseph Silverman, you say, citing Serge Lang, that it is possible to write endlessly on elliptic curves. Can you comment on why the theory of elliptic curves is so rich and how it interacts and makes contact with so many different branches of mathematics? For one thing, they are very concrete objects. An elliptic curve is described by a cubic polynomial in two variables so they are very easy to experiment with. On the other hand, elliptic curves illustrate very deep notions. They are the first non-trivial examples of abelian varieties. An elliptic curve is an abelian variety of dimension one, so you can get into this more advanced subject very easily by thinking about elliptic curves. On the other hand, they are algebraic curves. They are curves of John Tate 131 genus one, the first example of a curve which isn’t birationally equivalent to a projective line. The analytic and algebraic relations which occur in the theory of elliptic curves and elliptic functions are beautiful and unbelievably fascinating. The modularity theorem stating that every elliptic curve over the rational field can be found in the Jacobian variety of the curve which parametrizes elliptic curves with level structure its conductor is mindboggling. By the way, by my count about one quarter of Abel’s published work is devoted to elliptic functions.

Among the Abel Prize laureates so far, you are probably the one whose contributions would have been closest to Abel’s own interests. Could we challenge you to make an historical sweep, to put Abel’s work in some perspective and to compare it to your research? In modern parlance, Abel studied the multiplication-by-n map for elliptic equal parts, and studied the algebraic equations that arose. He also studied complex multiplication and showed that, in this case, it gave rise to a commutative Galois group. These are very central concepts and observations, aren’t they? Yes, absolutely, yes. Well, there’s no comparison between Abel’s work and mine. I am in awe of what I know of it. His understanding of algebraic equations, and of elliptic integrals and the more general, abelian integrals, at that time in history is just amazing. Even more for a person so isolated. I guess he could read works of Legendre, and other great predecessors, but he went far beyond. I don’t really know enough to say more. Abel was a great analyst and a great algebraist. His work contains the germs of many important modern developments.

Could you comment on how the concept of “good reduction” for an elliptic curve is so crucial, and how it arose? If one has an equation with integer coefficients it is completely natural, at least since Gauss [1777–1855], to consider the equation mod p for a prime p, which is an equation over the finite fieldp F with p elements. If the original equation is the equation of an elliptic curve E over the rational number field then the reduced equation may or may not define an elliptic curve over Fp. If it does, we say E has “good reduction at p”. This happens for all but a finite set of “bad primes for E”, those dividing the discriminant of E.

The Hasse Principle in the study of Diophantine equations says, roughly speaking: if an equation has a solution in p-adic numbers then it can be solved in the rational numbers. It does not hold in general. There is an example for this failure given by the Norwegian mathematician Ernst Selmer [1920–2006]… Yes. The equation 3x3 + 4y3 + 5z3 = 0.

Exactly! The extent of the failure of the Hasse Principle for curves of genus 1 is quantified by the Shafarevich–Tate group. The so-called Selmer groups are 132 Abel Prize 2010 related groups, which are known to be finite, but as far as we know the Sha- farevich–Tate group is not known to be finite. It is only a conjecture that it is always finite. What is the status concerning this conjecture? The conjecture that the Shafarevich group Sha is finite should be viewed as part of the conjecture of Birch and Swinnerton–Dyer. That conjecture, BSD for short, involves the L-function of the elliptic curve, which is a function of a complex variable s. Over the rational number field, L(s) is known to be defined near s= 1, thanks to the modularity theorem of A. Wiles, R. Taylor, et al. If L(s) either does not vanish or has a simple zero at s = 1, then Sha is finite and BSD is true, thanks to the joint work of B. Gross and D. Zagier on Heegner points, and the work of Kolyvagin on Euler systems. So, by three big results which are the work of many people, we know a very special circumstance in which Sha is finite. If L(s) has a higher order zero at s = 1, we know nothing, even over the field of rational numbers. Over an imaginary quadratic field we know nothing, period.

Do you think that this group is finite? Yes. I firmly believe the conjecture is correct. But who knows? The curves of higher rank, or whose L-functions have a higher order zero – BSD says the order of the zero is the rank of the curve – one knows nothing about.

What is the origin of the Tate Conjecture? Early on I somehow had the idea that the special case about endomorphisms of abelian varieties over finite fields might be true. A bit later I realized that a generalization fit perfectly with the function field version of the Birch and Swinnerton–Dyer conjecture. Also it was true in various particular examples which I looked at, and gave a heuristic reason for the Sato–Tate distribution. So it seemed a reasonable conjecture.

In the arithmetic theory of elliptic curves, there have been major breakthroughs like the Mordell–Weil theorem, Faltings’ proof of the Mordell conjecture, using the known reduction to a case of the Tate conjecture. Then we have Wiles’ breakthrough proving the Shimura–Taniyama–Weil conjecture. Do you hope the next big breakthrough will come with the Birch and Swinnerton–Dyer conjecture? Or the Tate conjecture, maybe? Who knows what the next big breakthrough will be, but certainly the Birch and Swinnerton–Dyer conjecture is a big challenge, and also the modularity, i.e. the Shimura–Taniyama–Weil idea, which is now seen as part of the Langlands program. If the number field is not totally real we don’t know much about either of these problems. There has been great progress in the last thirty years, but it is just the very beginning. Proving these things for all number fields and for all orders of vanishing, to say nothing of doing it for abelian varieties of higher dimension, will require much deeper insight than we have now.

Is there any particular work from your hand that you are most proud of, that you think is your most important contribution? John Tate 133

I don’t feel that any one of my results stands out as most important. I certainly enjoyed working out the proofs in my thesis. I enjoyed very much proving a very special case of the so-called Tate conjecture, the result about endomorphisms of abelian varieties over finite fields. It was great to be able to prove at least one non-trivial case and not have only a conjecture! That’s a case that is useful in cryptography, especially elliptic curves over finite fields. Over number fields, even finitely generated fields, that case of my conjecture was proved by Faltings, building on work of Zarhin over function fields, as the first step in his proof of the Mordell conjecture. I enjoyed very much the paper which I dedicated to Jean-Pierre Serre on the K2 groups of number fields. I also had fun with a paper on residues of differentials on curves giving a new definition of residue and a new proof that the sum of the residues is zero, even though I failed to see a more important aspect of the construction.

Applied number theory

Number theory stretches from the mysteries of the prime numbers to the way we save, transmit and secure information on modern computers. Can you comment on the amazing fact that number theory, in particular the arithmetic of elliptic curves, has been put to use in practical applications? It certainly is amazing to me. When I first studied and worked on elliptic curves I had no idea that they ever would be of any practical use. I did not foresee that. It is the high speed computers which made the applications possible, but of course many new ideas were needed also.

And now it’s an industry: elliptic curves, cryptography, intelligence and communication! It’s quite remarkable. It often happens that things which are discovered just for their own interest and beauty later turn out to be useful in practical affairs.

We interviewed Jacques Tits a couple of years ago. His comment was that the Monster group, the biggest of all the sporadic simple groups, is so beautiful that it has to have some application in physics or whatever. That would be interesting!

Collaboration and teaching

You have been one of the few non-French members of the Bourbaki group, the group of mathematicians that had the endeavour of putting all existing mathematics into a rigid format. Can you explain what this was all about and how you got involved? I would not say it was about putting mathematics in a rigid format. I view Bourbaki as a modern Euclid. His aim was to write a coherent series of books which would 134 Abel Prize 2010 contain the fundamental definitions and results of all mathematics as of mid-twentieth century. I think he succeeded pretty well, though the books are somewhat unbalanced – weak in classical analysis and heavy on Lie Theory. Bourbaki did a very useful service for a large part of the mathematics community just by establishing some standard notations and conventions. The presentation is axiomatic and severe, with no motivation except for the logic and beauty of the development itself. I was always a fan of Bourbaki. That I was invited to collaborate may have been at Serge Lang’s suggestion, or perhaps Jean- Pierre Serre’s also. As I mentioned, I am not a very prolific writer. I usually write a few pages and then tear them up, and start over, so I never was able to contribute much to the writing. Perhaps I helped somewhat in the discussion of the material. The conferences were enjoyable, all over France, in the Alps and even on Corsica. It was a lot of fun.

You mentioned Jean-Pierre Serre, who was the first Abel Prize laureate. He was one of the driving forces in the Bourbaki project after the Second World War. We were told that he was – as Serge Lang – instrumental in getting some of your results published in the form of lecture notes and text books. Do you have an ongoing personal relation with Jean-Pierre Serre? Yes. I’m looking forward to meeting him next week when we will both be at Harvard for a conference in honour of Dick Gross on his 60th birthday. Gross was one of my PhD students. I think Serre was a perfect choice for the first Abel Prize laureate.

Another possible choice would have been Alexander Grothendieck [1928– 2014]. But he went into reclusion. Did you meet him while you were in Paris or maybe at Harvard? I met him in Paris. I had a wonderful year. Harvard had the enlightened policy of giving a tenure track professor a year’s sabbatical leave. I went to Paris for the academic year ’57–’58 and it was a great experience. I met Serre, I met Grothendieck and I was free from any duty. I could think and I could learn. Later, they both visited Harvard several times so I saw them there too. It’s great good fortune to be able to know such people.

Did you follow Grothendieck’s program reconstructing the foundations of algebraic geometry closely? Well, yes, to the extent I could. I felt “ah, at last, we have a good foundation for algebraic geometry”. It just seemed to me to be the right thing. Earlier I was always puzzled, do we have affine varieties, projective varieties? But it wasn’t a category. Grothendieck’s schemes, however, did form a category. And breaking away from a ground field to a ground ring, or even a ground scheme, so that the foundations could handle not only polynomial equations, but also Diophantine equations and reduction mod p, was just what number theorists needed. John Tate 135

We have a question of a more general and philosophical nature. A great mathematician once mentioned that it is essential to possess a certain naivety in order to be able to create something really new in mathematics. One can do impressive things requiring complicated techniques, but one rarely makes original discoveries without being a bit naive. In the same vein, André Weil claimed that breakthroughs in mathematics are typically not done by people with long experience and lots of knowledge. New ideas often come without that baggage. Do you agree? I think it’s quite true. Most mathematicians do their best work when they are young and don’t have a lot of baggage. They haven’t worn grooves in their brains that they follow. Their brains are fresher, and certainly it’s important to think for oneself rather than just learning what others have done. Of course, you have to build on what has been done before or else it’s hopeless; you can’t rediscover everything. But one should not be prejudiced by the past work. I agree with the point of view you describe.

Did you read the masters of number theory early in your career? I’ve never been such a good reader. My instincts have been to err on the side of trying to be independent and trying to do things myself. But as I said, I was very fortunate to be in contact with brilliant people, and I learned very much from personal conversations. I never was a great reader of the classics. I enjoyed that more as I got older.

You have had some outstanding students who have made important contributions to mathematics. How did you attract these students in the first place, and how did you interact with them, both as students and later? I think we were all simply interested in the same kind of mathematics. You know, with such gifted students there is usually no problem; after getting to know them and their interests you suggest things to read and think about, then just hear about progress and problems, offering support and encouragement as they find their way.

Did you give them problems to work on or did they find the problems themselves? It varies. Several found their own problem. With others I made somewhat more specific suggestions. I urged Dick Gross to think about a problem which I had been trying unsuccessfully to solve, but very sensibly he wrote a thesis on a quite different subject of his own choosing. I was fortunate to have such able students. I continued to see many of them later and many are good friends.

You have taught mathematics for more than 60 years, both at Harvard and at Austin, Texas. How much did you appreciate this aspect of your professional duties? Is there a particular way of teaching mathematics that you prefer? I always enjoyed teaching at all levels. Teaching a subject is one of the best ways to learn it thoroughly. A few times, I’ve been led to a good new idea in preparing a 136 Abel Prize 2010 lecture for an advanced course. That was how I found my definition of the Néron height, for example.

Work style

Would you consider yourself mainly as a theory builder or as a problem solver? I suppose I’m a theory builder or maybe a conjecture maker. I’m not a conjecture prover very much, but I don’t know. It’s true that I’m not good at solving problems. For example, I would never be good in the Math Olympiad. There speed counts and I am certainly not a speedy worker. That’s one pleasant thing in mathematics: it doesn’t matter how long it takes if the end result is a good theorem. Speed is an advantage, but it is not essential.

But you are persistent. You have the energy to stay with a problem. At least, I did at one time.

May we ask you a question that we, in various ways, have asked almost everybody in previous interviews? Looking back on how you came up with new concepts, or made a breakthrough in an area you had been working on for some time, did that usually happen when you were concentrated and worked intensely on the problem or did it happen in a more relaxed situation? Do you have concrete examples? The first thing I did after my thesis was the determination of the higher-dimensional cohomology groups in class field theory. I had been working on that for several months, off and on. This was at the time of the seminar after my thesis at Princeton. One evening I went to a party and had a few drinks. I came home after midnight and thought I would think a little about the problem. About one or two in the morning I saw how to do it!

So this was a “Poincaré-moment”? In a way. I think that, like him, I had put the work aside for a longer time when this happened. I remember what it was: I had been invited to give some talks at MIT on class field theory and I thought “what am I going to say?” So it was after a party, motivated by needing something to say at MIT that this idea struck me. It was very fortunate. But it varies. Sometimes I’ve had an idea after talking to someone, and had the impression the person I was talking to had the idea and told me about it. The PhD thesis of my student Jonathan Lubin was on what should be called the Lubin groups. They somehow have been called the Lubin–Tate groups. Incidentally, I think it’s useful in math that theorems or ideas have two names so you can identify them. If I say Serre’s Theorem, my God, that doesn’t say too much. But anyway, they are called Lubin-Tate groups and it occurred to me, just out of the blue, that they might be useful in class field theory. And then we worked it out and indeed they were. One John Tate 137 gets ideas in different ways and it’s a wonderful feeling for a few minutes, but then there is a let-down, after you get used to the idea.

Group cohomology had been studied in various guises, long before the notion of group cohomology was formulated in the 1940s. You invented what is called Tate cohomology groups, which are widely used in class field theory, for instance. Could you elaborate? In connection with class field theory it suddenly dawned on me that if the group is finite – the operating group G – then one could view the homology theory of that group as negative dimensional cohomology. Usually the homology and the cohomology are defined in non-negative dimensions, but suddenly it became clear to me that for a finite group you could glue the two theories together. The i’th homology group can be viewed as a (1-i)’th cohomology group and then you can glue these two sequences together so that the cohomology goes off to plus infinity and the homology goes off, with renumbering, to minus infinity and you fiddle a little with the joining point and then you have one theory going from negative infinity to plus infinity.

Was this a flash of insight? Perhaps. There was a clue from the finite cyclic case, where there is periodicity; a periodicity of length two. For example, H 3 is isomorphic to H 1, the H 5 is isomorphic to H 3, etc., and it’s obvious that you could go on to infinity in both directions. Somehow it occurred to me that one could do that for an arbitrary finite group. I don’t remember exactly how it happened in my head.

The roles of mathematics

Can we speculate a little about the future development of mathematics? When the Clay Millennium Prizes for solving outstanding problems in mathematics were established back in the year 2000, you presented three of these problems to the mathematical public. Not necessarily restricting to those, would you venture a guess about new trends in mathematics: the twenty-first century compared to the twentieth century? Are there trends that are entirely new? What developments can we expect in mathematics and particularly in your own field, number theory? We certainly have plenty of problems to work on. One big difference in mathematics generally is the advent of high-speed computers. Even for pure math, that will increase the experimental possibilities enormously. It has been said that number theory is an experimental science and until recently that meant experimenting by looking at examples by hand and discovering patterns that way. Now we have a zil- lion-fold more powerful way to do that, which may very well lead to new ideas even in pure math, but certainly also for applications. Mathematics somehow swings between the development of new abstract theories and the application of these to more concrete problems and from concrete 138 Abel Prize 2010 problems to theories needed to solve them. The pendulum swings. When I was young better foundations were being developed, things were becoming more func- torial, if you will, and a very abstract point of view led to much progress. But then the pendulum swung the other way to more concrete things in the 1970s and 1980s. There were modular forms and the Langlands program, the proof of the Mordell conjecture, and of Fermat’s last theorem. In the first half of my career, theoretical physics and mathematics were not so close. There was the time when the development of mathematics went in the abstract direction, and the physicists were stuck. But now in the last thirty years they have come together. It is hard to tell whether string theory is math or physics. And non-commutative geometry has both sides. Who knows what the future will be? I don’t think I can contribute much in answering that question. Maybe a younger person would have a better idea.

Are you just as interested in mathematics now as you were when you were young? Well, not as intensely. I’m certainly still very much interested, but I don’t have the energy to really go so deeply into things.

But you try to follow what is happening in your field? Yes, I try. I’m in awe of what people are doing today.

Your teacher Emil Artin, when asked about whether mathematics was a science would rather say: “No. It’s an art.” On the other hand, mathematics is connected to the natural sciences, to computing and so on. Perhaps it has become more important in other fields than ever; the mutual interaction between science and engineering on one side and mathematics on the other has become more visible. Is mathematics an art, is it more to be applied in science or is it both? It’s both, for heaven’s sake! I think Artin simply was trying to make a point that there certainly is an artistic aspect to mathematics. It’s just beautiful. Unfortunately it’s only beautiful to the initiated, to the people who do it. It can’t really be understood or appreciated much on a popular level the way music can. You don’t have to be a composer to enjoy music, but in mathematics you do. That’s a really big drawback of the profession. A non-mathematician has to make a big effort to appreciate our work; it’s almost impossible. Yes, it’s both. Mathematics is an art, but there are stricter rules than in other arts. Theorems must be proved as well as formulated; words must have precise meanings. The happy thing is that mathematics does have applications which ena- ble us to earn a good living doing what we would do even if we weren’t paid for it. We are paid mainly to teach the useful stuff.

Public awareness of mathematics

Have you tried to popularize mathematics yourself? John Tate 139

When I was young I tried to share my enthusiasm with friends, but I soon realized that’s almost impossible.

We all feel the difficulty communicating with the general audience. This interview is one of the rare occasions providing public attention for mathematics! 2 Do you have any ideas about how From left to right: Martin Raussen, Christian Skau and mathematicians can make John Tate. (Photo: Eirik Furu Baardsen) themselves and what they do more well-known? How can we increase the esteem of mathematics among the general public and among politicians? Well, I think prizes like this do some good in that respect. And the Clay Prizes likewise. They give publicity to mathematics. At least people are aware. I think the appreciation of science in general and mathematics in particular varies with the country. What fraction of the people in Norway would you say have an idea about Abel?

Almost everyone in Norway knows about Abel, but they do not know anything about Lie [1842–1899]. And not necessarily anything about Abel’s work, either. They may know about the quintic. I see. And how about Sylow [1832–1918]?

He is not known either. Abel’s portrait has appeared on stamps and also on bills, but neither Lie’s nor Sylow’s. I think in Japan, people are more aware. I once was in Japan and eating alone. A Japa- nese couple came and wanted to practise their English. They asked me what I did. I said I was a mathematician, but could not get the idea across until I said: “Like Hironaka.” Wow! It’s as though in America I’d said “Like Babe Ruth” [1895–1948], or Michael Jor- dan, or Tiger Woods. Perhaps Hironaka’s name is, like Abel’s, the only one known, but in America I don’t think any mathematician’s name would get any response.

Private interests

Our last question: what other interests do you have in life? What are you occupied with when you are not thinking about mathematics? Certainly that happens once in a while, as well?

2 The interview was broadcast on Norwegian television. See http://www.abelprisen.no/en/mul- timedia/. 140 Abel Prize 2010

I’m certainly not a Renaissance man. I don’t have wide knowledge or interests. I have enjoyed very much the outdoors, hiking and also sports. Basketball was my favourite sport. I played on the Southeast Methodist church team as a teenager and we won the Minneapolis church league championship one year. There were several of us who went to church three out of four Sundays during a certain period in the winter, in order to play on the team. In the Navy I coached a team from the mine- sweeping research base which beat Coca-cola for the Panama City league championship. Anyway, I have enjoyed sport and the outdoors. I like to read a reasonable amount and I enjoy music, but I don’t have a really deep or serious hobby. I think I’m more concentrated in mathematics than many people. My feeling is that to do some mathematics I just have to concentrate. I don’t have the kind of mind that absorbs things very easily.

We would like to thank you very much for this interview, as well as on behalf of the Norwegian, Danish and European mathematical societies. Thank you very much! Well, thank you for not asking more difficult questions! I have enjoyed talking with you.

Abel Laureate John Tate taking part in mathematical games together with local schoolchildren at the University of Agder. (Photo: Tor Martin Lien) Abel Prize 2011: John Milnor

The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2011 to

John Milnor Institute for Mathematical Sciences, Stony Brook University, New York, USA

“for pioneering discoveries in topology, geometry and algebra.”

All of Milnor’s works display marks of great research: profound insights, vivid imagination, elements of surprise, and supreme beauty. Milnor’s discovery of exotic smooth spheres in seven dimensions was completely unexpected. It signaled the arrival of differential topology and an explosion of work by a generation of brilliant mathematicians; this explosion has lasted for decades and changed the landscape of mathematics. With Michel Kervaire, Milnor went on to give a complete inventory of all the distinct differentiable structures on spheres of all dimensions; in particular they showed that the 7-dimensional sphere carries exactly 28 distinct differentiable structures. They were among the first to identify the special Abel Laureate John Milnor. nature of four-dimensional manifolds, fore- (Photo: Knut Falch) shadowing fundamental developments in topology. Milnor’s disproof of the long-standing Hauptvermutung overturned expectations about combinatorial topology dating back to Poincaré. Milnor also discovered homeomorphic smooth manifolds with nonisomorphic tangent bundles, for which he developed the theory of microbundles. In three-manifold theory, he proved an elegant unique factorization theorem.

* 20.2.1931

141 142 Abel Prize 2011

Outside topology, Milnor made significant contributions to differential geometry, algebra, and dynamical systems. In each area Milnor touched upon, his insights and approaches have had a profound impact on subsequent developments. His monograph on isolated hypersurface singularities is considered the single most influential work in singularity theory; it gave us the Milnor number and the Milnor fibration. Topologists started to actively use Hopf algebras and coalgebras after the definitive work by Milnor and J. C. Moore. Milnor himself came up with new insights into the structure of the Steenrod algebra (of cohomology operations) using the theory of Hopf algebras. In algebraic K-theory, Milnor introduced the degree two functor; his celebrated conjecture about the functor — eventually proved by Voevodsky — spurred new directions in the study of motives in algebraic geometry. Milnor’s introduction of the growth invariant of a group linked combinatorial group theory to geometry, prefigur- ing Gromov’s theory of hyperbolic groups. More recently, John Milnor turned his attention to dynamical systems in low dimensions. With Thurston, he pioneered ”kneading theory” for interval maps, laying down the combinatorial foundations of interval dynamics, creating a focus of intense research for three decades. The Milnor−Thurston conjecture on entropy monotonicity prompted efforts to fully understand dynamics in the real quadratic family, bridging real and complex dynamics in a deep way and triggering exciting advances. Milnor is a wonderfully gifted expositor of sophisticated mathematics. He has often tackled difficult, cutting-edge subjects, where no account in book form existed. Adding novel insights, he produced a stream of timely yet lasting works of masterly lucidity. Like an inspired musical composer who is also a charismatic performer, John Milnor is both a discoverer and an expositor.

Abel Laureate John Milnor with his wife Dusa McDuff, King Harald of Norway and Tora Aasland, the Norwegian Minister of Education and Research. (Photo: Kyrre Lien) John Milnor 143

Professor John Milnor – on behalf of the Norwegian and Danish Mathematical Societies, we would like to congratulate you for being selected as the Abel Prize Laureate in 2011. Thank you very much!

Student at Princeton University

What kindled your interest in mathematics and when did you discover that you had an extraordinary aptitude for mathematics? I can place that quite clearly. The first time that I developed a particular interest in mathematics was as a freshman at Princeton University. I had been rather socially maladjusted and did not have too many friends but when I came to Princeton, I found myself very much at home in the atmosphere of the mathematics common room. People were chatting about mathematics, playing games and one could come by at any time and just relax. I found the lectures very interesting. I felt more at home there than I ever had before and I have stayed with mathematics ever since.

You were named a Putnam Fellow as one of the top scorers of the Putnam competition in mathematics in 1949 and 1950. Did you like solving mathematics problems and puzzles? I think I always approached mathematics as interesting problems to be solved so I certainly found that congenial.

Your first important paper was accepted already in 1949 and published in 1950 in the prestigious journal Annals of Mathematics. You were only 18 years of age at the time and this is rather exceptional. The title of the paper was “On the Total Curvature of Knots”. Could you tell us how you got the idea for that paper? I was taking a course in differential geometry under Albert Tucker (1905–1995]. We learned that Werner Fenchel [1905–1988], and later Karol Borsuk [1905–1982], had proved the following statement: the total curvature of a closed curve in space is always at least 2p with equality only if the curve bounds a convex subset of some plane. Borsuk, a famous Polish topologist, had asked what one could say about total curvature if the curve was knotted? I thought about this for a few days and came up with a proof that the total curvature is always greater than 4p. (I think I did a poor job explaining the proof in the published paper but one has to learn how to explain mathematics.) The Hungarian mathematician István Fáry [1922–1984] had produced a similar proof at more or less the same time; but this was still a wonderful introduction to mathematics.

That was quite an achievement! When you started your studies at Princeton in 1948 you met John Nash [1928–2015], three years your senior, who was a PhD student. John Nash is well-known through the book and movie ‘A Beautiful 144 Abel Prize 2011

Mind’. Did you have any interaction with him? And how was it to be a Prince- ton student? As I said, I spent a great deal of time in the common room, and so did Nash. He was a very interesting character and full of ideas. He also used to wander in the cor- ridors whistling things like Bach which I had never really heard before – a strange way to be introduced to classical music! I saw quite a bit of him over those years and I also became interested in game theory in which he was an important contributor. He was a very interesting person.

You played Kriegspiel, Go and a game called Nash at Princeton? That is true. Kriegspiel is a game of chess in which the two players are back to back and do not see each other’s boards. There is a referee who tells whether the moves are legal or not. It is very easy for the referee to make a mistake and it often happened that we could not finish because he got confused. In that case we said that the referee won the game! It was a marvellous game. The game of Go was also very popular there. My first professor Ralph Fox [1913–1973] was an expert in Go. So I learned something of it from him and also from many other people who played. The game that we called Nash had actually been developed earlier in Denmark by Piet Hein [1905–1996] but Nash invented it independently. This game, also called Hex, is based on topology. It is very interesting from a mathematical point of view. It is not hard to prove that the first player will always win if he plays correctly but there is no constructive proof. In fact, when you play, it often happens that the first player does not win.

You even published some papers on game theory with John Nash? We often talked about game theory but there was only one joint paper. Together with G. Kalish and E. D. Nering, we carried out an experiment with a group of people playing a many-person game. This experiment convinced me that many-person game theory is not just a subject of mathematics. It is also about social interactions and things far beyond mathematics so I lost my enthusiasm for studying it mathematically. One paper written on my own described a theoretical model for the game of Go. This was further developed by Olof Hanner [1922–2015], and much later by Berle kamp and Wolfe [1927–2016]. (John Conway’s construction of “surreal numbers” is closely related.)

Knot theory

You wrote your PhD thesis under the supervision of Ralph Fox; the title of the thesis was “Isotopy of Links”. Did you get the idea to work on this topic yourself? And what was the impact of this work? Fox was an expert in knot theory so I learned a great deal about knots and links from him. There were many people in the department then that were active in this area, although there were also other people at the department that considered it a low-class John Milnor 145 subject and not very interesting. I think it’s strange that, although it wasn’t considered a very central subject then, it’s today a subject which is very much alive and active. As one example, I often saw a quiet, Greek gentleman Christos Papakyriakopou- lus [1914–1976] around the common room but I never got to know him very well. I had no idea he was doing important work but Fox had managed to find money to support him for many years, while he did research more or less by himself. He finally succeeded in solving a very important problem in knot theory which, perhaps, was the beginning of a rebirth of the study of three dimensional manifolds as a serious part of mathematics. A paper in 1910 by Max Dehn [1878–1952] had claimed to prove a simple property about knots. Essentially it said that if the fundamental group of the complement of a knot is cyclic then the knot can be un-knotted. This proof by Max Dehn had been accepted for almost 20 years until Hellmuth Kneser [1898–1973] in 1929 pointed out there was a big gap in the argument. This remained a famous unsolved problem until 1957, when Papakyriakopoulus developed completely new methods and managed to give a proof of “Dehn’s Lemma” and related theorems. That was a big step in mathematics and an example of a case in which someone working in isolation made tremendous progress. There are relatively few examples of that. Andrew Wiles’ proof of Fermat’s last theorem is also an example of someone who had been working by himself and surprised everyone when he came up with the proof. Another example is Grigori Perelman in Russia who was working very much by himself and produced a proof of the Poincaré hypothesis. These are isolated examples. Usually mathematicians work in a much more social context, communicating ideas to each other. In fact, ideas often travel from country to country very rapidly. We are very fortunate that mathematics is usually totally divorced from political situations. Even at the height of the Cold War, we received information from the Soviet Union and people in the Soviet Union were eagerly reading papers from outside. Mathematics was much more open than most scientific subjects.

As a footnote to what you said: Max Dehn was a student of David Hilbert [1862–1943] and he solved Hilbert’s 3rd problem about three-dimensional polyhedra of equal volume, showing that you cannot always split them up into congruent polyhedra. No wonder people trusted his proof because of his name. It’s a cautionary tale because we tend to believe in mathematics that when something is proved, it stays proved. Cases like Dehn’s Lemma, where a false proof was accepted for many years, are very rare.

Manifolds

For several years after your PhD your research concentrated on the theory of manifolds. Could you explain what a manifold is and why manifolds are important? In low dimensions manifolds are things that are easily visualized. A curve in space is an example of a one-dimensional manifold; the surface of a sphere or of a doughnut 146 Abel Prize 2011 are examples of two-dimensional manifolds. But for mathematicians the dimensions one and two are just the beginning; things get more interesting in higher dimensions. Also, for physicists manifolds are very important and it is essential for them to look at higher dimensional examples. For example, suppose you study the motion of an airplane. To describe just the position takes three coordinates but then you want to describe what direction it is going in, the angle of its wings and so on. It takes three coordinates to describe the point in space where the plane is centred and three more coordinates to describe its orientation, so already you are in a six-dimensional space. As the plane is moving, you have a path in sixdimensional space and this is only the beginning of the theory. If you study the motion of the particles in a gas, there are enormously many particles bouncing around and each one has three coordinates describing its position and three coordinates describing its velocity, so a system of a thousand particles will have six thousand coordinates. Of course, much larger numbers occur; so mathematicians and physicists are used to working in large dimensional spaces.

The one result that made you immediately famous at age 25 was the discovery of different exotic structures on the seven-dimensional sphere. You exhibited smooth manifolds that are topologically equivalent to a seven-dimensional sphere but not smoothly equivalent, in a differentiable sense. Would you explain this result and also describe to us how you came up with the idea? It was a complete accident, and certainly startled me. I had been working on a project of understanding different kinds of manifolds from a topological point of view. In particular, I was looking at some examples of seven-dimensional manifolds which were constructed by a simple and well understood construction. They were explicit smooth objects which I would have thought were well understood but looking at them from two different points of views, I seemed to find a complete contradiction. One argument showed that these manifolds were topological spheres and another very different argument showed that they couldn’t be spheres. Mathematicians get very unhappy when they have apparently good proofs of two contradictory statements. It’s something that should never happen. The only way I could get out of this dilemma was by assuming there was an essential difference between the concept of a topological sphere (homeomorphic to the standard sphere) and the concept of a differentiable sphere (diffeomorphic to the standard sphere). This was something which hadn’t been expected and I am not aware that anybody had explicitly asked the question; we just assumed the answer was obvious. For some purposes one assumed only the topology and for other purposes one assumed the differentiable structure; but no one had really considered the possibility that there was a real difference. This result awakened a great deal of interest and a need for further research to understand exactly what was going on.

You were certainly the driving force in this research area and you applied techniques both from differential geometry and topology, and also from algebraic John Milnor 147 topology, to shed new light on manifolds. It is probably fair to say that the work of European mathematicians, and especially French mathematicians like René Thom and Jean-Pierre Serre, who, by the way, received the first Abel Prize in 2003, made very fundamental contributions and made your approach possible. How did the collaboration over the Atlantic work at the time? It was very easy to travel back and forth and I found French mathematicians very welcoming. I spent a great deal of time at the IHES near Paris. I hardly knew Serre (until much later) but I admired him tremendously, and still do. His work has had an enormous influence. René Thom I got to know much better. He was really marvellous. He had an amazing ability to combine geometric arguments with hard algebraic topology to come up with very surprising conclusions. I was a great admirer of Thom and found he was also extremely friendly.

Building on the work of, among others, Frank Adams [1930–1989] from Brit- ain and Stephen Smale from the United States, you, together with the French mathematician Michel Kervaire [1927–2007], were able to complete, to a certain extent, the classification of exotic structures on spheres. There are still some open questions concerning the stable homotopy of spheres but at least up to those, we know what differentiable structures can be found on spheres. That’s true, except for very major difficulties in dimension four, and a few problems in high dimensions (notably, the still unsolved “Kervaire Problem” in dimension 126). There are very classical arguments that work in dimensions one and two. Dimension three is already much more difficult but the work of Bill Thurston [1946–2012] and Grisha Perelman has more or less solved that problem. It was a tremendous surprise when we found, in the 60s, that high dimensions were easier to work with than low dimensions. Once you get to a high enough dimension, you have enough room to move around so that arguments become much simpler. In many cases, one can make such arguments work even in dimension five but dimension four is something else again and very difficult: neither high dimensional methods nor low dimensional methods work.

One seems to need much more hard pure analysis to work in dimensions three and four. Well, yes and no. Michael Freedman first proved the topological Poincaré hypothesis in dimension four and that was the very opposite of analysis. It was completely by methods of using very wild topological structures with no differentiability. But the real breakthrough in understanding differential 4-manifolds was completely based on methods from mathematical physics: methods of gauge theory, and later Seiberg–Witten theory. Although motivated by mathematical physics, these tools turned out to be enormously useful in pure mathematics.

Terminology in manifold theory is graphic and down to earth. Some techniques are known as ‘plumbing’. Also ‘surgery’ has become a real industry in mathe- 148 Abel Prize 2011 matics and you have written a paper on ‘killing’, but of course just homotopy groups. May we ask to what extent you are responsible for this terminology? To tell the truth, I’m not sure. I probably introduced the term ‘surgery’, meaning cutting up manifolds and gluing them together in a different way (the term ‘spher- ical modification’ is sometimes used for the same thing). Much later, the idea of quasi-conformal surgery has played an important role in holomorphic dynamics. Simple graphic terminology can be very useful but there are some words that get used so much that one loses track of what they mean (and they may also change their meaning over the years). Words like ‘regular’ or ‘smooth’ are very dangerous. There are very many important concepts in mathematics and it is important to have a terminology which makes it clear exactly what you are talking about. The use of proper names can be very useful because there are so many possible proper names. An appropriate proper name attached to a concept often pins it down more clearly than the use of everyday words. Terminology is very important; it can have a very good influence if it’s successfully used and can be very confusing if badly used.

Another surprising result from your hand was a counterexample to the so- called Hauptvermutung, the “main conjecture” in combinatorial topology, dating back to Steinitz [1871–1928] and Tietze [1880–1964] in 1908. It is concerned with triangulated manifolds or, more generally, triangulated spaces. Could you explain what you proved at the time? One of the important developments in topology in the early part of the 20th century was the concept of homology, and later cohomology. In some form, they were already introduced in the 19th century but there was a real problem making precise definitions. To make sense of them, people started by cutting a topological space up into linear pieces called simplexes. It was relatively easy to prove that homology was well defined on that level, and well behaved if you cut the simplexes into smaller ones, so the natural conjecture was that you really were doing topology when you defined things this way. If two simplicial complexes were homeomorphic to each other then you should be able to cut them up in pieces that corresponded to each other. This was the first attempt to prove that homology was topologically invariant; but nobody could quite make it work. Soon they developed better methods and got around the problem. But the old problem of the Hauptvermutung, showing that you could always find isomorphic subdivisions, remained open. I ran into an example where you could prove that it could not work. This was a rather pathological example, not about manifolds; but about ten years later, counterexamples were found even for nicely triangulated manifolds. A number of people worked on this but the ones who finally built a really satisfactory theory were Rob Kirby and my student Larry Siebenmann.

Over a long period of years after your thesis work, you published a paper almost every year, sometimes even several papers, that are known as landmark papers. They determined the direction of topology for many years ahead. This includes, apart from the themes we have already talked about, topics in knot John Milnor 149 theory, three dimensional manifolds, singularities of complex hypersurfaces, Milnor fibrations, Milnor numbers, complex cobordism and so on. There are also papers of a more algebraic flavour. Are there any particular papers or particular results you are most fond or proud of? It’s very hard for me to answer; I tend to concentrate on one subject at a time so that it takes some effort to remember precisely what I have done earlier.

Geometry, topology and algebra

Mathematics is traditionally divided into algebra, analysis and geometry/ topology. It is probably fair to say that your most spectacular results belong to geometry and topology. Can you tell us about your working style and your intuition? Do you think geometrically, so to say? Is visualization important for you? Very important! I definitely have a visual mind so it’s very hard for me to carry on a mathematical conversation without seeing anything written down.

On the other hand, it seems to be a general feature, at least when you move into higher dimensional topology, that real understanding arises when you find a suitable algebraic framework which allows you to formulate what you are thinking about. We often think by analogies. We have pictures in small dimensions and must try to decide how much of the picture remains accurate in higher dimensions and how much has to change. This visualization is very different from just manipulating a string of symbols.

Certainly, you have worked very hard on algebraic aspects of topology and also algebraic questions on their own. While you developed manifold theory, you wrote, at the same time, papers on Steenrod algebras, Hopf algebras and so on. It seems to us that you have an algebraic mind as well? One thing leads to another. If the answer to a purely topological problem clearly requires algebra then you are forced to learn some algebra. An example: in the study of manifolds one of the essential invariants – perhaps first studied by Henry Whitehead [1904–1960]– was the quadratic form of a four-dimensional manifold, or more generally a 4k-dimensional manifold. Trying to understand this, I had to look up the research on quadratic forms. I found this very difficult until I found a beautiful exposition by Jean-Pierre Serre which provided exactly what was needed. I then discovered that the theory of quadratic forms is an exciting field on its own. So just by following my nose, doing what came next, I started studying properties of quadratic forms. In these years, topological K-theory was also developed, for example by Michael Atiyah, and was very exciting. There were beginnings of algebraic analogues. Grothendieck [1928–2014] was one of the first. Hyman Bass developed a theory of algebraic K-theory and I pursued that a bit 150 Abel Prize 2011 further and discovered that there were relations between the theory of quadratic forms and algebraic K-theory. John Tate was very useful at that point, helping me work out how these things corresponded.

John Tate was last year’s Abel Prize winner, by the way. I made a very lucky guess at that point, conjecturing a general relationship between algebraic K-theory, quadratic forms and Galois cohomology. I had very limited evidence for this but it turned out to be true and much later was proved by Vladimir Voevodsky. It’s very easy to make guesses but it feels very good when they turn out to be correct.

That’s only one of the quite famous Milnor conjectures. Well, I also had conjectures that turned out to be false.

Algebraic K-theory is a topic you already mentioned and we guess your interest in that came through Whitehead groups and Whitehead torsion related to K1. That is certainly true.

It is quite obvious that this is instrumental in the theory of non-simply connected manifolds through the s-cobordism theorem. That must have aroused your interest in general algebraic K-theory where you invented what is called Milnor K-theory today. Dan Quillen [1940–2011] then came up with a competing or different version with a topological underpinning… Topological K-theory worked in all dimensions, using Bott periodicity properties, so it seemed there should be a corresponding algebraic theory. Hyman Bass had worked out a complete theory for K0 and K1 and I found an algebraic version of K2. Quillen, who died recently after a long illness, provided a satisfactory theory of Kn for all values of n. Quillen’s K2 was naturally isomorphic to my K2, although our motivations and expositions were different. I did construct a rather ad hoc definition for the higher Kn. This was in no sense a substitute for the Quillen K-theory. However, it did turn out to be very useful for certain problems so it has kept a separate identity.

Giving rise to motivic cohomology, right? Yes, but only in the sense that Voevodsky developed motivic cohomology in the process of proving conjectures which I had posed.

You introduced the concept of the growth function for a finitely presented group in a paper from 1968. Then you proved that the fundamental group of a negatively curved Riemannian manifold has exponential growth. This paved the way for a spectacular development in modern geometric group theory and eventually led to Gromov’s hyperbolic group theory. Gromov, by the way, received the Abel Prize two years ago. Could you tell us why you found this concept so important? John Milnor 151

I have been very much interested in the relation between the topology and the geometry of a manifold. Some classical theorems were well-known. For example, Preissman had proved that if the curvature of a complete manifold is strictly negative then any Abelian subgroup of the fundamental group must be cyclic. The growth function seemed to be a simple property of groups which would reflect the geometry in the fundamental group. I wasn’t the first to notice this. Albert Schwarz in Russia had done some similar work before me but I was perhaps better known and got much more publicity for the concept. I can bring in another former Abel Prize winner Jacques Tits, who proved what is now called the “Tits alternative” for finitely generated subgroups of algebraic groups. He proved that either there was a free subgroup or the group was virtually solvable. All the finitely generated groups I was able to construct had this property: either they contained a noncyclic free subgroup or else they contained a solvable subgroup of finite index. Such groups always have either polynomial growth or exponential growth. The problem of groups of intermediate growth remained unsolved for many years until Grigorchuk in Russia found examples of groups that had less than exponential growth but more than polynomial growth. It is always nice to ask interesting questions and find that people have interesting answers.

Dynamics

We jump in time to the last thirty years in which you have worked extensively on real and complex dynamics. Roughly speaking, this is the study of iterates of a continuous or holomorphic function and the associated orbits and stability behaviour. We are very interested to hear why you got interested in this area of mathematics? I first got interested under the influence of Bill Thurston, who himself got interested from the work of Robert May in mathematical ecology. Consider an isolated population of insects where the numbers may vary from year to year. If there get to be too many of these insects then they use up their resources and start to die off but if they are very few they will grow exponentially. So the curve which describes next year’s population as a function of this year’s will have positive slope if the population is small and negative slope if the population gets too big. This led to the study of dynamical properties of such “unimodal” functions. When you look at one year after another, you get a very chaotic looking set of population data. Bill Thurston had gotten very interested in this problem and explained some of his ideas to me. As frequently happened in my interactions with Bill, I first was very dubious and found it difficult to believe what he was telling me. He had a hard time convincing me but finally we wrote a paper together explaining it.

This was a seminal paper. The first version of this paper dates from around 1977. The manuscript circulated for many years before it was published in the 152 Abel Prize 2011

Springer Lecture Notes in 1988. You introduced a new basic invariant that you called the ‘kneading matrix’ and the associated ‘kneading determinant’. You proved a marvellous theorem connecting the kneading determinant with the zeta function associated to the map, which counts the periodic orbits. Browsing through the paper it seems to us that it must have been a delight to write it up. Your enthusiasm shines through! You said that the zeta function describes periodic orbits, which is true but it omits a great deal of history. Zeta functions were first made famous by Riemann’s [1826– 1866] zeta function (actually first studied by Euler [1707–1783]). Zeta functions are important in number theory but then people studying dynamics found that the same mathematical formalism was very useful for counting periodic orbits. The catalyst was André Weil [1906–1998] who studied an analogue of the Riemann zeta function for curves over a finite field, constructed by counting periodic orbits of the Frobenius involution. So there is a continuous history here from pure number theory, starting with Euler and Riemann, and then André Weil, to problems in dynamics in which one studies iterated mappings and counts how many periodic orbits there are. This is typical of something that makes mathematicians very happy: techniques that are invented in one subject turn out to be useful in a completely different subject.

You must have been surprised that the study of a continuous map from an interval into itself would lead to such deep results? Well, it was certainly a very enjoyable subject.

Your work with Bill Thurston has been compared to Poincaré’s [1854–1912] work on circle diffeomorphisms 100 years earlier which led to the qualitative theory of dynamical systems and had a tremendous impact on the subject.

Use of computers in mathematics

This leads to another question. There is a journal called Experimental Mathe- matics. The first volume appeared in 1992 and the first article was written by you. It dealt with iterates of a cubic polynomial. The article included quite a lot of computer graphics. You later published several papers in this journal. What is your view on computers in mathematics? I was fascinated by computers from the very beginning. At first one had to work with horrible punch cards. It was a great pain; but it has gotten easier and easier. Actually, the biggest impact of computers in mathematics has been just to make it easier to prepare manuscripts. I always have had a habit of rewriting over and over, so in the early days I drove the poor secretaries crazy. I would hand in messy longhand manuscripts. They would present a beautiful typescript. I would cross out this, change that and so on. It was very hard on them. It has been so much easier since one can edit manuscripts on the computer. John Milnor 153

Of course, computers also make it much easier to carry out numerical experiments. Such experiments are nothing new; Gauss [1777–1855] carried out many numerical experiments but it was very difficult at his time. Now it’s so much easier. In particular, in studying a difficult dynamical system it can be very helpful to run the system (or perhaps a simplified model of it) on a computer. Hopefully this will yield an accurate result. But it is dangerous. It is very hard to be sure that round-off errors by the computer, or other computing errors, haven’t produced a result which is not at all accurate. It becomes a kind of art to understand what the computer can do and what the limitations are but it is enormously helpful. You can get a fast idea of what you can expect from a dynamical system and then try to prove something about it using the computer result as an indication of what to expect. At least, that’s in the best case. There’s also the other case where all you can do is to obtain the computer results and hope that they are accurate.

In a sense, this mathematical discipline resembles what the physicists do when they plan their experiments, and when they draw conclusions from the results of their experiments… There is also the intermediate stage of a computer assisted proof where (at least if you believe there are no mistakes in the computer program or no faults in the hardware) you have a complete proof. But the assumption that there are no mistakes is a very important one. Enrico Bombieri had an experience with this. He was using a fancy new high-speed computer to make experiments in number theory. He found that in some cases the result just seemed wrong. He traced it back, and traced it back, and finally found that there was a wiring mistake in the hardware!

Do you have examples from your own experience where all experiments you have performed indicate that a certain conjecture must be true but you don’t have a way to prove it in the end? In my experience, computer experiments seldom indicate that something is definitely true. They often show only that any possible exception is very hard to find. If you verify a number theoretical property for numbers less than 1010, who knows what would happen for 1011? In dynamics, there may be examples where the behaviour changes very much as we go to higher dimensions. There is a fundamental dogma in dynamics, saying that we are not interested in events which happen with probability zero. But perhaps something happens with probability 10–10. In that case, you will never see it on a computer.

Textbooks and expository articles

You have written several textbooks which are legendary in the sense that they are lucid and lead the reader quickly to the point, seemingly in the shortest possible way. The topics of your books deal with differential topology, algebraic 154 Abel Prize 2011

K-theory, characteristic classes, quadratic forms and holomorphic dynamics. Your books are certainly enjoyable reading. Do you have a particular philosophy when you write mathematical textbooks? I think most textbooks I have written have arisen because I have tried to understand a subject. I mentioned before that I have a very visual memory and the only way I can be convinced that I understand something is to write it down clearly enough so that I can really understand it. I think the clarity of writing, to the extent it exists, is because I am a slow learner and have to write down many details to be sure that I’m right, and then keep revising until the argument is clear.

Apart from your textbooks and your research contributions, you have written many superb expository and survey articles which are a delight to read for every mathematician, expert or non-expert. Two questions come to mind. Do you enjoy writing articles of an historical survey type? You certainly have a knack for it. Do you think it is important that articles and books on mathematics of a popular and general nature are written by prominent mathematicians like yourself? The answer to your first question is certainly yes. Mathematics has a rich and interesting history. The answer to the second question is surely no. I don’t care who writes an article or a book. The issue is: is it clearly written, correct and useful.

Are you interested in the history of mathematics also – following how ideas develop? I certainly enjoy trying to track down just when and how the ideas that I work with originated. This is, of course, a very special kind of history, which may concentrate on obscure ideas which turned out to be important, while ignoring ideas which seemed much more important at the time. History to most scientists is the history of the ideas that worked. One tends to be rather bored by ideas that didn’t work. A more complete history would describe how ideas develop and would be interested in the false leads also. In this sense, the history I would write is very biased, trying to find out where the important ideas we have today came from – who first discovered them. I find that an interesting subject. It can be very difficult to understand old papers because terminology changes. For example, if an article written 100 years ago describes a function as being ‘regular’, it is hard to find out precisely what this means. It is always important to have definitions which are clearly written down so that, even if the terminology does change, people can still understand what you were saying.

Is it also important to communicate that to a wider mathematics audience? It is important to communicate what mathematics is and does to a wide audience. However, my own expositions have always been directed to readers who already have a strong interest in mathematics. In practice, I tend to write about what interests me, in the hope that others will also be interested. John Milnor 155

Academic work places

You started your career at Princeton University and you were on the staff for many years. After some intermediate stages in Los Angeles and at MIT, you went back to Princeton but now to the Institute for Advanced Study. Can you compare the Institute and the University and the connections between them? They are alike in some ways. They have close connections; people go back and forth all the time. The big difference is that at the university you have contin- ual contact with students, both in teaching and with the graduate students, and there is a fair amount of continuity since the students stay around, at least for a few years. The institute is much more peaceful, with more opportunity for work and more idyllic circumstances, but there is a continually rotating population, so almost before you get to know people, the year is over and they move on. So it’s unsatisfactory in that way. But they are both wonderful institutions and I was very happy at both.

In the late 80s you left for Stony Brook, to the State University of New York, where you got in contact with students again, as an academic teacher. Yes, that was certainly one strong motivation. I felt that the institute was a wonderful place to spend some years but for me it was, perhaps, not a good place to spend my life. I was too isolated, in a way. I think the contact with young people and students and having more continuity was important to me so I was happy to find a good position in Stony Brook. There were also domestic reasons: my wife was at Stony Brook and commuting back and forth, which worked very well until our son got old enough to talk. Then he started complaining loudly about it.

A colleague of mine and I had an interview with Atle Selberg in Princeton in 2005. He told us, incidentally, that he thought Milnor would never move from the institute because his office was so messy that just to clean it up would take a tremendous effort. But you moved in the end… I don’t know if the office ever got cleaned up. I think it was moved into boxes and stored in our garage.

Development of mathematics

Are there any mathematicians that you have met personally during your lifetime who have made a special, deep impression on you? There are many, of course. There were certainly the professors at Princeton. Ralph Fox, Norman Steenrod [1910–1971] and Emil Artin [1898–1962] all made a strong impression on me. Henry Whitehead, I remember, invited a group of very young topologists to Oxford. This was a wonderful experience for me when I was young. 156 Abel Prize 2011

I mentioned René Thom. More recently Adrien Douady [1935–2006] was a very important influence. He was an amazing person, always full of life and willing to talk about any mathematical subject. If you had a question and emailed him, you would always get an answer back within a day or so. These are the names that occur most prominently to me.

When we observe mathematics as a whole, it has changed during your lifetime. Mathematics has periods in which internal development is predom- inant and other periods where a lot of momentum comes more from other disciplines, like physics. What period are we in currently? What influences from the outside are important now and how would you judge future developments? I think the big mystery is how the relation between mathematics and biology will develop.

You mentioned ecology as an example. Yes, but that was a discussion of a very simplified mathematical model. It’s clear that most biological problems are so complex that you can never make a total mathematical model. This is part of the general problem in applied mathematics; most things that occur in the real world are very complicated. The art is to realise what the essential variables are, in order to construct a simplified model that can still say something about the actual more complex situation. There has recently been tremendous success in the understanding of large data sets (also in statistical analysis). This is not a kind of mathematics I have ever done but, nevertheless, it’s very important. The question of what kind of mathematics will be useful in biology is still up in the air, I think.

Work style

You have proved many results that are described as breakthroughs by mathematicians all around. May we ask you to recall some of the instances when an idea struck you that all of a sudden solved a problem you had been working on? Did that rather occur when you had been working on it very intensely or did it often happen in a relaxed atmosphere? Here is one scenario. After a lot of studying and worrying about a question, one night you go to sleep wondering what the answer is. When you wake up in the morning, you know the answer. That really can happen. The other more common possibility is that you sit at the desk working and finally something works out. Mathematical conversations are definitely very important. Talking to people, reading other people’s work and getting suggestions are usually very essential.

Talking, very often, makes ideas more clear. John Milnor 157

Yes, in both directions. If you are explaining something to someone else, it helps you understand it better. And certainly, if someone is explaining something to you, it can be very important.

Is the way you do mathematics today any different from how you did mathematics when you were 30 or 40? Probably, yes.

How many hours per day do you work on mathematics? I don’t know. I work a few hours in the morning, take a nap and then work a few hours in the afternoon. But it varies. When I was younger I probably worked longer hours.

Do you subscribe to Hardy [1877–1947] when he said that mathematics is a young man’s game? You seem to be a counterexample! What can you say? Whatever age, do the best you can!

In an article around 15 years ago, you described several areas in mathematics that you first had judged as of minor interest but which later on turned out to be fundamental to solve problems that you had been working on yourself. I think Michael Freedman’s work was one of the examples you mentioned. Do you have more examples and is there a general moral? I think that one of the joys about mathematics is that it doesn’t take an enormous grant and an enormous machine to carry it out. One person working alone can still make a big contribution. There are many possible approaches to most questions so I think it’s a big mistake to have everything concentrated in a few areas. The idea of having many people working independently is actually very useful because it may be that the good idea comes from a totally unexpected direction. This has happened often. I am very much of the opinion that mathematics should not be directed from above. People must be able to follow their own ideas.

This leads to a natural question: what is mathematics to you? What is the best part of being a mathematician? It is trying to understand things, trying to explain them to yourself and to others, to interchange ideas and watch how other people develop new ideas. There is so much going on that no one person can understand all of it; but you can admire other people’s work even if you don’t follow it in detail. I find it an exciting world to be in.

What’s the worst part of being a mathematician, if there is any? Is competition part of it? Competition can be very unpleasant if there are several people fighting for the same goal, especially if they don’t like each other. If the pressure is too great and if the reward for being the successful one is too large, it distorts the situation. I think, in 158 Abel Prize 2011 general, most mathematicians have a fair attitude. If two different groups produce more or less the same results more or less at the same time, one gives credit to everyone. I think it’s unfortunate to put too much emphasis on priority. On the other hand, if one person gets an idea and other people claim credit for it, that becomes very unpleasant. I think the situation in mathematics is much milder than in other fields, like biology where competition seems to be much more ferocious.

Do you have the same interest in mathematics now as you had when you were young? I think so, yes.

Prizes

You received the Fields Medal back in 1962, particularly for your work on manifolds. This happened in Stockholm at the International Congress and you were only 31 years old. The Fields Medal is the most important prize given to mathematicians, at least to those under the age of 40. The Abel Prize is relatively new and allows us to honour mathematicians regardless of age. Receiving the Fields Medal almost 50 years ago, do you remember what you felt at the time? How did receiving the Fields Medal influence your academic career? Well, as you say, it was very important. It was a recognition and I was certainly honoured by it. It was a marvellous experience going to Stockholm and receiving it. The primary motive is to understand mathematics and to work out ideas. It’s gratifying to receive such honours but I am not sure it had a direct effect.

Did you feel any extra pressure when you wrote papers after you received the Fields Medal? No, I think I continued more or less as before.

You have won a lot of prizes throughout your career: the Fields Medal, the Wolf Prize and the three Steele Prizes given by the American Mathematical Society. And now you will receive the Abel Prize. What do you feel about getting this prize on top of all the other distinctions you have gotten already? It is surely the most important one. It is always nice to be recognised for what you have done; but this is an especially gratifying occasion.

What do you generally feel about prizes to scientists as a means of raising public awareness? It is certainly very successful at that. I’m not sure I like getting so much attention but it doesn’t do me much harm. If this is a way of bringing attention to mathematics, I’m all in favour. The danger of large prizes is that they will lead to the situations I described in biology. The competition can become so intense, it becomes poison- ous; but I hope that will never happen in mathematics. John Milnor 159

Personal interests

Having talked about mathematics all the time, may we finish this interview by asking about other things you are interested in: your hobbies, etc? I suppose I like to relax by reading science-fiction or other silly novels. I certainly used to love mountain climbing, although I was never an expert. I have also enjoyed ski- From left to right: Martin Raussen, Christian Skau and ing. Again I was not an expert John Milnor. (Photo: Eirik Furu Baardsen) but it was something I enjoyed doing… I didn’t manage it this winter but I hope I will be able to take up skiing again.

What about literature or music? I enjoy music but I don’t have a refined musical ear or a talent for it. I certainly enjoy reading although, as I said, I tend to read non-serious things for relaxation more than trying to read serious things. I find that working on mathematics is hard enough without trying to be an expert in everything else.

We would like to thank you very much for this most interesting interview. This is, of course, on the behalf of the two of us but also on behalf of the Danish, Norwegian and the European Mathematical Societies. Thank you very much!

Abel Laureate John Milnor and Ragni Piene, chair of the Abel committee, at the award ceremony in Oslo. (Photo: Kyrre Lien)

Abel Prize 2012: Endre Szemerédi

The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2012 to

Endre Szemerédi Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Buda- pest, and Department of Computer Science, Rutgers, The State University of New Jersey, USA

“for his fundamental contributions to discrete mathematics and theoretical computer science, and in recognition of the profound and lasting impact of these contributions on additive number theory and ergodic theory.”

Discrete mathematics is the study of structures such as graphs, sequences, permutations, and geometric configurations. The mathematics of such structures forms the foundation of theoretical computer science and information theory. For instance, communication networks such as the internet can be described and analyzed using the tools of graph theory, and the design of efficient computational algorithms relies crucially on insights from discrete mathematics. The Abel Laureate Endre Szemerédi. combinatorics of discrete structures is also (Photo: Knut Falch) a major component of many areas of pure mathematics, including number theory, probability, algebra, geometry, and analysis. Endre Szemerédi has revolutionized discrete mathematics by introducing ingenious and novel techniques, and by solving many fundamental problems. His work has brought combinatorics to the center-stage of mathematics, by revealing its deep connections to such fields as additive number theory, ergodic theory, theoretical computer science, and incidence geometry.

* 21.8.1940

161 162 Abel Prize 2012

In 1975, Endre Szemerédi first attracted the attention of many mathematicians with his solution of the famous Erdős–Turán conjecture, showing that in any set of integers with positive density, there are arbitrarily long arithmetic progressions. This was a surprise, since even the case of progressions of lengths 3 or 4 had earlier required substantial effort, by Klaus Roth and by Szemerédi himself, respectively. A bigger surprise lay ahead. Szemerédi’s proof was a masterpiece of combinatorial reasoning, and was immediately recognized to be of exceptional depth and importance. A key step in the proof, now known as the Szemerédi Regularity Lemma, is a structural classification of large graphs. Over time, this lemma has become a central tool of both graph theory and theoretical computer science, leading to the solution of major problems in property testing, and giving rise to the theory of graph limits. Still other surprises lay in wait. Beyond its impact on discrete mathematics and additive number theory, Szemerédi’s theorem inspired Hillel Furstenberg to develop ergodic theory in new directions. Furstenberg gave a new proof of Szemerédi’s theorem by establishing the Multiple Recurrence Theorem in ergodic theory, thereby unexpectedly linking questions in discrete mathematics to the theory of dynamical systems. This fundamental connection led to many further developments, such as the Green-Tao theorem asserting that there are arbitrarily long arithmetic progressions of prime numbers. Szemerédi has made many additional deep, important, and influential contributions to both discrete mathematics and theoretical computer science. Examples in discrete mathematics include the Szemerédi–Trotter theorem, the Ajtai–Komlós–Sze- merédi semi-random method, the Erdős–Szemerédi sum-product theorem, and the Balog–Szemerédi–Gowers lemma. Examples in theoretical computer science include the Ajtai–Komlós–Szemerédi sorting network, the Fredman–Komlós–Szemerédi hashing scheme, and the Paul–Pippenger–Szemerédi–Trotter theorem separating deterministic and non-deterministic linear time. Szemerédi’s approach to mathematics exemplifies the strong Hungarian problem-solving tradition. Yet, the theoretical impact of his work has been a game-changer.

Abel Laureate Endre Szemerédi with his Majesty King Harald of Norway. (Photo: Erlend Aas) Endre Szemerédi 163

Professor Szemerédi, first of all we would like to congratulate you as the 10th Abel Prize recipient! You will receive the prize tomorrow from His Majesty, the King of Norway.

Youth

You were born in Budapest, Hungary, in 1940 during the Second World War. We have heard that you did not start out studying mathematics; instead, you started in medical school and only later on shifted to mathematics. Were you nevertheless interested in mathematical problems as a child or teenager? Did you like to solve puzzles? I have always liked mathematics and it actually helped me to survive in a way: When I was in elementary school, I was very short and weak and the stronger guys would beat me up. So I had to find somebody to protect me. I was kind of lucky, since the strongest guy in the class did not understand anything about mathematics. He could never solve the homework exercises, let alone pass the exam. So I solved the homework exercises for him and I sat next to him at the exam. Of course, we cheated and he passed the exam. But he was an honest person and he always protected me afterwards from the other big guys; so I was safe. Hence my early interest in mathematics was driven more by necessity and self-interest than by anything else. In elementary school I worked a lot with mathematics but only on that level, solving elementary school exercises. In high school, I was good at mathematics. However, I did not really work on specific problems and, if I remember correctly, I never took part in any competitions. In Hungary there are different kinds of competitions. There is also a monthly journal KöMaL, where you may send in solutions to problems that are posed. At the end of the year the editors will add up points you get for good solutions. I never took part in this, the main reason being that my father wanted me to be a physician. At the time, this was the most recognised profession, prestigiously and also financially. So I studied mainly biology and some physics but I always liked mathematics. It was not hard for me to solve high school exercises and to pass the exams. I even helped others, sometimes in an illegal way, but I did not do more mathematics than that. My education was not the usual education you get in Hungary if you want to be a mathematician. In Hungary we have two or three extremely good elite high schools. The best is Fazekas, in Budapest; they produce every year about five to ten mathematicians who, by the time they go to the university, know a lot. I was not among those. This is not a particular Hungarian invention; also in the US, there are special schools concentrating on one subject. I can name a lot of mathematicians that are now considered to be the best ones in Hungary. Most of them (90%) finished the school at Fazekas. In Szeged, which is a town with about 200,000 inhabitants, there are two specialist schools also producing some really good mathematicians. One of those mathematicians was a student of Bourgain at the Institute for Advanced Study in Princeton, who just recently de- 164 Abel Prize 2012 fended his thesis with a stunning result. But again, I was not among those highly educated high school students.

Is it correct that you started to study mathematics at age 22? Well, it depends on how you define “started”. I dropped out of medical school after half a year. I realised that, for several reasons, it was not for me. Instead I started to work at a machine-making factory, which actually was a very good experience. I worked there slightly less than two years. In high school my good friend Gábor Ellmann was by far the best mathematician. Perhaps it is not proper to say this in this kind of interview but he was tall. I was very short in high school – at least until I was seventeen. I am not tall now but at the time I was really short and that actually has its disadvantages. I do not want to elaborate. So I admired him very much because of his mathematical ability and also because he was tall. It was actually quite a coincidence that I met him in the centre of the town. He was to date a girlfriend but he was 15 minutes late so she had left. He was standing there and I ran into him and he asked me what I was doing. Gábor encouraged me to go to Eötvös University and he also told me that our mathematics teacher at high school, Sándor Bende, agreed with his suggestion. As always, I took his advice; this was really the reason why I went to university. Looking back, I have tried to find some other reason but so far I have not been successful. At that time in Hungary you studied mathematics and physics for two years, and then one could continue to study physics, mathematics and pedagogy for three years in order to become a maths-physics teacher. After the third year they would choose 15 out of about 200 students who would specialise in mathematics.

Turán and Erdo˝s

We heard that Paul Turán was the first professor in mathematics that made a lasting impression on you. That’s true. In my second year he gave a full-year lecture on number theory which included elementary number theory, a little bit of analytic number theory and algebraic number theory. His lectures were perfect. Somehow he could speak to all different kinds of students, from the less good ones to the good ones. I was so impressed with these lectures that I decided I would like to be a mathematician. Up to that point I was not sure that I would choose this profession, so I consider Paul Turán to be the one who actually helped me to decide to become a mathematician. He is still one of my icons. I have never worked with him; I have only listened to his lectures and sometimes I went to his seminars. I was not a number theorist and he mainly worked in analytic number theory.

By the way, Turán visited the Institute for Advanced Study in Princeton in 1948 and he became a very good friend of the Norwegian mathematician Atle Selberg. Endre Szemerédi 165

Yes, that is known in Hungary among the circle of mathematicians.

May we ask what other professors at the university in Budapest were important for you; which of them did you collaborate with later on? Before the Second World War, Hungarian mathematics was very closely connected to German mathematics. The Riesz brothers [1880–1956, 1886–1969], as well as Haar [1885–1933] and von Neumann [1903–1957] and many others actually went to Germany after they graduated from very good high schools in Hungary. Actually, my wife Anna’s father studied there almost at the same time as von Neumann and, I guess, the physicist Wigner [1902–1995]. After having finished high school he, and also others, went to Germany. And after having finished university education in Germany, most of them went to the US. I don’t know the exact story but this is more or less the case. After the Second World War, we were somehow cut off from Germany. We then had more connections with Russian mathematics. In the late 50s, Paul Erdős [1913–1996], the leading mathematician in discrete mathematics and combinatorics – actually, even in probability theory he did very good and famous work – started to visit Hungary, where his mother lived. We met quite often. He was a specialist in combinatorics. At the time combinatorics had the reputation that you didn’t have to know too much. You just had to sit down and meditate on a problem. Erdős was outstanding in posing good problems. Well, of course, as it happens to most people he sometimes posed questions which were not so interesting. But many of the problems he posed, after being solved, had reper- cussions in other parts of mathematics – also in continuous mathematics, in fact. In that sense Paul Erdős was the most influential mathematician for me, at least in my early mathematical career. We had quite a lot of joint papers.

Twenty-nine joint papers, according to Wikipedia… Maybe, I’m not sure. In the beginning I almost exclusively worked with Paul Erdős. He definitely had a lasting influence on my mathematical thinking and mathematical work.

Was it usually Erdős who posed the problems or was there an interaction from the very start? It was not only with me, it was with everybody. It was usually he who came up with the problems and others would work on them. Probably for many he is considered to be the greatest mathematician in that sense. He posed the most important problems in discrete mathematics which actually affected many other areas in mathematics. Even if he didn’t foresee that solving a particular problem would have some effect on something else, he had a very good taste for problems. Not only the solution but actually the methods used to obtain the solution often survived the problem itself and were applied in many other areas of mathematics.

Random methods, for instance? Yes, he was instrumental in introducing and popularising random methods. Actu- ally, it is debatable who invented random methods. The Hungarian mathematician 166 Abel Prize 2012

Szele [1918–1955] used the so-called random method – it was not a method yet – to solve a problem. It was not a deterministic solution. But then Paul Erdős had a great breakthrough result when he gave a bound on the Ramsey number, still the central problem in Ramsey Theory. After that work there has been no real progress. A little bit, yes, but nothing really spectacular. Erdős solved the problem using random methods. Specifically, he proved that by 2-colouring the edges of a complete graph with n vertices randomly, then almost certainly there will not be more than 2log n vertices so that all the connecting edges are of the same colour. In the US, where I usually teach undergraduate courses, I present that solution. The audience is quite diverse; many of them do not understand the solution. But the solution is actually simple and the good students do understand it. We all know it is extremely important – not only the solution but the method. Then Erdős systemat- ically started to use random methods. To that point they just provided a solution for a famous problem but then he started to apply random methods to many problems, even deterministic ones. And, of course, his collaboration with Rényi on the random graph is a milestone in mathematics; it started almost everything in random graph theory.

And that happened around 1960? Yes. It was in the 60s and it is considered to be the most influential paper in random graph theory. Their way of thinking and their methods are presently of great help for many, many mathematicians who work on determining the properties of real- life, large-scale networks and to find random methods that yield a good model for real-life networks.

Moscow: Gelfond and Gelfand

You did your graduate work in Moscow in the period 1967–1970 with the eminent mathematician Israel Gelfand as your supervisor. He was not a specialist in combinatorics. Rumours would have it that you, in fact, intended to study with another Russian mathematician, Alexander Gelfond, who was a famous number theorist. How did this happen and whom did you actually end up working with in Moscow? This can be taken, depending how you look at it, as a joke or it can be taken seriously. As I have already told you, I was influenced by Paul Turán, who worked in analytic number theory. He was an analyst; his mathematics was much more concrete than what Gelfand and the group around him studied. At the time, this group consisted of Kazhdan, Margulis, Manin, Arnold and others, and he had his famous Gelfand seminar every week that lasted for hours. It was very frightening sitting there and not understanding anything. My education was not within this area at all. I usually had worked with Erdős on elementary problems, mainly within graph theory and combinatorics; it was very hard for me! Endre Szemerédi 167

I wanted to study with Gelfond but by some unfortunate misspelling of the name I ended up with Gelfand. That is the truth.

But why couldn’t you swap when you realised that you had got it wrong? I will try to explain. I was a so-called candidate student. That meant that you were sent to Moscow – or to Warsaw for that matter – for three years. It had already been decided who would be your supervisor and the system was quite rigid, though not entirely. I’m pretty sure that if you put a lot of effort into it, you could change your supervisor, but it was not so easy. However, it was much worse if you decided after half a year that it was not the right option for you, and to go home. It was quite a shameful thing to just give up. You had passed the exams in Hungary and kind of promised you were going to work hard for the next three years. I realised immediately that this was not for me and Gelfand also realised it and advised me not to do mathematics anymore, telling me: “Just try to find another profession; there are plenty in the world where you may be successful.” I was 27 years old at the time and he had all these star students aged around 20; and 27 was considered old! But in a sense, I was lucky: I went to Moscow in the Fall of 1967 and, in the Spring next year, there was a conference on number theory in Hungary – in Debre- cen, not Budapest. I was assigned to Gelfond; it was customary that every guest had his own Hungarian guide. I had a special role too, because Gelfond was supposed to buy clothes and shoes which were hard to get in Russia at the time for his wife. So I was in the driving seat because I knew the shops pretty well.

You spoke Russian then? Well, my Russian was not that good. I don’t know if I should tell this in this interview but I failed the Russian exam twice. Somehow I managed to pass the final exam and I was sent to Russia. My Russian was good enough for shopping but not good enough for having more complex conversations. I only had to ask Gelfond for the size of the shoes he wanted for his wife and then I had a conversation in Hungarian with the shopkeepers. I usually don’t have good taste but because I had to rise to the occasion, so to say, I was very careful and thought about it a lot. Later Gelfond told me that his wife was very satisfied. He was very kind and said that he would arrange the switch of supervisors! This happened in the Spring of 1968 but unfortunately he died that summer of a heart attack, so I stayed with Gelfand for a little more than a year after that. I could have returned to Hungary but I didn’t want that; when I first agreed to study there, I felt I had to stay. They, i.e. Gelfand and the people around him, were very understanding when they realised that I would never learn what I was supposed to. Actually my exam consisted of two exercises about representation theory taken from Kirillov’s book, which they usually give to third-year students. I did it but there was an error in my solution. My supervisor was Bernstein, as you know a great mathematician and a very nice guy, too. He found the error in the solution but he 168 Abel Prize 2012 said that it was the effort that I had put into it that was important, rather than the result – and he let me pass the exam. To become a candidate you had to write a dissertation and Gelfand let me write one about combinatorics. This is what I did. So, in a way, I finished my study in Moscow rather successfully. I did not learn anything but I got the paper showing that I had become a candidate. At this time there was a hierarchy in Hungary: doctorate of the university, then candidate, doctorate of the academy, then corresponding member of the academy and then member of the academy. I achieved becoming a candidate of mathematics.

You had to work entirely on your own in Moscow? Yes, since I worked in combinatorics.

Gelfond must have realised that you were a good student. Did he communicate this to Gelfand in any way? That I don’t know. I only know that Gelfand very soon realised my lack of mathematical education. But when Gelfond came to Hungary, he talked to Turán and Erdős and also to Hungarian number theorists attending that meeting, and they were telling him: “Here is this guy who has a very limited background in mathematics.” This may be the reason why Gelfond agreed to take me as his student. But unfortunately he died early.

Hungarian mathematics

We would like to come back to Hungarian mathematics. Considering the Hun- garian population is only about ten million people, the list of famous Hungar- ian mathematicians is very impressive. To mention just a few, there is János Bolyai [1802–1860] in the 19th century, one of the fathers of non-Euclidean geometry. In the 20th century there is a long list, starting with the Riesz brothers, Frigyes and Marcel, Lipót Fejér [1880–1959], Gábor Szegő [1895–1985], Alfréd Haar, Tibor Radó [1895–1965], John von Neumann, perhaps the most ingenious of them all, Paul Turán, Paul Erdős, Alfréd Rényi, Raoul Bott [1923– 2005] (who left the country early but then became famous in the United States). Among those still alive, you have Peter Lax, who won the Abel Prize in 2005, Bela Bollobás, who is in Great Britain, László Lovász and now you. It’s all very impressive. You have already mentioned some facts that may explain the success of Hungarian mathematics. Could you elaborate, please? We definitely have a good system to produce elite mathematicians, and we have always had that. At the turn of the century – we are talking about the 19th century and the beginning of the 20th century – we had two or three absolutely outstanding schools, not only the so-called Fasori where von Neumann and Wigner studied but also others. We were able to produce a string of young mathematicians, some of whom later went abroad and became great mathematicians – or great physicists, Endre Szemerédi 169 for that matter. In that sense I think the educational system was extremely good. I don’t know whether the general education was that good but definitely for mathematics and theoretical physics it was extremely good. We had at least five top schools that concentrated on these two subjects; and that is already good enough to produce some great mathematicians and physicists. Back to the question of whether the Hungarians are really so good or not. Definitely, in discrete mathematics there was a golden period. This was mainly because of the influence of Erdős. He always travelled around the world but he spent also a lot of time in Hungary. Discrete mathematics was certainly the strongest group. The situation has changed now. Many Hungarian students go abroad to study at Princeton, Harvard, Oxford, Cambridge or Paris. Many of them stay abroad but many of them come home and start to build schools. Now we cover a much broader spectrum of mathematics, like algebraic geometry, differential geometry, low-dimensional topology and other subjects. In spite of being myself a mathematician working in discrete mathematics who practically doesn’t know anything about these subjects, I am very happy to see this development.

You mentioned the journal KöMaL that has been influential in promoting mathematics in Hungary. You told us that you were not personally engaged, but this journal was very important for the development of Hungarian mathematics; isn’t that true? You are absolutely right. This journal is meant for a wide audience. Every month the editors present problems, mainly from mathematics but also from physics. At least in my time, in the late 50s, it was distributed to every high school and a lot of the students worked on these problems. If you solved the problems regularly then by the time you finished high school you would almost know as much as the students in the elite high schools. The editors added the points you got from each correct solution at the end of the year, giving a bonus for elegant solutions. Of course, the winners were virtually always from one of these elite high schools. But it was intended for a much wider audience and it helped a lot of students, not only mathematicians. In particular, it also helped engineers. People may not know this but we have very good schools for different kinds of engineering, and a lot of engineering students-to-be actually solved these problems. They may not have been among the best but it helped them to develop a kind of critical thinking. You just don’t make a statement but you try to see connections and put them together to solve the problems. So by the time they went to engineering schools, which by itself required some knowledge of mathematics, they were already quite well educated in mathematics because of KöMaL. KöMaL plays an absolutely important role and, I would like to emphasise, not only in mathematics but more generally in natural sciences. Perhaps even students in the humanities are now working on these problems. I am happy for that and I would advise them to continue to do so (of course not to the full extent because they have many other things to study). 170 Abel Prize 2012

Important methods and results

We would now like to ask you some questions about your main contributions to mathematics. You have made some groundbreaking – and we don’t think that this ad- jective is an exaggeration – discoveries in combinatorics, graph theory and combinatorial number theory. But arguably, you are most famous for what is now called the Szemerédi theorem, the proof of the Erdős-Turán conjecture from 1936. Your proof is extremely complicated. The published proof is 47 pages long and it has been called a masterpiece of combinatorial reasoning. Could you explain first of all what the theorem says, the history behind it and why and when you got interested in it? Yes, I will start in a minute to explain what it is but I suspect that not too many people have read it. I will explain how I got to the problem. But first I want to tell how the whole story started. It started with the theorem of van der Waerden [1903–1996]: you fix two numbers, say five and three. Then you consider the integers up to a very large number, from 1 to n, say. Then you partition this set into five classes, and then there will always be a class containing a three-term arithmetic progression. That was a fundamental result of van der Waerden, of course not only with three and five but with general parameters. Later, Erdős and Turán meditated over this result. They thought that maybe the reason why there is an arithmetic progression is not the partition itself; if you partition into five classes then one class contains at least one fifth of all the numbers. They made the conjecture that what really counts is that you have dense enough sets. That was the Erdős-Turán conjecture: if your set is dense enough in the interval 1 to n – we are of course talking about integers – then it will contain a long arithmetic progression. Later Erdős formulated a very brave and much stronger conjecture: let’s consider an infinite sequence of positive integers,a 1 < a2 < … such that the sum of the inverses {1/ai} is divergent. Then the infinite sequence contains arbitrarily long arithmetic progressions. Of course, this would imply the absolutely fundamental result of Green and Tao about arbitrarily long arithmetic progressions within the primes because for the primes we know that the sum of the inverses is divergent. That was a very brave conjecture; it isn’t even solved for arithmetic progressions of length k = 3. But now, people have come very close to proving it: Tom Sanders proved that if we have a subset between 1 and n containing at least n over log n(log log n)5 elements then the subset contains a 3-term arithmetic progression. Unfortunately, we need a little bit more but we are getting close to solving Erdős’s problem for k = 3 in the near future, which will be a great achievement. If I’m not mistaken, Erdős offered 3,000 USD for the solution of the general case a long time ago. If you consider inflation, that means quite a lot of money.

Erdős offered 1,000 USD for the problem you solved, and that’s the highest sum he ever paid, right? Endre Szemerédi 171

Erdős offered $1,000 as well for a problem in graph theory that was solved by Vojtěch Rödl and Péter Frankl. These are the two problems I know about.

Let us get back to how you got interested in the problem. That was very close to the Gelfand/Gelfond story, at least in a sense. At least the message is the same: I overlooked facts. I tried to prove that if you have an arithmetic progression then it cannot happen that the squares are dense inside of it; specifically, it cannot be that a positive fraction of the elements of this arithmetic progression are squares. I was about 25 years old at the time and at the end of my university studies. At that time I already worked with Erdős. I very proudly showed him my proof because I thought it was my first real result. Then he pointed out two, well not errors but deficiencies in my proof. Firstly, I had assumed that it was known 1 that r4(n) = o(n) , i.e. that if you have a set of positive upper density then it has to contain an arithmetic progression of length four, or for that matter of any length. I assumed that that was a true statement. Then I used that there are no four squares that form an arithmetic progression. However, Erdős told me that the first statement was not known; it was an open problem. The other one was already known to Euler [1707–1783], which was 250 years before my time. So I had assumed something that is not known and, on the other hand, I had proved something that had been proven 250 years ago! The only way to try to correct something so embarrassing was to start working on the arithmetic progression problem. That was the time I started to work on r4(n) and, more generally, on rk(n). First I took a look at Klaus Roth’s [1925–2015] proof from 1953 of r3(n) being less than n divided by log log n . I came up with a very elementary proof for r3(n) = o(n) so that even high school students could understand it easily. That was the starting point. Later I proved also thatr 4(n) = o(n). Erdős arranged for me to be invited to Nottingham to give a talk on that result. But my English was virtually non-existent. Right now you can still judge that there is room for improvement of my English, but at the time it was almost non-existent. I gave a series of lectures; Peter Elliot and Eduard Wirsing, both extremely strong mathematicians, wrote a paper based almost entirely on my pictures on the blackboard. Perhaps they understood some easy words in English that I used. Anyway, they helped to write up the paper for me. A similar thing happened when I solved rk(n) = o(n) for general k. Then my good friend András Hajnal [1931–2016] helped me to write up the paper. That is actually an understatement. The truth is that he listened to my explanations and he then wrote up the paper. I am very grateful to Peter Elliot, Edward Wirsing and to my good friend András for their invaluable help.

When did all this happen? It was in 1973. The paper appeared in Acta Arithmetica in 1975. There is a controversial issue – well, maybe controversial is too strong a word – about the proof. It is

1 rk(n) denotes the proportion of elements between 1 and n that a subset must contain in order for it to contain an arithmetic progression of length k. 172 Abel Prize 2012 widely said that one of the main tools in the proof is the so-called regularity lemma, which is not true in my opinion. Well, everybody forgets about the proofs they produced 30 years ago. But I re-read my paper and I couldn’t find the regularity lemma. There occurs a lemma in the proof which is similar to the regularity lemma, so maybe that lemma, which is definitely not the regularity lemma, inspired me later to prove the regularity lemma. The real story is that I heard Bollobás’ lectures from 1974 about strengthen- ing the Erdős–Stone theorem. The Erdős–Stone theorem from the 40s was also a breakthrough result but I don’t want to explain it here. Then Bollobás and Erdős strengthened it. I listened to Bollobás’ lectures and tried to improve their result. Then it struck me that a kind of regularity lemma would come in handy and this led me to proving the regularity lemma. I am very grateful to Vasek Chvátal who helped me to write down the regularity paper. Slightly later the two of us gave a tight bound for the Erdős-Stone theorem.

We’ve seen that people refer to it in your proof of the Erdős-Turán conjecture as a weakened form of the regularity lemma. Yes, weaker; but similar in ideology, so to speak.

Connections to ergodic theory

Your proof of the Szemerédi Theorem is the beginning of a very exciting story. We have heard from a reliable source that Hillel Furstenberg at the Hebrew Univer- sity in Jerusalem first learned about your result when somebody gave a collo- quium talk there in December 1975 and mentioned your theorem. Following the talk, there was a discussion in which Furstenberg said that his weak mixing of all orders theorem, which he already knew, would prove the ergodic version of the Szemerédi Theorem in the weak mixing case. Since the Kronecker (or compact) case is trivial, one should be able to interpolate between them so as to get the full ergodic version. It took a couple of months for him to work out the details which became his famous multiple recurrence theorem in ergodic theory. We find it very amazing that the Szemerédi Theorem and Furstenberg’s Multiple Recurrence Theorem are equivalent, in the sense that one can deduce one theorem from the other. We guess it is not off the mark to say that Fursten- berg’s proof gave a conceptual framework for your theorem. What are your comments? As opposed to me, Furstenberg is an educated mathematician. He is a great mathematician and he already had great results in ergodic theory; he knew a lot. He proved that a measure-preserving system has a multiple recurrence property; this is a far-reaching generalisation of a classical result by Poincaré [1854–1912]. Using his result, Furstenberg proved my result on the k-term arithmetic progressions. So that is the short story about it. But I have to admit that his method is much stronger because it could be generalised to a multi-dimensional setting. Together Endre Szemerédi 173 with Katznelson he proved that in 1978. They could actually also prove the density Hales–Jewett theorem but it took more than ten years. Then Bergelson and Leibman proved a polynomial version of the arithmetic progression result, much stronger than the original one. I doubt that you can get it by elementary methods but that is only my opinion. I will bet that they will not come up with a proof of the polynomial version within the next ten years by using elementary methods. But then very interesting things happened. Tim Gowers started the so-called Polymath Project: many people communicated with each other on the internet and decided that they would try to give a combinatorial proof of the Hales–Jewett density theorem using only elementary methods. After two months, they come up with an elementary proof. The density Hales–Jewett theorem was considered to be by far the hardest result proved by Furstenberg and Katznelson and its proof is very long. The elementary proof of the density Hales–Jewett theorem is about 25 pages long. There is now a big discussion among mathematicians whether one can use this method to solve other problems. Joint papers are very good, when a small group of mathematicians cooperate. But the Polymath Project is different: hundreds of people communicate. You may work on something your whole life, then a hundred people appear and many of them are ingenious. They solve your problem and you are slightly disappointed. Is this a good thing? There is a big discussion among mathematicians about this method. I am for it. I will soon turn 72 years old, so I believe I can evaluate it without any self-interest.

Still, all this started with your proof of the Erdős-Turán conjecture. You mentioned Green-Tao. An important ingredient in their proof of the existence of arithmetic progressions of arbitrary length within the primes is a Szemeré- di-type argument involving so-called pseudo-primes, whatever that is. So the ramifications of your theorem have been impressive. In their abstract they say that the three main ingredients in their proof are the Goldstone–Yıldırım result which gives an estimate for the difference of consecutive primes, their transference principle and my theorem on arithmetic progressions.

By the way, according to Green and Tao one could have used the Selberg sieve instead. You are right. However, in my opinion the main revolutionary new idea is their transference principle that enables us to go from a dense set to a sparse set. I would like to point out that later, while generalising their theorem, they did not have to use my theorem. Terry Tao said that he read all the proofs of the Szemerédi theorem and compared them, and then he and Ben Green meditated on it. They were probably more inspired by Furstenberg’s method, the ergodic method. That is at least my take on this thing but I am not an expert on ergodic theory.

But Furstenberg’s theorem came after and was inspired by yours. So however you put it, it goes back to you. 174 Abel Prize 2012

Yes, that is what they say.

We should mention that Tim Gowers also gave a proof of the Szemerédi Theorem. He started with Roth’s method, which is an estimation of exponential sums. Roth proved in his paper that r3(n) is less than n divided by log log n. Tim Gowers’ fundamental work did not only give an absolutely strong bound for the size of a set A in the interval [1,n] not containing a k-term arithmetic progression; he also invented methods and concepts that later became extremely influential. He introduced a norm (actually, several norms), which is now called the Gowers norm. This norm controls the randomness of a set. If the Gowers norm is big, he proved that it is correlated with a higher order phase function, which is a higher order polynomial. Gowers, and independently Rödl, Naegle, Schacht and Skokan, proved the hypergraph regularity lemma and the hypergraph counting lemma, which are main tools in additive combinatorics and in theoretical computer science.

We should mention that Gowers received the Fields Medal in 1998 and that Terence Tao got it in 2006. Also, Roth was a Fields Medal recipient back in 1958.

Random graphs and the regularity lemma

Let’s get back to the so-called Szemerédi regularity theorem. You have to explain the notions of random graphs and extremal graphs because they are involved in this result. How can we imagine a random graph? I will talk only about the simplest example. You have n points and the edges are just the pairs, so each edge connects two points. We say that the graph is complete if you include all the edges, but that is, of course, not an interesting object. In one model of the random graph, you just close your eyes and with probability ½, you choose an edge. Then you will eventually get a graph. That is what we call a random graph, and most of them have very nice properties. You just name any configuration – like 4-cyclesC 4 for instance, or the complete graph K4 – then the number of such configurations is as you would expect. A random graph has many beautiful properties and it satisfies almost everything. Extremal graph theory is about finding a configuration in a graph. If you know that your graph is a random graph, you can prove a lot of things. The regularity lemma is about the following. If you have any graph – unfortunately we have to assume a dense graph, which means that you have a lot of edges – then you can break the vertex set into a relatively small number of disjoint vertex sets, so that if you take almost any two of these vertex sets, then between them the so-called bipartite graph will behave like a random graph. We can break our graph into not too many pieces, so we can work with these pieces and we can prove theorems in extremal graph theory. Endre Szemerédi 175

We can also use it in property testing, which belongs to theoretical computer science and many other areas. I was surprised that they use it even in biology and neuro- science but I suspect that they use it in an artificial way – that they could do without the regularity lemma. But I am not an expert on this so I can’t say this for sure.

The regularity lemma really has some important applications in theoretical computer science? Yes, it has; mainly in property testing but also in constructing algorithms. Yes, it has many important applications. Not only the original regularity lemma but, since this is 30 years ago, there have appeared modifications of the regularity lemma which are more adapted for these purposes. The regularity lemma is for me just a philosophy. Not an actual theorem. Of course, the philosophy is almost everything. That is why I like to say that in every chaos there is an order. The regularity lemma just says that in every chaos there is a big order.

Do you agree that the Szemerédi theorem, i.e. the proof of the Erdős–Turán conjecture, is your greatest achievement? It would be hard to disagree because most of my colleagues would say so. However, perhaps I prefer another result of mine with Ajtai and Komlós. In connection with a question about Sidon sequences we discovered an innocent looking lemma. Sup- pose we have a graph of n vertices in which a vertex is connected to at most d other vertices. By a classical theorem of Turán, we can always find at least n/d vertices such that no two of them are connected by an edge. What we proved was that under the assumption that the graph contains no triangle, a little more is true: one can find n/d times log d vertices with the above property. I am going to describe the proof of the lemma very briefly. We choose n/2d vertices of our graph randomly. Then we omit all the neighbours of the points in the chosen sets. This is, of course, a deterministic step. Then in the remaining vertex set we again choose randomly n/2d vertices and again deterministically omit the neighbours of the chosen set. It can be proved that this procedure can be repeated log d times and in the chosen set the average degree is at most 2. So in the chosen sets we can find a set of size at least n/4d such that no two points are connected with an edge. Because of the mixture of random steps and deterministic steps we called this new technique the “semirandom method”. Historically, the first serious instance of a result of extremal graph theory was the famous theorem of Ramsey [1903–1930], and, in a quantitative form, of Erdős and Szekeres [1911–2005]. This result has also played a special role in the development of the “random method”. Therefore it has always been a special challenge for combinatorialists to try to determine the asymptotic behaviour of the Ramsey functions R(k,n)2, as n (or both k and n) tend to infinity. It can be easily deduced

2 R(k,n) denotes the least positive integer N such that for any (red/blue)-coloring of the complete graph KN on N vertices, there exists either an entirely red complete subgraph on k vertices or an entirely blue complete subgraph on n vertices. 176 Abel Prize 2012 from our lemma that R(3,n) < cn2/log n, which solved a longstanding open problem of Erdős. Surprisingly, about 10 years later, Kim proved that the order of magnitude of our bound was the best possible. His proof is based on a brilliant extension of the “semirandom method”. The “semirandom method” has found many other applications. For instance, together with Komlós and Pintz I used the same technique to disprove a famous geometric conjecture of Heilbronn [1908–1975]. The conjecture dates back to the 40s. The setting is as follows: you have n points in the unit square and you consider the triangles defined by these points. Then the conjecture says that you can always choose a triangle of area smaller than a constant over n2. That was the Heilbronn conjecture. For the bound 1/n, this is trivial, and then Klaus Roth improved this to 1 over n(log log n)½. Later Wolfgang Schmidt improved it further to 1 over n(log n)½. Roth, in a very brilliant and surprising way, used analysis to prove that we can find a a triangle of area less than 1 over n(1+ ), where a is a constant. We then proved, using the semi-random method, that it is possible to put down n points such that the smallest area of a triangle is at least log n over n2, disproving the Heilbronn conjecture. Roth told us that he gave a series of talks about this proof.

Further research areas

It is clear from just checking the literature and talking with people familiar with graph theory and combinatorics, as well as additive number theory, that you – sometimes with co-authors – have obtained results that have been groundbreaking and have set the stage for some very important developments. Apart from the Szemerédi theorem and the regularity lemma that we have already talked about, here is a short list of important results that you and your co-authors have obtained: (i) The Szemerédi-Trotter theorem in the paper “Extremal problems in discrete geometry” from 1983. (ii) The Erdős–Szemerédi theorem on product-sum estimates, in the paper “On sums and products of integers” from 1983. (iii) The results obtained by AKS, which is the acronym for Miklós Ajtai, János Komlós and Endre Szemerédi. The “sorting algorithm” is among the highlights. Could you fill in some details, please? (i) Euclid’s system of axioms states some of the basic facts about incidences between points and lines in the plane. In the 1940s, Paul Erdős started asking slightly more complicated questions about incidences that even Euclid would have understood. How many incidences can occur among m points and n lines, where an “incidence” means that a line passes through a point? My theorem with Trotter confirmed Erdős’ rather surprising conjecture: the maximal number of incidences is much smaller in the real plane than in the projective one – much smaller than what we could deduce by simple combinatorial considerations. Endre Szemerédi 177

(ii) Together with Paul Erdős, we discovered an interesting phenomenon and made the first non-trivial step in exploring it. We noticed, roughly speaking, that a set of numbers may have nice additive properties or nice multiplicative properties but not both at the same time. This has meanwhile been generalised to finite fields and other structures by Bourgain, Katz, Tao and others. Their results had far-reaching consequences in seemingly unrelated fields of mathematics. (iii) We want to sort n numbers, that is, to put them in increasing order by using comparisons of pairs of elements. Our algorithm is non-adaptive: the next comparison never depends on the outcome of the previous ones. Moreover, the algorithm can efficiently run simultaneously on cn processors such that every number is processed by only one of them at a time. Somewhat surprisingly, our algorithm does not require more comparisons than the best possible adaptive non-parallel algorithm. It is well known that any sorting algorithm needs at least n log n comparisons.

What are, in your opinion, the most interesting and important open problems in combinatorics and graph theory? I admit that I may be somewhat conservative in taste. The problem that I would like to see solved is the very first problem of extremal graph theory: to determine the asymptotic behaviour of the Ramsey functions.

Combinatorics compared to other areas of mathematics

It is said, tongue in cheek, that a typical combinatorialist is a bright mathematician with an aversion to learning or embracing abstract mathematics. Does this description fit you? I am not sure. In combinatorics we want to solve a concrete problem, and by solving a problem we try to invent new methods. And it goes on and on. Sometimes we actually borrow from so-called well established mathematics. People in other areas of mathematics often work in ways that are different from how we do in combinatorics. Let’s exaggerate somewhat: they have big theories and they find sometimes a problem for the theory. In combinatorics, it is usually the other way around. We start with problems which actually are both relevant and necessary; that is, the combinatorics itself requires the solution of the problems; the problems are not randomly chosen. You then have to find methods which you apply to solve the problems and sometimes you might create theory. But you start out by having a problem; you do not start by having a theory and then finding a problem for which you can apply the theory. Of course, that happens from time to time but it is not the major trend. Now, in the computer era, it is unquestionable that combinatorics is extremely important. If you want to run programs efficiently, you have to invent algorithms in 178 Abel Prize 2012 advance and these are basically combinatorial in nature. This is perhaps the reason why combinatorics today is a little bit elevated, so to say, and that mathematicians from other fields start to realise this and pay attention. If you look at the big results, many of them have big theories which I don’t understand, but at the very root there is often some combinatorial idea. This discussion is a little bit artificial. It’s true that combinatorics was a second rated branch of mathematics 30 years ago but hopefully not any longer.

Do you agree with Bollobás who in an interview from 2007 said the following: “The trouble with the combinatorial problems is that they do not fit into the existing mathematical theories. We much more prefer to get help from ‘mainstream’ mathematics rather than to use ‘combinatorial’ methods only, but this help is rarely forthcoming. However, I am happy to say that the landscape is changing.” I might agree with that.

Gowers wrote a paper about the two cultures within mathematics. There are problem solvers and there are theory builders. His argument is that we need both. He says that the organising principles of combinatorics are less explicit than in core mathematics. The important ideas in combinatorial mathematics do not usually appear in the form of precisely stated theorems but more often as general principles of wide applicability. I guess that Tim Gowers is right. But there is interplay between the two disciplines. As Bollobás said, we borrow from the other branches of mathematics if we can, when we solve concrete discrete problems, and vice versa. I once sat in class when a beautiful result in analytic number theory was presented. I understood only a part of it. The mathematician who gave the talk came to the bottleneck of the whole argument. I realised that it was a combinatorial statement and if you gave it to a combinatorialist, he would probably have solved it. Of course, one would have needed the whole machinery to prove the result in question but at the root it actually boiled down to a combinatorial argument. A real interplay!

There is one question that we have asked almost all Abel Prize recipients; it concerns the development of important new concepts and ideas. If you recol- lect: would key ideas turn up when you were working hard at your desk on a problem or did they show up in more relaxed situations? Is there any pattern? Actually, both! Sometimes you work hard on a problem for half a year and nothing comes out. Then suddenly you see the solution, and you are surprised and slightly ashamed that you haven’t noticed these trivial things which actually finish the whole proof, and which you did not discover for a long time. But usually you work hard and step-by-step you get closer to the solution. I guess that this is the case in every science. Sometimes the solution comes out of the blue but sometimes several people are working together and find the solution. Endre Szemerédi 179

I have to tell you that my success ratio is actually very bad. If I counted how many problems I have worked on and in how many problems I have been successful, the ratio would be very bad.

Well, in all fairness this calculation should take into consideration how many problems you have tried to solve. Right, that is a nice remark.

You have been characterised by your colleagues – and this is meant as a huge compliment – as having an “irregular mind”. Specifically, you have been described as having a brain that is wired differently than most mathematicians. Many admire your unique way of thinking, your extraordinary vision. Could you try to explain to us how you go about attacking problems? Is there a particular method or pattern? I don’t particularly like the characterisation of having an “irregular mind”. I don’t feel that my brain is wired differently and I think that most neurologists would agree with me. However, I believe that having unusual ideas can often be useful in mathematical research. It would be nice to say that I have a good general approach of attacking mathematical problems. But the truth is that after many years of research I still do not have any idea what the right approach is.

Mathematics and computer science

We have already talked about connections between discrete mathematics and computer science – you are in fact a professor in computer science at Rutgers University in the US. Looking back, we notice that for some important mathematical theorems, like the solution of the four-colour problem for instance, computer power has been indispensable. Do you think that this is a trend? Will we see more results of this sort? Yes, there is a trend. Not only for this but also for other types of problems as well where computers are used extensively. This trend will continue, even though I am not a computer expert. I am at the computer science department but fortunately nobody asked me whether I could answer email, which I cannot! They just hired me because so-called theoretical computer science was highly regarded in the late 80s. Nowadays, it does not enjoy the same prestige, though the problems are very important, the P versus NP problem, for instance. We would like to understand computation and how fast it is; this is absolutely essential mathematics, and not only for discrete mathematics. These problems lie at the heart of mathematics, at least in my opinion.

May we come back to the P versus NP problem which asks whether every problem whose solution can be verified quickly by a computer can also be solved quickly by a computer. Have you worked on it yourself? 180 Abel Prize 2012

I am working on two problems in computer science. The first one is the following: assume we compute an n-variable Boolean function with a circuit. For most of the n-variable Boolean functions the circuit size is not polynomial. But to the best of my knowledge, we do not know a particular function which cannot be computed with a Boolean circuit of linear size and depth log n. I have no real idea how to solve this problem. The second one is the minimum weight spanning tree problem; again, so far I am unsuccessful. I have decided that now I will, while keeping up with combinatorics, learn more about analytic number theory. I have in mind two or three problems, which I am not going to tell you. It is not the Riemann hypothesis; that I can tell.

The P versus NP conjecture is on the Clay list of problems, the prize money for a solution being one million USD, so it has a lot of recognition. Many people believe that the P versus NP problem is the most important one in current mathematics, regardless of the Riemann hypothesis and the other big problems. We should understand computation. What is in our power? If we can check easily that something is true, can we easily find a solution? Most probably not! Almost everybody will bet that P is not equal to NP but not too much has been proved.

Soccer

You have described yourself as a sport fanatic. Yes, at least I was. I wanted to be a soccer player but I had no success.

We have to stop you there. In 1953, when you were 13 years old, Hungary had a fantastic soccer team; they were called “The Mighty Magyars”. They were the first team outside the British Isles that beat England at Wembley, and even by the impressive score of six to three. At the return match in Budapest in 1954 they beat England seven to one, a total humiliation for the English team. Some of these players on the Hungarian team are well known in the annals of soccer, names like Puskás [1927–2006], Hidegkuti [1922–2002], Czibor [1929–1997], Bozsik [1925–1978] and Kocsis [1929–1979]. Yes. These five were world class players.

We have heard that the Hungarian team, before the game in Budapest, lived at the same place as you did. Bozsik watched you play soccer and he said that you had real talent. Is this a true story? Yes, that is true except that they did not live at the same place. My mother died early; this is why we three brothers lived at a boarding school. That school was very close to the hotel where the Hungarian team lived. They came sometimes to our soccer field to relax and watch our games, and one time we had a very important game Endre Szemerédi 181 against the team that was our strongest competitor. You know, boarding schools were competing like everyone else. I was a midfielder like Bo- zsik. I was small and did not have the speed but I understood the Hungarian team’s strategy. They revolutionised the soccer game, foreshad- owing what was later called “Total Football”. They did not pass the ball to the nearest guy From left to right: Martin Raussen, Christian Skau and but rather they aimed the ball Endre Szemerédi. (Photo: Eirik Furu Baardsen) to create space and openings, often behind the other team’s defence. That was a completely different strategy than the standard one and therefore they were extremely effective. I studied this and I understood their strategy and tried to imitate it. Bozsik saw this and he understood what I was trying to do.

You must have been very proud. Yes, indeed I was very proud. He was nice and his praise is still something which I value very much.

Were you very disappointed with the World Cup later that year? As you very well know, the heavily favoured Hungarian team first beat West Germany eight to three in the preliminary round but then they lost two to three in the final to West Germany. Yes. It was very unfortunate. Puskás was injured, so he was not at his best, but we had some other problems, too. I was very, very sad and for months I practically did not speak to anybody. I was a real soccer fan. Much later, in 1995, a friend of mine was the ambassador for Hungary in Cairo and I visited him. Hidegkuti came often to the embassy because he was the coach for the Egyptian team. I tried to make him explain to me what happened in 1954 but I got no answer. By the way, to my big surprise I quite often guess correct results. Several jour- nalists came to me in Hungary for an interview after it was announced that I would receive the Abel Prize. The last question from one of them was about the impending European Cup quarter final match between Barcelona and Milan. I said that up to now I have answered your questions without hesitation but now I need three minutes. I reasoned that the defence of Barcelona was not so good (their defender Puyol is a bit old) but their midfield and attack is good, so: 3 to 1 to Barcelona. On the day the game was played, the paper appeared with my, as it turned out, correct prediction. I was very proud of this and people on these blogs wrote that I could be very rich if I would enter the odds prediction business! 182 Abel Prize 2012

We can at least tell you that you are by far the most sports interested person we have met so far in these Abel interviews! On behalf of the Norwegian, Danish and European mathematical societies, and on behalf of the two of us, thank you very much for this most interesting interview. Thank you very much. I am very happy for the possibility of talking to you.

The Award ceremony in The University aula. In front: President of The Norwegian Academy of Science and Letters Nils Chr. Stenseth and Abel Laureate Endre Szemerédi. Behind: Chair of the Abel Committee Ragni Piene. (Photo: Erlend Aas) Abel Prize 2013: Pierre Deligne

The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2013 to

Pierre Deligne Institute for Advanced Study, Princeton, New Jersey, USA

“for seminal contributions to algebraic geometry and for their transformative impact on number theory, representation theory, and related fields.”

Geometric objects such as lines, circles and spheres can be described by simple algebraic equations. The resulting fundamental connection between geometry and algebra led to the development of algebraic geometry, in which geometric methods are used to study solutions of polynomial equations, and, conversely, algebraic techniques are applied to analyze geometric objects. Over time, algebraic geometry has undergone several transformations and expansions, and has become a central subject with deep connections to almost every area of mathematics. Pierre Deligne played a crucial role in many of these developments. Deligne’s best known achievement is his spectacular solution of the last and deepest Abel Laureate Pierre Deligne. of the Weil conjectures, namely the analogue (Photo: Knut Falch) of the Riemann hypothesis for algebraic varieties over a finite field. Weil envisioned that the proof of these conjectures would require methods from algebraic topology. In this spirit, Grothendieck and his school developed the theory of l-adic cohomology, which would then become a basic tool in Deligne’s proof. Deligne’s brilliant work is a real

* 3.10.1944

183 184 Abel Prize 2013 tour de force and sheds new light on the cohomology of algebraic varieties. The Weil conjectures have many important applications in number theory, including the solution of the Ramanujan–Petersson conjecture and the estimation of exponential sums. In a series of papers, Deligne showed that the cohomology of singular, non-compact varieties possesses a mixed Hodge structure that generalized the classical Hodge theory. The theory of mixed Hodge structures is now a basic and powerful tool in algebraic geometry and has yielded a deeper understanding of cohomology. It was also used by Cattani, Deligne and Kaplan to prove an algebraicity theorem that provides strong evidence for the Hodge conjecture. With Beilinson, Bernstein and Gabber, Deligne made definitive contributions to the theory of perverse sheaves. This theory plays an important role in the recent proof of the fundamental lemma by Ngo. It was also used by Deligne himself to greatly clarify the nature of the Riemann–Hilbert correspondence, which extends Hilbert’s 21st problem to higher dimensions. Deligne and Lusztig used l-adic cohomology to construct linear representations for general finite groups of Lie type. With Mumford, Deligne introduced the notion of an algebraic stack to prove that the moduli space of stable curves is compact. These and many other contributions have had a profound impact on algebraic geometry and related fields. Deligne’s powerful concepts, ideas, results and methods continue to influence the development of algebraic geometry, as well as mathematics as a whole.

Pierre Deligne with King Harald. (Photo: Heiko Junge) Pierre Deligne 185

Dear Professor Deligne, first of all we would like to congratulate you as the eleventh recipient of the Abel Prize. It is not only a great honour to be selected as recipient of this prestigious prize, the Abel Prize also carries a cash-amount of 6 million NOK, that is around 1 million US$. We are curious to hear what you are planning to do with this money… I feel that this money is not really mine, but it belongs to mathematics. I have a responsibility to use it wisely and not in a wasteful way. The details are not clear yet, but I plan to give part of the money to the two institutions that have been most important to me: the Institut des Hautes Études Scientifiques (IHÉS) in Paris and to the Institute for Advanced Study (IAS) in Princeton. I would also like to give some money to support mathematics in Russia. First to the Department of Mathematics of the Higher School of Economics (HSE). In my opinion, it is one of the best places in Moscow. It is much smaller than the Faculty of Mechanics and Mathematics at the University, but has better people. The student body is small; only fifty new students are accepted each year. But they are among the best students. The HSE has been created by economists. They have done their best under difficult circumstances. The department of mathematics has been created five years ago, with the help of the Independent University of Moscow. It is giving prestige to the whole HSE. There I think some money could be well used. Another Russian institution I would like to donate some money to is the Dy- nasty Foundation, created by the Russian philanthropist Dmitry Zimin. For them, money is most likely not that important. It is rather a way for me to express my admiration for their work. It is one of the very few foundations in Russia that gives money to science; moreover, they do it in a very good way. They give money to mathematicians, to physicists and to biologists; especially to young people, and this is crucial in Russia! They also publish books to popularize science. I want to express my admiration for them in a tangible way.

The Abel Prize is certainly not the first important prize in mathematics that you have won. Let us just mention the Fields Medal that you received 35 years ago, the Swedish Crafoord Prize, the Italian Balzan Prize and the Israeli Wolf Prize. How important is it for you, as a mathematician, to win such prestigious prizes? And how important is it for the mathematical community that such prizes exist? For me personally, it is nice to be told that mathematicians I respect find the work I have done interesting. The Fields Medal possibly helped me to be invited to the Institute for Advanced Study. To win prizes gives opportunities, but they have not changed my life. I think prizes can be very useful when they can serve as a pretext for speaking about mathematics to the general public. I find it particularly nice that the Abel Prize is connected with other activities such as competitions directed towards children and the Holmboe Prize for high school teachers. In my experience, good high school teachers are very important for the development of mathematics. I think all these activities are marvellous. 186 Abel Prize 2013

Youth

You were born in 1944, at the end of the Second World War in Brussels. We are curious to hear about your first mathematical experiences: In what respect were they fostered by your own family or by school? Can you remember some of your first mathematical experiences? I was lucky that my brother was seven years older than me. When I looked at the thermometer and realized that there were positive and negative numbers, he would try to explain to me that minus one times minus one is plus one. That was a big surprise. Later when he was in high school he told me about the second degree equation. When he was at the university he gave me some notes about the third degree equation, and there was a strange formula for solving it. I found it very interesting. When I was a Boy Scout, I had a stroke of extraordinary good luck. I had a friend there whose father, Monsieur Nijs, was a high school teacher. He helped me in a number of ways; in particular, he gave me my first real mathematical book, namely Set Theory by Bourbaki, which is not an obvious choice to give to a young boy. I was 14 years old at the time. I spent at least a year digesting that book. I guess I had some other lectures on the side, too. Having the chance to learn mathematics at one’s own rhythm has the benefit that one revives surprises of past centuries. I had already read elsewhere how rational numbers, then real numbers, could be defined starting from the integers. But I remember wondering how integers could be defined from set theory, looking a little ahead in Bourbaki, and admiring how one could first define what it means for two sets to have the “same number of elements”, and derive from this the notion of integers. I was also given a book on complex variables by a friend of the family. To see that the story of complex variables was so different from the story of real variables was a big surprise: once differentiable, it is analytic (has a power series expansion), and so on. All those things that you might have found boring at school were giving me a tremendous joy. Then this teacher, Monsieur Nijs, put me in contact with professor Jacques Tits at the University of Brussels. I could follow some of his courses and seminars, though I still was in high school.

It is quite amazing to hear that you studied Bourbaki, which is usually considered quite difficult, already at that age. Can you tell us a bit about your formal school education? Was that interesting for you, or were you rather bored? I had an excellent elementary school teacher. I think I learned a lot more in elementary school than I did in high school: how to read, how to write, arithmetics and much more. I remember how this teacher made an experiment in mathematics which made me think about proofs, surfaces and lengths. The problem was to compare the surface of a half sphere with that of the disc with the same radius. To do so, he covered both surfaces with a spiralling rope. The half sphere required twice as much rope. This made me think a lot: how could one measure a surface with a Pierre Deligne 187 length? How to be sure that the surface of the half sphere was indeed twice that of the disc? When I was in high school, I liked problems in geometry. Proofs in geometry make sense at that age because surprising statements have not too difficult proofs. Once we were past the axioms, I enjoyed very much doing such exercises. I think that geometry is the only part of mathematics where proofs make sense at the high school level. Moreover, writing a proof is another excellent exercise. This does not only concern mathematics, you also have to write in correct French – in my case – in order to argue why things are true. There is a stronger connection between language and mathematics in geometry than for instance in algebra, where you have a set of equations. The logic and the power of language are not so apparent.

You went to the lectures of Jacques Tits when you were only 16 years old. There is a story that one week you could not attend because you participated in a school trip…? Yes. I was told this story much later. When Tits came to give his lecture he asked: Where is Deligne? When it was explained to him that I was on a school trip, the lecture was postponed to the next week.

He must already have recognised you as a brilliant student. Jacques Tits is also a recipient of the Abel Prize. He received it together with John Griggs Thomp- son five years ago for his great discoveries in group theory. He was surely an influential teacher for you? Yes; especially in the early years. In teaching, the most important can be what you don’t do. For instance, Tits had to explain that the centre of a group is an invariant subgroup. He started a proof, then stopped and said in essence: “An invariant subgroup is a subgroup stable by all inner automorphisms. I have been able to define the centre. It is hence stable by all symmetries of the data. So it is obvious that it is invariant.” For me, this was a revelation: the power of the idea of symmetry. That Tits did not need to go through a step-by-step proof, but instead could just say that symmetry makes the result obvious, has influenced me a lot. I have a very big respect for symmetry, and in almost every of my papers there is a symmetry-based argument.

Can you remember how Tits discovered your mathematical talent? That I cannot tell, but I think it was Monsieur Nijs who told him to take good care of me. At that time, there were three really active mathematicians in Brussels: apart from Tits himself, professors Franz Bingen and Lucien Waelbroeck. They organised a seminar with a different subject each year. I attended these seminars and I learned about different topics such as Banach algebras, which were Waelbroeck’s speciality, and algebraic geometry. Then, I guess, the three of them decided it was time for me to go to Paris. Tits introduced me to Grothendieck and told me to attend his lectures as well as Serre’s. That was an excellent advice. 188 Abel Prize 2013

This can be a little surprising to an outsider. Tits being interested in you as a mathematician, one might think that he would try to capture you for his own interests. But he didn’t? No. He saw what was best for me and acted accordingly.

Algebraic geometry

Before we proceed to your career in Paris, perhaps we should try to explain to the audience what your subject algebraic geometry is about. When Fields medalist Tim Gowers had to explain your research subjects to the audience during the Abel Prize announcement earlier this year, he began by confessing that this was a difficult job for him. It is difficult to show pictures that illustrate the subject, and it is also difficult to explain some simple applications. Could you, nevertheless, try to give us an idea what algebraic geometry is about? Perhaps you can mention some specific problems that connect algebra and geometry with each other. In mathematics, it is always very nice when two different frames of mind come together. Descartes wrote: ”La géométrie est l’art de raisonner juste sur des figures fausses” (Geometry is the art of correct reasoning on false figures). “Figures” is plural: it is very important to have various perspectives and to know in which way each is wrong. In algebraic geometry, you can use intuitions coming both from algebra – where you can manipulate equations – and from geometry, where you can draw pictures. If you picture a circle and consider the equation x² + y² = 1, different images are evoked in your mind, and you can try to play one against the other. For instance, a wheel is a circle and a wheel turns; it is interesting to see what the analogue is in algebra: an algebraic transformation of x and y maps any solution of x² + y² = 1 to another. This equation describing a circle is of the second degree. This implies that a circle will have no more than two intersections points with a line. This is a property you also see geometrically, but the algebra gives more. For instance, if the line has a rational equation and one of the intersection points with the circle x² + y² = 1 has rational coordinates, then the other intersection point will also have rational coordinates. Algebraic geometry can have arithmetical applications. When you consider polynomial equations, you can use the same expressions in different number systems. For instance, on finite sets on which addition and multiplication are defined, these equations lead to combinatorial questions: you try to count the number of solutions. But you can continue to draw the same pictures, keeping in mind a new way in which the picture is false, and in this way you can use geometrical intuition while looking at combinatorial problems. I have never really been working at the centre of algebraic geometry. I have mostly been interested in all sorts of questions that only touch the area. But algebraic geometry touches many subjects! As soon as a polynomial appears, one can try to think about it geometrically; for example in physics with Feynman integrals, Pierre Deligne 189 or when you consider an integral of a radical of a polynomial expression. Algebraic geometry can also contribute to the understanding of integer solutions of polynomial equations. You have the old story of elliptic functions: to understand how elliptic integrals behave, the geometrical interpretation is crucial.

Algebraic geometry is one of the main areas in mathematics. Would you say that to learn algebraic geometry requires much more effort than other areas in mathematics, at least for a beginner? I think it’s hard to enter the subject because one has to master a number of different tools. To begin with, cohomology is now indispensable. Another reason is that Algebraic geometry developed in a succession of stages, each with its own language. First, the Italian school which was a little hazy, as shown by the infamous saying: “In Algebraic geometry, a counterexample to a theorem is a useful addition to it”. Then Zariski [1899–1986] and Weil [1906–1998] put things on a better footing. Later Serre and Grothendieck gave it a new language which is very powerful. In this language of schemes one can express a lot; it covers both arithmetical applications and more geometrical aspects. But it requires time to understand the power of this language. Of course, one needs to know a number of basic theorems, but I don’t think that this is the main stumbling block. The most difficult is to understand the power of the language created by Grothendieck and how it relates to our usual geometrical intuition.

Apprentice in Paris

When you came to Paris you came in contact with Alexander Grothendieck and Jean-Pierre Serre. Could you tell us about your first impression of these two mathematicians? I was introduced to Grothendieck by Tits during the Bourbaki seminar of Novem- ber 1964. I was really taken aback. He was a little strange, with his shaved head, a very tall man. We shook hands, but did nothing more until I went to Paris a few months later to attend his seminar. That was really an extraordinary experience. In his way, he was very open and kind. I remember the first lecture I attended. In it, he used the expression “cohomology object” many times. I knew what cohomology was for abelian groups, but I did not know the meaning of ”cohomology object”. After the lecture I asked him what he meant by this expression. I think that many other mathematicians would have thought that if you didn’t know the answer, there wouldn’t be any point to speak to you. This was not his reaction at all. Very patiently he told me that if you have a long exact sequence in an abelian category and you look at the kernel of one map, you divide by the image of the previous one and so on…. I recognized quickly that I knew about this in a less general context. He was very open to people who were ignorant. I think that you should not ask him the same stupid question three times, but twice was all right. 190 Abel Prize 2013

I was not afraid to ask completely stupid questions, and I have kept this habit until now. When attending a lecture, I usually sit in front of the audience, and if there is something I don’t understand, I ask questions even if I would be supposed to know what the answer was. I was very lucky that Grothendieck asked me to write up talks he had given the previous year. He gave me his notes. I learned many things, both the content of the notes, and also a way of writing mathematics… This both in a prosaic way, namely that one should write only on one side of the paper and leave some blank space so he could make comments, but he also insisted that one was not allowed to make any false statement. This is extremely hard. Usually one takes shortcuts; for instance, not keeping track of signs. This would not pass muster with him. Things had to be correct and precise. He told me that my first version of the redaction was much too short, not enough details… It had to be completely redone. That was very good for me. Serre had a completely different personality. Grothendieck liked to have things in their natural generality; to have an understanding of the whole story. Serre appre- ciates this, but he prefers beautiful special cases. He was giving a course at Collège de France on elliptic curves. Here, many different strands come together, including automorphic forms. Serre had a much wider mathematical culture than Gro- thendieck. In case of need, Grothendieck redid everything for himself, while Serre could tell people to look at this or that in the literature. Grothendieck read extremely little; his contact with classical Italian geometry came basically through Serre and Dieudonné. I think Serre must have explained him what the Weil conjectures were about and why they were interesting. Serre respected the big constructions Groth- endieck worked with, but they were not in his taste. Serre preferred smaller objects with beautiful properties such as modular forms, to understand concrete questions, for instance congruences between coefficients. Their personalities were very different, but I think that the collaboration between Serre and Grothendieck was very important and it enabled Grothendieck to do some of his work.

You told us that you needed to go to Serre’s lectures in order to keep your feet on the ground? Yes, because it was a danger in being swept away in generalities with Grothendieck. In my opinion, he never invented generalities that were fruitless, but Serre told me to look at different topics that all proved to be very important for me.

The Weil Conjectures

Your most famous result is the proof of the third – and the hardest – of the so- called Weil conjectures. But before talking about your achievement, can you try to explain why the Weil conjectures are so important? Pierre Deligne 191

There were some previous theorems of Weil about curves in the one-dimensional situation. There are many analogies between algebraic curves over finite fields and the rational numbers. Over the rational numbers, the central question is the Riemann hypothesis. Weil had proved the analogue of the Riemann hypothesis for curves over finite fields, and he had looked at some higher-dimensional situations as well. This was at the time where one started to understand the cohomology of simple algebraic varieties, like the Grassmannians. He saw that some point-counting for objects over finite fields reflected what happened over the complex numbers and the shape of the related space over the complex numbers. As Weil looked at it, there are two stories hidden in the Weil conjectures. First, why should there be a relation between apparently combinatorial questions and geometric questions over the complex numbers. Second, what is the analogue of the Riemann hypothesis? Two kinds of applications came out of these analogies. The first started with Weil himself: estimates for some arithmetical functions. For me, they are not the most important. Grothendieck’s construction of a formalism explaining why there should be a relation between the story over the complex numbers, where one can use topology, and the combinatorial story, is more important. Secondly, algebraic varieties over finite fields admit a canonical endomorphism, the Frobenius. It can be viewed as a symmetry, and this symmetry makes the whole situation very rigid. Then one can transpose this information back into the geometric world over the complex numbers, it yields constraints on what will happen in classical algebraic geometry, and this is used in applications to representation theory and the theory of automorphic forms. It was not obvious at first that there would be such applications, but for me they are the reason why the Weil conjecture is important.

Grothendieck had a program on how to prove the last Weil conjecture, but it didn’t work out. Your proof is different. Can you comment on this program? Did it have an influence on the way you proved it? No. I think that the program of Grothendieck was, in a sense, an obstruction to finding the proof, because it made people think in just a certain direction. It would have been more satisfying if one had been able to do the proof following the program, because it would have explained a number of other interesting things as well. But the whole program relied on finding enough algebraic cycles on algebraic varieties; and on this question one has made essentially no progress since the 70s. I used a completely different idea. It is inspired by the work of Rankin and his work on automorphic forms. It still has a number of applications, but it did not realize the dream of Grothendieck.

We heard that Grothendieck was glad that the Weil conjecture was proved, of course, but still he was a little disappointed? Yes. And with very good reason. It would have been much nicer if his program had been realized. He did not think that there would be another way to do it. When he 192 Abel Prize 2013 heard I had proved it, he felt I must have done this and that, which I hadn’t. I think that’s the reason for the disappointment.

You have to tell us about the reaction of Serre when he heard about the proof. I wrote him a letter when I did not have a complete proof yet, but a test case was clear. I think he got it just before he had to go to the hospital for an operation of a torn tendon. He told me later that he went into the operation theatre in a euphoric state because he knew now that the proof was roughly done.

Several famous mathematicians have called your proof of the last Weil conjecture a marvel. Can you describe how you got the ideas that led to the proof? I was lucky that I had all the tools needed at my disposal at the same time and that I understood that those tools would do it. Parts of the proof have since been simplified by Gérard Laumon, and a number of these tools are no more needed. At the time, Grothendieck had ideas for putting into a purely algebraic framework the work of Solomon Lefsc hetz [1884–1972] from the 20s about families of hyperplane sections of an algebraic variety. Of particular interest was a statement of Lefschetz, later proved by William Hodge [1903–1975], the so-called hard Lefschetz theorem. Lefschetz’ approach was topological. In contrast to what one might think, if arguments are topological there is a better chance to translate them into abstract algebraic geometry than if they are analytic, such as the proof given by Hodge. Grothendieck asked me to look at the 1924 book L’analysis situs et la géométrie algébrique” by Lefschetz. It is a beautiful and very intuitive book, and it contained some of the tools I needed. I was also interested in automorphic forms. I think it is Serre who told me about an estimate due to Robert Rankin. I looked carefully at it. Rankin was getting some non-trivial estimates for coefficients of modular forms by proving for some related L-functions what was needed to apply results of Landau [1877–1938], in which the location of the poles of an L-function gave information on the poles of the local factors. I saw that the same tool, in a much less sophisticated way, just using that a sum of squares is positive, could be used here because of the control the work of Grothendieck gave on poles. This was enough. The poles were much easier to understand than the zeros and it was possible to apply Rankin’s idea. I had all these tools at my disposal, but I cannot tell how I put them together.

A little bit about subsequent work

What is a motive? A surprising fact about algebraic varieties is that they give rise not to one, but to many cohomology theories. Among them the l-adic theories, one for each prime l different from the characteristic, and in characteristic zero, the algebraic de Rham cohomology. These theories seem to tell the same story, over and over again, each in a different language. The philosophy of motives is that there should exist a universal Pierre Deligne 193 cohomology theory, with values in a category of motives to be defined, from which all these theories could be derived. For the first cohomology group of a projective non-singular variety, the Picard variety plays the role of a motivic H 1: the Picard variety is an abelian variety, and from it the H 1 in all available cohomology theories can be derived. In this way, abelian varieties (taken up to isogeny) are a prototype for motives. A key idea of Grothendieck is that one should not try to define what a motive is. Rather, one should try to define the category of motives. It should be an abelian category with finite dimensional rational vector spaces as Hom groups. Crucially, it should admit a tensor product, needed to state a Künneth theorem for the universal cohomology theory, with values in the category of motives. If only the cohomology of projective non-singular varieties is considered, one speaks of pure motives. Grothendieck proposed a definition of a category of pure motives, and showed that if the category defined had a number of properties, mod- elled on those of Hodge structures, the Weil conjectures would follow. For the proposed definition to be viable, one needs the existence of “enough” algebraic cycles. On this question almost no progress has been made.

What about your other results? Which of those that you worked on after the proof of the Weil conjecture are you particularly fond of? I like my construction of a so-called mixed Hodge structure on the cohomology of complex algebraic varieties. In its genesis, the philosophy of motives has played a crucial role, even if motives don’t appear in the end result. The philosophy suggests that whenever something can be done in one cohomology theory, it is worthwhile to look for a counterpart in other theories. For projective non-singular varieties, the role played by the action of Galois is similar to the role played by the Hodge decomposition in the complex case. For instance, the Hodge conjecture, expressed using the Hodge decomposition, has as counterpart the Tate conjecture, expressed using the action of Galois. In the l-adic case, cohomology and action of Galois remain defined for singular or non-compact varieties. This forces us to ask: what is the analogue in the complex case? One clue is given by the existence, in l-adic cohomology, of an increasing filtration, the weight filtration W, for which the i-th quotient Wi/Wi–1 is a subquotient of the cohomology of a projective non-singular variety. We hence expect in the complex case a filtration W such that the i-th quotient has a Hodge decomposition of weight i. Another clue, coming from works of Griffiths and Grothendieck, is that the Hodge filtration is more important than the Hodge decomposition. Both clues force the definition of mixed Hodge structures, suggest that they form an abelian category, and suggest also how to construct them.

What about the Langlands program? Have you been involved in it? I have been very interested in it, but I have contributed very little. I have only done some work on GL(2), the linear group in two variables. I tried to understand things. A somewhat remote application of the Weil conjecture has been used in Ngo’s 194 Abel Prize 2013 recent proof of what is called the fundamental lemma. I didn’t do a lot of work myself, though I had a lot of interest in the Langlands program.

French, American and Russian mathematics

You have already told us about the two institutions you mainly have worked for, namely the IHÉS in Paris and then, since 1984, the IAS in Princeton. It would be interesting for us to hear what your motives were for leaving IHÉS and moving to Princeton. Moreover, we would like to hear what unites the two institutions and how they differ, in your opinion. One of the reasons I left, was that I don’t think it’s good to spend all of one’s life in the same place. Some variation is important. I was hoping to have some contact with Harish-Chandra [1923–1983] who had done some beautiful work in representation theory and automorphic forms. That was a part of the Langlands program that I am very interested in, but unfortunately Harish-Chandra died shortly before I arrived at Princeton. Another reason was that I had imposed on myself to give seminars, each year on a new subject, at the IHÉS in Bures. That became a little too much. I was not really able to both give the seminars and to write them down, so I did not impose the same obligation on myself after I came to Princeton. These are the main reasons why I left the IHÉS for IAS in Princeton. Concerning the difference between the two institutions, I would say that the Institute for Advanced Study is older, bigger and more stable. Both are very similar in the way that there are many young visitors that come there. So they are not places where you can fall asleep since you will always be in contact with young people who will tell you that you are not as good as you think you are. In both places there are physicists, but I think the contact with them was more fruitful for me in Princeton than it was in Bures. In Princeton, there have been common seminars. One year was very intense, with both mathematicians and physi cists participating. This was due mainly to the presence of Edward Witten. He has received the Fields Medal even though he is a physicist. When Witten asks me questions, it’s always very interesting to try to answer them, but it can be frustrating as well. Princeton is also bigger in the sense that it has not only maths and physics, but also the School of Historical Studies and the School of Social Sciences. There is no real scientific interaction with these Schools but it is pleasant to be able to go and hear a lecture about, for instance, ancient China. One good feature about Bures which you do not have in Princeton is the following: In Bures, the cafeteria is too small. So you sit where you can and you don’t get to choose the people you are sitting with. I was often sitting next to an analyst or a physicist and such random informal interactions are very useful. In Princeton, there is one table for the mathematicians, another for the astronomers, the ordinary physicists and so on. You will not be told to go away if you sit down at the wrong table, but still there is segregation. Pierre Deligne 195

The Institute for Advanced Study has a big endowment, while the IHÉS had none, at least when I was there. This didn’t affect the scientific life. Sometimes it created instability, but the administration was usually able to hide the difficulties from us.

Apart from your connections with French and US mathematics, you have also had a very close contact with Russian mathematics for a long time, even from long before the fall of the iron curtain. In fact, your wife is the daughter of a Russian mathematician. How did your contact with Russian mathematics develop? Grothendieck or Serre told Manin, who was in Moscow at the time, that I had done some interesting work. The Academy invited me to a conference for I. M. Vinogra- dov [1891–1983], a terribly anti-Semitic person, by the way. I came to Russia, and I found a beautiful culture for mathematics. At that time mathematics was one of the few subjects where the communist party could not meddle, as it did not understand it at all, and this turned it into a space of freedom. We would go to somebody’s home and sit by the kitchen table to discuss mathematics over a cup of tea. I fell in love with the atmosphere and this enthusiasm for mathematics. Moreover, Russian mathematics was one of the best in the world at that time. Today there are still good mathematicians in Russia, but there has been a catastrophic emigration. Furthermore, among those wanting to stay, many need to spend at least half of the time abroad, just to make a living.

You mentioned Vinogradov and his anti-Semitism. You talked to somebody and asked whether he was invited? It was Piatetskii-Shapiro [1929–2009]. I was completely ignorant. I had a long discussion with him. For me it was obvious that someone like him should be invited by Vinogradov, but I was explained that that was not the case. After this introduction to Russian mathematics, I still have some nostalgia for the beautiful memories of being in Moscow and speaking with Yuri Manin, Ser- gei Bernstein [1880–1968] or being at the Gelfand seminar. There was a tradition, which still exists, of a strong connection between the university and the secondary education. People like Andrey Kolmogorov [1903–1987] had a big interest in secondary education (perhaps not always for the best). They have also the tradition of Olympiads and they are very good at detecting promising people in mathematics early on in order to help them. The culture of seminars is in danger because it’s important that the head of the seminars is working full time in Moscow and that is not always the case. There is a whole culture which I think it’s important to preserve. That is the reason why I used half of the Balzan Prize to try to help young Russian mathematicians.

That was by a contest that you arranged. Yes. The system is falling apart at the top because there is no money to keep people, but the infrastructure was so good that the system continues to produce very good 196 Abel Prize 2013 young mathematicians. One has to try to help them and make it possible for them to stay somewhat longer in Russia so that the tradition can continue.

Competition and collaboration in mathematics

Some scientists and mathematicians are very much driven by the aim to be the first to make major discoveries. That seems not to be your main driving force? No. I don’t care at all.

Do you have some comments on this culture in general? For Grothendieck it was very clear: he once told me that mathematics is not a competition sport. Mathematicians are different and some will want to be the first, especially if they are working on very specific and difficult questions. For me it’s more important to create tools and to understand the general picture. I think mathematics is much more a collective enterprise of long duration. In contrast to what happens in physics and biology, mathematical articles have long and useful lives. For instance, the automatic evaluation of people using bibliographic criteria is particularly perverse in mathematics, because those evaluation methods take only account of papers published during the last three or five years. This does not make sense in mathematics. In a typical paper of mine, I think at least half of the papers cited can be twenty to thirty years old. Some will even be two hundred years old.

You like to write letters to other mathematicians? Yes. Writing a paper takes a lot of time. Writing it is very useful, to have everything put together in a correct way, and one learns a lot doing so, but it’s also somewhat painful. So in the beginning of forming ideas, I find it very convenient to write a letter. I send it, but often it is really a letter to myself. Because I don’t have to dwell on things the recipient knows about, some short-cuts will be all right. Sometimes the letter, or a copy of it, will stay in a drawer for some years, but it preserves ideas and when I eventually write a paper, it serves as a blue-print.

When you write a letter to someone and that person comes with additional ideas, will that result in a joint paper? That can happen. Quite a lot of my papers are by me alone and some are joint work with people having the same ideas. It is better to make a joint paper than having to wonder who did what. There are a few cases of genuine collaborations where different people have brought different intuitions. This was the case with George Lusztig. Lusztig had the whole picture of how to use l-adic cohomology for group representations, but he did not know the techniques. I knew the technical aspect of l-adic cohomology and I could give him the tools he needed. That was real collaboration. A joint paper with Morgan, Griffiths and Sullivan was also a genuine collaboration. Pierre Deligne 197

Also with Bernstein, Beilinson and Gabber: we put together our different un- derstandings.

Work style, pictures, and even dreams

Your CV shows that you haven’t taught big classes of students a lot. So, in a sense, you are one of the few full-time researchers in mathematics. Yes. And I find myself very lucky to have been in this position. I never had to teach. I like very much to speak with people. In the two institutions where I have worked young people come to speak with me. Sometimes I answer their questions, but more often I ask them counter-questions which sometimes are interesting, too. So this aspect of teaching with one-to-one contact, trying to give useful information and learning in the process, is important to me. I suspect it must be very painful to teach people who are not interested, but are forced to learn math because they need the grade to do something else. I would find that repulsing.

What about your mathematical work style? Are you most often guided by examples, specific problems and computations, or are you rather surveying the landscape and looking for connections? First I need to get some general picture of what should be true, what should be accessible and what tools can be used. When I read papers I will not usually remember the details of the proofs, but I will remember which tools were used. It is important to be able to guess what is true and what is false in order not to do completely useless work. I don’t remember statements which are proved, but rather I try to keep a collection of pictures in my mind. More than one picture, all false but in different ways, and knowing in which way they are false. For a number of subjects, if a picture tells me that something should be true, I take it for granted and will come back to the question later on.

What kind of pictures do you have of these very abstract objects? Sometimes very simple things! For instance, suppose I have an algebraic variety, and hyperplane sections, and I want to understand how they are related, by looking at a pencil of hyperplane sections. The picture is very simple. I draw it in my mind something like a circle in the plane and a moving line which sweeps it. Then I know how this picture is false: the variety is not one-dimensional, but higher dimensional and when the hyperplane section degenerates, it is not just two intersection points coming together. The local picture is more complicated, like a conic which becomes a quadratic cone. These are simple pictures put together. When I have a map from some space to another I can study properties it has. Pictures can then convince me that it is a smooth map. Besides having a collection of pictures, I also have a collection of simple counter-examples, and statements that 198 Abel Prize 2013

I hope to be true have to be checked against both the pictures and the counter-examples.

So you think more in geometric pictures than algebraically? Yes.

Some mathematicians say that good conjectures, or even good dreams, are at least as important as good theorems. Would you agree? Absolutely. The Weil conjectures, for instance, have created a lot of work. Part of the conjecture was the existence of a cohomology theory for algebraic systems, with some properties. This was a vague question, but that is all right. It took over twenty years of work, even a little more, in order to really get a handle on it. Another example of a dream is the Langlands program which has involved many people over fifty years, and we have now only a slightly better grasp of what is happening. Another example is the philosophy of motives of Grothendieck about which very little is proved. There are a number of variants taking care of some of the ingredients. Sometimes, such a variant can be used to make actual proofs, but more often the philosophy is used to guess what happens, and then one tries to prove it in another way. These are examples of dreams or conjectures that are much more important than specific theorems.

Have you had a “Poincaré moment” at some time in your career where you, in a flash, saw the solution of a problem you had worked on in a long time? The closest I have been to such a moment must have been while working on the Weil conjecture when I understood that perhaps there was a path using Rankin against Grothendieck. It took a few weeks after that before it really worked, so it was a rather slow development. Perhaps also for the definition of mixed Hodge structures, but also in this case, it was a progressive process. So it was not a complete solution in a flash.

When you look back on fifty years of doing mathematics, how have your work and your work style changed over the years? Do you work as persistently as you did in your early years? I am not as strong as I was earlier, in the sense that I cannot work as long or as intensively as I did. I think I have lost some of my imagination but I have much more technique that can act as a substitute to some extent. Also the fact that I have contact with many people, gives me access to some of the imagination I am lacking myself. So when I bring my technique to bear, the work can be useful, but I’m not the same as when I was thirty.

You have retired from your professorship at IAS rather early… Yes, but that’s purely formal. It means I receive retirement money instead of a salary; and no School meetings for choosing next year’s members. So that’s all for the best, it gives me more time for doing mathematics. Pierre Deligne 199

Hopes for the future

When you look at the development of algebraic geometry, number theory and the fields that are close to your heart, are there any problems or areas where you would like to see progress soon? What would be particularly significant, in your opinion? Whether or not it’s within reach in ten years, I have abso- From left to right: Pierre Deligne, Martin Raussen and lutely no idea; as it should be… Christian Skau. (Photo: Anne-Marie Astad) but I would very much like to see progress in our understanding of motives. Which path to take and what are the correct questions, is very much in the air. Grothendieck’s program relied on proving the existence of algebraic cycles with some properties. To me this looks hopeless, but I may be wrong. The other kind of question for which I would really like to see some progress is connected with the Langlands program, but that is a very long story… In yet another direction, physicists regularly come up with unexpected conjectures, most often using completely illegal tools. But so far, whenever they have made a prediction, for instance a numerical prediction on the number of curves with certain properties on some surface – and these are big numbers, in the millions perhaps – they were right! Sometimes previous computations by mathematicians were not in accordance with what the physicists were predicting, but the physicists were right. They have put their fingers on something really interesting, but we are, so far, unable to capture their intuition. Sometimes they make a prediction and we work out a very clumsy proof without real understanding. That is not how it should be. In one of the seminar programs that we had with the physicists at IAS, my wish was not to have to rely on Ed Witten but instead to be able to make conjectures myself. I failed! I did not understand enough of their picture to be able to do that, so I still have to rely on Witten to tell me what should be interesting.

What about the Hodge conjecture? For me, this is a part of the story of motives, and it is not crucial whether it is true or false. If it is true, that’s very good and it solves a large part of the problem of constructing motives in a reasonable way. If one can find another purely algebraic notion of cycles for which the analogue of the Hodge conjecture holds, and there are a number of candidates, this will serve the same purpose, and I would be as happy as if the Hodge conjecture were proved. For me it is motives, not Hodge, that is crucial. 200 Abel Prize 2013

Private interests – and an old story

We have the habit of ending these interviews by asking questions that are outside of mathematics. Could you tell us a little bit about your private interests outside your profession? We know about your interest in nature and in garden- ing, for example. These are my main interests. I find the earth and nature so beautiful. I don’t like just to go and have a look at a scenery. If you really want to enjoy the view from a mountain, you have to climb it by feet. Similarly, to see the nature, you have to walk. As in mathematics, in order to take pleasure in nature – and the nature is a beautiful source of pleasure – one has to do some work. I like to bicycle because that’s also a way to look around. When distances are a little bigger than what is convenient by feet, this is another way of enjoying the nature.

We heard that you also build igloos? Yes. Unfortunately, there’s not enough snow every year and even when there is, snow can be tricky. If it’s too powdery, it’s impossible to do anything; likewise if it’s too crusty and icy. So there is maybe just one day, or a few hours each year when building an igloo is possible, and one has to be willing to do the work of packing the ice and putting the construction together.

And then you sleep in it? And then I sleep in the igloo, of course.

You have to tell us what happened when you were a little child. Yes. I was in Belgium at the sea-side for Christmas, and there was much snow. My brother and sister, who are much older than me, had the nice idea to build an igloo. I was a little bit in the way. But then they decided I might be useful for one thing: if they grabbed me by my hands and feet, I could be used to pack the snow.

Thank you very much for granting us this interview. These thanks come also on behalf of the Norwegian, the Danish and the European mathematical societies that we represent. Thank you very much! Thank you. Abel Prize 2014: Yakov G. Sinai

The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2014 to

Yakov G. Sinai Princeton University, New Jersey, USA and Landau Institute for Theoretical Phys- ics, The Russian Academy of Sciences, Moscow, Russia

“for his fundamental contributions to dynamical systems, ergodic theory, and mathematical physics.”

Ever since the time of Newton, differential equations have been used by mathematicians, scientists and engineers to explain natural phenomena and to predict how they evolve. Many equations incorporate stochastic terms to model unknown, seemingly random, factors acting upon that evolution. The range of modern applications of deterministic and stochastic evolution equations encompasses such diverse issues as plane- tary motion, ocean currents, physiological cycles, population dynamics, and electrical networks, to name just a few. Some of these phenomena can be foreseen with great accu- racy, while others seem to evolve in a chaotic, unpredictable way. Now it has become Abel Laureate Yakov G. Sinai. clear that order and chaos are intimately (Photo: Knut Falch) connected: we may find chaotic behavior in deterministic systems, and conversely, the statistical analysis of chaotic systems may lead to definite predictions. Yakov Sinai made fundamental contributions in this broad domain, discovering surprising connections between order and chaos and developing the use of probability and measure theory in the study of dynamical systems. His achievements include

* 21.9.1935

201 202 Abel Prize 2014 seminal works in ergodic theory, which studies the tendency of a system to explore all of its available states according to certain time statistics; and statistical mechanics, which explores the behavior of systems composed of a very large number of particles, such as molecules in a gas. Sinai’s first remarkable contribution, inspired by Kolmogorov, was to develop an invariant of dynamical systems. This invariant has become known as the Kolmog- orov–Sinai entropy, and it has become a central notion for studying the complexity of a system through a measure-theoretical description of its trajectories. It has led to very important advances in the classification of dynamical systems. Sinai has been at the forefront of ergodic theory. He proved the first ergodicity theorems for scattering billiards in the style of Boltzmann, work he continued with Bun- imovich and Chernov. He constructed Markov partitions for systems defined by iter- ations of Anosov diffeomorphisms, which led to a series of outstanding works showing the power of symbolic dynamics to describe various classes of mixing systems. With Ruelle and Bowen, Sinai discovered the notion of SRB measures: a rather general and distinguished invariant measure for dissipative systems with chaotic behavior. This versatile notion has been very useful in the qualitative study of some archetypal dynamical systems as well as in the attempts to tackle real-life complex chaotic behavior such as turbulence. Sinai’s other pioneering works in mathematical physics include: random walks in a random environment (Sinai’s walks), phase transitions (Pirogov–Sinai theory), one-dimensional turbulence (the statistical shock structure of the stochastic Burgers equation, by E–Khanin–Mazel–Sinai), the renormalization group theory (Bleher– Sinai), and the spectrum of discrete Schrödinger operators. Sinai has trained and influenced a generation of leading specialists in his research fields. Much of his research has become a standard toolbox for mathematical physicists. His works had and continue to have a broad and profound impact on mathematics and physics, as well as on the ever-fruitful interaction of these two fields.

Yakov Sinai and Crown Prince Haakon. (Photo: Håkon Mosvold Larsen) Yakov G. Sinai 203

Professor Sinai – first of all we would like to express our congratulations. You have been selected as the 12th winner of the Abel Prize and you will receive the prize tomorrow. We are curious; did you have any expectations beforehand? How did you receive the information? In early March this year I got to know that the Abel Committee were interested in taking my photograph. A friend of mine told me this and I thought this must mean something because this had never happened before. And then there was a telephone call from the Norwegian Academy of Science and Letters informing me about the prize.

And this was on the same day that the prize was announced here in Oslo? Yes. That happened on March 26.

Youth

You were born in Moscow in 1935 into a family of scientists. Your parents were both biologists and your grandfather was well known in mathematics. We suppose this had important consequences for the development of your interests? Definitely, yes. How could I say no to this question? Everything was about mathematics and mathematical events. But, at that time, I preferred to play volleyball. The influence of mathematics was not as direct as you may think. I participated in many Olympiads in mathematics during my school years but never had any success and never won any awards. I say this to young people who have never won in Olym- piads; there may be compensation in the future. At this time, my grandfather was of a great age and he did not have the energy to push me into mathematics. And I also have a half-brother, G. I. Barrenblatt, who worked at Moscow State University and who was convinced that I should pursue a career in mathematics.

Do you remember when you found out that you had an exceptional talent for mathematics? If at all, it happened very late. I was a graduate student when I brought my paper on entropy to my advisor A. N. Kolmogorov [1903–1987] and he said: “At last you can compete with my other students.” But I am not sure that he was right and that I have an exceptional talent for mathematics.

You must have entered school at about the same time that Nazi Germany in- vaded Russia. How did the war influence your first years at school? I entered school in 1943 after my family returned from the evacuation of Moscow. At that time boys and girls studied separately; at the end of each year, we had about 10 exams. Before the evacuation, life was different. It was forbidden to leave windows open in the apartments in Moscow because it had to be dark. In 1943, 204 Abel Prize 2014 windows were allowed to be open again. In Moscow there were no clear signs of war. But life was hard because of the time of Stalin. People had to behave in a special way.

And that also influenced life at school? It was everywhere; you could be expelled from school or even sent to prison for being controversial.

Were there teachers with a lot of influence on you, in particular in mathematics? We had a very good teacher in mathematics at our high school. His name was Vasily Alekseevich Efremov and he was a great old-style schoolteacher. He always brought us his problems in accurate handwriting on a piece of paper which he distributed among the students. Because of the well organised and inspiring work, mathematics was very popular among us. We discussed and tried to solve his problems. At this time I was not among the best in the class. There were definitely other students who were much better than me.

What was your age at this point? This was still in high school just before I entered university. Thus, I was probably 16 or 17 years old.

Student at MSU-Mech-Mat

You entered the Faculty of Mechanics and Mathematics at Moscow State Uni- versity in 1952 still a teenager. How was it to study at this famous institution as such a young student? We had a number of very good professors there. For example, the lecture course in analysis was given by M. A. Lavrentiev [1900–1980], who was a very famous scientist at the time. He was also involved in administration but was a great teacher and his lectures were very interesting. We also had a very good lecturer in classical mechanics, Chetaev. I was his student in the second year. Moreover, we had lectures in geometry given by Bachvalov [1934–2005], who was famous in Russia but not so much known in the West. There is a story about him: when we entered the university on 1 September, he came into the room and said: “Let’s continue.” And that was the beginning of his lecture course. In algebra, we had lectures by Dynkin [1924–2014], who was an excellent teacher for those who had started to study. These were lectures at a very high level. Dyn- kin used to hand out interesting problems for the enthusiastic students. Among such students in my year I could particularly mention I. Girsanov, who became a famous probabilist, and L. Seregin.

Was it Dynkin who inspired your first paper in mathematics? Yakov G. Sinai 205

Yes. I was a student of Dynkin during the second and third years and I wrote the first paper under his supervision. I solved a problem that he formulated for me; this became my first published paper when I was a student in the third year. I loved the work I did and still do. Dynkin wanted me to work on problems on Markov processes in the style of Feller. The papers by Feller [1906–1970] became very popular in Moscow at that time and Dynkin suggested that I should continue along this line. However, I was not very excited and interested in it.

To what extent were mathematics and mechanics integrated in the curriculum? These were independent parts of the curriculum. Everybody could attend lectures within each branch. I was attending lectures in mathematics and mechanics but also, to a minor degree, some lectures in physics. But on the whole it was mainly in the mathematics department.

We imagine that besides Dynkin, Kolmogorov and Gelfand [1913–2009] must have been very important figures for you? Kolmogorov had many students and I became one of them. His students had complete freedom to work on any problem. Kolmogorov loved to discuss their results with them. There were several cases when Kolmogorov wrote their papers in order to teach them how to write mathematical texts. Kolmogorov organised a seminar, which was initially a seminar on random processes and later became a seminar on dynamical systems and ergodic theory. I began to attend, together with other mathematicians like Arnold [1937–2010], Alek- seev, Tikhomirov and others. Later I became a student of Kolmogorov. At that time he was also interested in problems of entropy in different compact and functional spaces. Questions of this type were very much discussed at that time.

But Gelfand tried to recruit you as a graduate student as well? Yes. Gelfand organised his famous seminar, which was attended by many mathematicians of different generations. I took part in it for many years. It happened, if I remember it correctly, in 1955 when Gelfand was writing a famous volume in his series of books on distributions. Gelfand was interested in probability theory and he wanted me to become his student. We had some discussions about it and I told him that I wanted very much to work on problems related to probability theory but I had already written a paper under the supervision of Dynkin. He asked me: “Do you want to have probability theory as an appetiser or as a dessert?” I answered: “I want it as a main course.” That was the end of the story… This did not mean that our contact came to an end; we met many times, especially when he worked on problems in representation theory, which were connected with problems in ergodic theory, like the theory of horocycles and others. We discussed this many times. I attended Gelfand’s seminars for many years because Gelfand had the remarkable ability to explain difficult mathematical topics in a clear and simple way. 206 Abel Prize 2014

Dynamical systems. Entropy and chaos

Could you explain what a dynamical system is? We understand dynamical systems as objects that describe all types of evolution. The most interesting case is non-linear dynamical systems, when the equation for the dynamics of the evolution is non-linear. There can be many different phenomena, which require deep analysis.

And among these dynamical systems, what is an ergodic system? I have a very good example for an ergodic system which I always explain to my students. Suppose you want to buy a pair of shoes and you live in a house that has a shoe store. There are two different strategies: one is that you go to the store in your house every day to check out the shoes and eventually you find the best pair; another is to take your car and to spend a whole day searching for footwear all over town to find a place where they have the best shoes and you buy them immediately. The system is ergodic if the result of these two strategies is the same. The entropy characterises the growth of the number of possibilities in dynamics. I heard the first explanation of this role of entropy from I. M. Gelfand.

Ergodic theory originally came from physics, in particular from the study of Hamiltonian equations. Can you explain in general terms what chaos is and how one can measure it? This is the subject of my lecture, which I will give the day after tomorrow, but I can summarise it briefly here. The main question concerns the difference between chaos theory and probability theory. In probability theory one deals with statistical experiments, say you toss a coin 100 times. One can have many different series resulting from this experiment and study the result. If you consider the problem of chaos and for example want to measure the temperature at the same point you make the measurement during the year, you now have only one realisation of the temperature. You cannot have a hundred realisa- tions of the temperature at a given place and at a given time. So the theory of chaos studies the series when the results of measurements have a limit as time tends to infinity and how to describe this limit. The existence of the limit actually follows from some hypothesis about the equations of motion. This gives the existence of the distribution, which determines the value of all kinds of averages (or, it is better to say, the existence of the averages and also finding their values). Then, the question is: what are the equations of motion which determine the distribution and these averages? The basic statement in chaos theory is that the dynamics must be unstable. Instability means that small perturbations of the initial conditions lead to large perturbations in the dynamics after some time. Then there is a mathematical theory that says that if the system is unstable the time averages exist and there is a possibility of calculating them. This is the general description of what is done in chaos. A more precise description requires more mathematics. Yakov G. Sinai 207

How do you measure chaos? Does entropy come into the picture here? If we understand chaos as mentioned already, i.e. as the existence of time averages and also properties related to mixing, then there is a natural description of chaos in terms of some special distribution. Entropy is used in the theory of unstable systems and it characterises how many types of dynamics a given system can have. It is certainly a very useful notion because the positivity of the entropy determines other properties of the systems that can be studied. Physicists always expressed their hope that entropy would allow them to understand turbulence (see, for example, the paper by B. Chirikov and the books by G. Zaslavski [1935–2008], R. Sagdeev and others). It is hard to say that this hypothesis is true. On the other hand there are many situations in physics where systems have small entropy.

Definition of entropy for dynamical systems

Kolmogorov had come up with the definition of entropy for Bernoulli shifts but then he changed it to a definition that was not invariant. Then you came with the correct definition. What is now called the Kolmogorov–Sinai theorem gives an efficient way to compute the entropy. Kolmogorov started his seminar with von Neumann’s [1903–1957] theory of dynamical systems with pure point spectrum, which he explained in a purely probabilistic way. Later I found this approach in the book by Blanc-Lapierre and Fortet.1 Everything in Kolmogorov’s seminars was very exciting. At that time we believed that the main problem in ergodic theory was to extend the theory of von Neu- mann to systems with continuous spectrum that can be constructed in terms of the second homology group of the spectrum with coefficients in the ring of bounded operators. It did not work but the idea remained. At that time, Kolmogorov spent his time primarily on problems in information theory and the concept of dimension of linear spaces. I do not know how it happened but one day Kolmogorov came to his lecture and presented his definition of entropy. Using modern terminology, one can say that he gave the definition of entropy for Bernoulli shifts and thus proposed a new invariant for this class of dynamical systems. It was certainly a great result. Kolmog- orov wrote his text. He submitted it for publication and left for Paris where he spent the whole semester. As is known, the text that was submitted for publication was different from what he explained in class. In his paper he introduced a new class of systems which he called quasi-regular. Later they were called K-systems (K for Kolmogorov). For this class of systems he introduced the notion of entropy. While Kolmogorov was away, I was thinking about a definition of entropy that could work for all dynamical systems. Later it appeared in my paper on entropy.

1 A. Blanc-Lapierre, R. Fortet, Théorie des fonctions aléatoires, Paris: Masson et Cie., 1953. 208 Abel Prize 2014

At that time, there was a clear feeling that for dynamical systems appearing in probability theory, the entropy is usually positive, while for dynamical systems generated by ODEs it should be zero. Thus, there seemed to be a possibility to distinguish dynamical systems in probability theory from dynamical systems in analysis.

How about your connection with Rokhlin? The story about my connection with Rokhlin [1919–1984], who later became a close friend of mine, started when Kolmogorov’s paper on entropy appeared in 1958. At that time, Rokhlin lived in a small provincial town Kolomna not far from Moscow. He had a very good graduate student, Leonid Abramov. There are several general theorems that were proven by Abramov, like the entropy of special flows, and other things like Abramov’s formula, etc. When Rokhlin heard about the paper by Kol- mogorov, he sent Abramov to Moscow to find out what had been really done, what was the situation and if possible to bring the text. When Abramov came to Moscow, he found me, we talked a lot and I taught him everything I knew. Abramov then invited me to Kolomna to talk to Rokhlin and I accepted the invitation. I remember my first visit to Kolomna very well. Rokhlin had an apartment there, which was very orderly; everything was very accurate and he was dressed very well. We began to talk and he made a very strong impression on me. Rokhlin had great experience in ergodic theory because he had already published several papers in this field. His doctoral thesis was also about this subject. Rokhlin formulated a number of interesting problems in ergodic theory. Some of them were connected with Rokhlin’s theory of measurable partitions. This theory became very useful in ergodic theory because through it one can understand con- ditional probabilities in probability theory much better. One of the problems that I began to work on under the influence of Rokhlin was the calculation of entropy for group automorphisms of the two-dimensional torus. At that time it was not known that Kolmogorov’s definition had to be modified; the analysis was rather difficult and I could not achieve anything. Following the ideology of that time, I tried to prove that the entropy was zero but all my attempts failed. Then I visited Kolmogorov and showed him my drawings. He said that it was clear in this case that the entropy must be positive. After that I proved the result. At that time there was no question about publication of my paper because Kol- mogorov’s paper on entropy had been published and it was not clear why another definition of entropy was needed. However, after some time, Rokhlin pointed out his result about the deficiencies in the definition by Kolmogorov. It became clear that I had to publish my paper with the definition and the calculation of the entropy for the automorphism which I had already done at that time. This was the beginning of my contact with Rokhlin. After that, he organised a seminar on ergodic theory in Moscow, which was attended by Arnold, Anosov [1936–2014], Alexeev and others. In parallel, he had a seminar in topology where Novikov was the central figure. Yakov G. Sinai 209

Later Rokhlin moved to Leningrad (Saint Petersburg) and I used to go there to give talks at his seminar on later results.

Billiard systems

You then came up with an extremely interesting example of an ergodic system: the so-called billiards. Can you explain what these are? A billiard, as people know, is the motion of a ball on the billiard table. An interesting mathematical theory arises if you allow the table to have a more or less arbitrary form. A natural question, which was actually raised by the Russian physicist Krylov long before the theory of entropy appeared, was: Which billiard systems have the same instability as the dynamics of particles moving in a space of negative curvature? Particles moving in a space with negative curvature yield the best example of unstable systems. The theory of billiards says that if the boundary of the table is con- cave then the system is unstable in the sense we previously described. If we consider two initial conditions with different values of the velocities then the corresponding trajectories diverge exponentially. If you consider a typical unstable billiard, namely the usual square billiard with a circle removed, then the difference between this billiard and the usual billiard is that for the unstable billiard the particles come to the holes much faster than for the usual billiard.

This may become a little technical now. You proved a very important result about systems with positive entropy. Given a system with positive entropy you can find a Bernoulli shift – which is a so-called factor – with the same entropy. This implies that if you have two Bernoulli shifts with the same entropy, they are what is called weakly isomorphic. Ornstein proved later that entropy is a complete invariant for Bernoulli shifts. It follows then from the work of Orn- stein that the billiard example is the most chaotic system and is actually a Bernoulli flow, right? From Ornstein’s theorem it follows that if we have two ergodic billiard systems with the same value of entropy then they are isomorphic. This is a remarkable and great result.

So coin tossing is, in a sense, similar to the deterministic billiard system – an amazing fact. My result says that if you have a system with positive entropy there could be subsys- tems that move like Bernoulli shifts.

What about billiard systems in higher dimensions? Is anything known there? A lot of things are known. We have, for example, the result from the Hungarian mathematician Nándor Simányi who is in Alabama now. He studied multi-dimensional dynamical systems that eventually become unstable and have positive entropy and are ergodic. 210 Abel Prize 2014

You introduced Markov partitions in your study of Anosov diffeomorphisms. This led to what later became known as the Sinai–Ruelle–Bowen measure, also referred to as the SRB-measure. Would you please explain? First of all, there was my paper where I constructed this measure for the case of the so-called Anosov systems, or just hyperbolic systems. Then there was a paper by Bowen [1947–1978] and Ruelle where they extended this construction to systems considered by Smale, that is, Axiom A systems with hyperbolic behaviour. These measures are important if you study irreversible processes in these systems. Suppose you start with some non-equilibrium distribution and consider the evolution and you ask how a non-equilibrium distribution converges to the equilibrium one. The result of the theory says that the evolution is in a sense very non-uniform, along some directions the expansion is very small and all the time averages behave very well and converge to a limit. But along other directions this convergence is very erratic and hence it can only be studied using probability theory. So the measures, which are called SRB-measures, are the ones which are smooth along some unstable directions and are very irregular along other directions. This is a class of measures that appears in the theory of evolution of distributions in the case of chaotic systems.

Are the SRB-measures related to Gibbs measures? Yes. These measures are examples of Gibbs measures. But the Gibbs measures are a much more general object.

Mathematics and physics

Let’s go back to more general questions, starting with the interplay between mathematics and physics. May we begin with the physicist Eugene Wigner [1902–1995], who in 1960 published the paper “The unreasonable effectiveness of mathematics in the natural sciences”, in which he gave many examples showing how mathematical formalism advanced physical theory to an extent that was truly amazing. Do you have a similar experience? My impression is that this effectiveness of mathematics is no longer a surprise for people. There are so many cases, for instance the fact that string theory is practically a mathematical theory for physics. Some time ago Joel Lebowitz organised a discussion about this phrase of Wigner – in particular how it can be that mathematics is so effective. The conclusion was that this is just a well-established fact. In my generation, there was a group of young mathematicians who decided to study physics seriously. However, there were different points of view of how to do mathematical physics. F. A. Berezin [1931–1980] always stressed that mathematicians should prove only results that are interesting for physicists. R. L. Dobrush- in [1929–1995] and I always tried to find in physical results some possibilities for mathematical research. Yakov G. Sinai 211

On the other hand, there seems to be influence going in the opposite direction. Physicists have had a noteworthy impact on questions in quantum geometry and sometimes even in number theory. They have come up with formalisms that were not really developed in mathematics but nevertheless led to correct predictions which could be verified only after lengthy mathematical development. So mathematics is effective but you can say that it is not effective enough.

You published in 2006 an article with the title “Mathematicians and Physicists = Cats and Dogs?” What is the main message of that paper? I wanted to show examples where mathematicians and physicists look at the same problems differently. One example for this is the following story: my student Piro- gov and I worked on problems in the theory of phase transitions in statistical physics. We proved several theorems and I went to meet the famous Russian physicist Ilya Lifshitz [1917–1982] to show him our theory; Lifshitz replaced Lev Landau [1908–1968] when Landau had his car accident, severely incapacitating him. When I presented the theory he stopped me and said: “It’s very simple what you are talking about.” He started to write formulas which eventually gave our results. I left him very much embarrassed and I started to think why this had happened. I realised that the final result of our theory was an obvious statement for him. He certainly did not know how to prove it but he did not need the proof. He just used it as an obvious fact.

There is a famous quotation of the great Gauss: “Now I have the result. The only thing remaining is the proof.” So intuition does play an important role in mathematics… I can also tell the following story, again connected to Gelfand. I explained to him a theorem, which we obtained together with Robert Minlos. And Gelfand said: “This is obvious. All physicists know this.” So we asked him if it was so obvious, should we write a text of 200 pages with complete proofs? He looked at us and said: “Certainly, yes!”

A Jewish mathematician in the Soviet Union

May we continue with a political question? You mentioned that being at school in the time of Stalin was not easy; life was still difficult for you when you entered university and started your career. You came from a Jewish family; in the Soviet Union, at least sometimes, a latent anti-Semitism prevailed… I can mention two cases in my career when I encountered anti-Semitism. The first one was the entrance examination, which I failed. The influence of my grandfather, who was Head of the Chair of Differential Geometry, and the help from the Pres- ident of Moscow University, I. G. Pertrovsky, were needed in order to give me the 212 Abel Prize 2014 possibility of being admitted to the University. This was a clear sign that things were not simple. The other case arose with my entrance examination to graduate school. This exam was about the history of the Communist Party; I was very bad in this topic and failed the exam (I don’t want to discuss the details). But P. S. Alexandrov – who was Head of the Mathematical Department at Moscow State University – together with Kolmogorov, visited the Head of the Chair of the History of the Party and asked her to allow me to have another attempt. She gave permission and I got a B on the second attempt, which was enough to enter the graduate school. The result was not clear a-priori and it could have gone either way.

In spite of these obstructions, it is quite obvious that many famous Russian mathematicians were and are of Jewish origin. This is quite amazing – can you offer any explanation? First something trivial: Jews had more traditions in learning than other nations. They study the Bible, the Talmud and other religious books and spend a lot of time doing this, which is conducive to learning. At that time, following the Jewish religion was strictly forbidden. People still did, however, but under very high pressure. If you do something under pressure you work more. There is some kind of conservation law. This is my opinion of why Jews could succeed.

You had to be much better in order to get the same opportunities? I think it would be wrong to say that we had this feeling. We certainly tried to prepare for all exams and competitions; the result was not clear a-priori but there was always a hope that something could come out of it.

Perhaps another reason is (especially under Stalin but also later) that a lot of very intelligent people were attracted to the natural sciences because there were fewer restrictions than in, say, history or political science… That is certainly true. I can give you one example: at the time, Mech-Mat, the Fac- ulty of Mechanics and Mathematics, had many graduate students that came from other countries. The rule was that they could only have advisors who were members of the Party. But there were students who wanted to work with Arnold, with me or perhaps with some other people. The way out of this situation was the following: there were a number of people in the Party who became the students’ official advisors but the students actually worked with professors and mathematicians who were not members of the Party. East and West

You told us that you were not allowed to travel for many years, and this happened to a lot of Russian mathematicians at the time. Did these obstructions hamper or delay progress in science? Did it have the effect that Russian mathematics did not get recognition in the West that it deserved? Yakov G. Sinai 213

It is very difficult to answer your question because you are asking what would have happened if something didn’t happen. It is just impossible to say. It certainly caused harm but it is not clear how big it was.

Arnold was rather adamant about the lack of recognition. As a consequence of bad communication between East and West, results by Russian mathematicians during the isolation period were sometimes later rediscovered in the West. Therefore Russian mathematicians did not get the credit they deserved. I have perhaps a special point of view concerning this. The question is whether some results can be stolen or not. My point of view, to which many people probably won’t agree, is that if a result can be stolen, it is not a very good result.

Tell us about the Landau Institute for Theoretical Physics in the Russian Acad- emy of Sciences – your workplace for many years. For many years the Landau Institute was the best institution in Russia. It was organised after Lev Landau’s untimely death as a result of a car accident. Its director I. M. Khalatnikov had a remarkable talent to find gifted people all over Russia and to invite them to the institute. After several years, the Landau Institute had a very strong group of physicists like Abrikosov [1928–2017], Gorkov [1929–2016], Dzy- aloshinski, A. B. Migdal [1911–1991], Larkin [1932–2005], Zakharov, Polyakov, A. A. Migdal and many others. The group of mathematical physicists was headed by S. P. Novikov and was much smaller. It turned out that there was a big area of theoretical physics in which mathematicians and physicists could understand each other very well. They could even work on similar problems. Among these mathematicians I can name Novikov, Krichever, Khanin, Shabat and Bogoyavlenskiy. Sometimes we invited physicists to explain to us their results in our seminars. The tradition of discussing problems of mutual interest still prevails.

You moved in 1993 from the Landau Institute to Princeton University while still maintaining your position in Moscow. Why was it so attractive for you to go to the USA? That is an easy question. First of all I had many friends at Princeton. When we met we always had many points for discussion and common interests. Another reason was that many people had escaped from Russia so the situation there was no longer what it was before. In previous times everybody was in Moscow and St. Petersburg and you could call everyone to ask questions or to have discussions. Now that became impossible. The working conditions were better in the West and in particular at Princeton.

You have now been in the USA for more than 20 years and you must know the American system almost as well as the Russian one. Could you tell us about how they compare from your perspective? 214 Abel Prize 2014

Concerning academic lives, it seems to me that they are more or less similar. How- ever, I must stress that I was never a member of any scientific committee at Mech- Mat at Moscow State University and I was never invited to participate in any organ- isational meetings. Now I am Chairman of the Scientific Council at the Institute of Transmission of Information.

Teaching and collaboration

You have been teaching courses and seminars for almost all your career. Do you have a particular technique or philosophy? First of all, I like to teach undergraduate courses rather than graduate courses for the following reason: when you teach undergraduate courses you can easily see how your students become cleverer and more educated as they absorb new notions and connections and so on. When you teach graduate courses, the subject matter is usually a narrow piece of work and students are mostly interested in some special issues that are needed for their theses. For me, that is less attractive. My basic principle is as follows: if people do not understand my explanations then this is my fault. I always ask students to ask questions as much as possible. Students who have asked me many questions during the lecture course have better chances for a good mark.

You have an impressive list of students that have done well after graduation under your supervision. Grigory Margulis, just to mention one name, won the Fields medal in 1978 and he will give one of the Abel lectures related to your work later this week… I think the reason for this is not because of me but because of the types of problems we worked on. We did very interesting mathematics and formulated interesting problems that students were attracted to. This is my explanation. Many students preferred to work independently and I was never against this.

You are a very good example of the fact that mathematicians can flourish in late age as well. We came across a paper on number theory that you published this year together with two of your students. You have also published other papers related to number theory so you must have kept an interest in that aspect of ergodic theory? Yes, definitely. In the field we are working in there are many problems that are more natural for ergodic theory than for number theory. I don’t want to be specific but we had a paper that was more natural for an ergodic theorist than for a number theorist, so we could get the results more easily.

Many joint papers appear on your list of publications. Apparently you like to have a lot of collaborators. Yakov G. Sinai 215

Well, I would say that they like it! And I’m not against it. It has never happened that I have asked someone to be my co-author. I can only talk about some problems and explain why they are interesting. But you are right, I have had many co-authors. I very much liked collaborating with Dong Li, who is now a professor at the University of British Columbia. When we work on the same problem we call each other many times a day. There are many others of my students with whom I liked to work. It’s different to work with different people. Certainly I can work with Russian mathematicians as well as with mathematicians from other countries. Sometimes I like to work alone but with age, I need co-authors.

You have only published one paper with Kolmogorov but you have mentioned that you would have liked to publish more papers with him. At a certain time, Kolmogorov decided that the Soviet Union did not have enough applied statistics. He worked on theoretical statistics and found many beautiful and deep results but he was not satisfied with the fact that the theorems in applied statistics were not used for practical purposes. He found a problem related to the motion of the rotational axis of the earth that could be studied with the help of mathematical statistics. French observatories published data about the axis of rotation every two weeks and Kolmogorov wanted to construct statistical criteria that could predict this motion. He wanted us to work on this problem and invited a very good geophysicist Yevgeny Fyodorov [1910–1981], who was one of the main experts in this field. We were sitting there – Kolmogorov and Fyodorov were present. Kolmogorov said: “Look at these people; they prefer to write a paper for Doklady instead of doing something useful.” (Doklady was the leading Russian journal.) In our joint paper (by M. Arató [1931–2015], A. Kolmogorov and myself) written on this occasion, practically everything was done and written by Kolmogorov. Later, M. Arató wrote a big monograph on that subject. In other cases, I often tried to explain my latest results to Kolmogorov. Some- times his reaction was unexpected: “Why did you work on that problem? You are already a grown-up?” But usually his reaction was very friendly. I regret very much that we never worked together; perhaps the reason is a difference in style.

Wasn’t it Kolmogorov who said that he spent a maximum of two weeks on a problem? Kolmogorov used to stress that he did not have papers on which he worked for a long time. He mostly prepared his papers, including the proof and the text, in just two weeks and this was a major difference in our approaches. Kolmogorov was a person with a strong temperament and he could not do anything slowly. I worked on some of my papers for years.

He was a towering figure, not only in Russian mathematics but worldwide in the 20th century. 216 Abel Prize 2014

Yes, definitely. Can I tell you one more story about him? It was when Kolmogo- rov was close to 80. I asked him how it happened that he was a pure mathematician, even though he worked on concrete physical problems like turbulence. He answered that he was studying the results of concrete experiments. He had a lot of papers with results from experiments lying on the floor. He was studying them and in this way he came up with his hypotheses on turbulence.

So his intuition was motivated by physical considerations? Yes. He subscribed to physical journals and one could say he was into physics in a big way.

Is that also true for you? Do you think mainly in terms of algebraic or analytic formulas? Or is it geometric intuition or even a mixture of all of that? It depends on the problem. I can come to the conclusion that some problem must have a specific answer. I just told a journalist the story about a problem in which I knew there should be a definite answer. I worked on this problem for two years and at the end of that time I discovered that the answer was one-half! In general, I probably prefer to develop theories, sometimes to find the right concepts rather than solving specific problems.

Have you had what we sometimes call a Poincaré moment, where all of sudden you see the proof? Ideas often come unexpectedly, sometimes like revelations. But it happens only after a long period, maybe years, of difficult work. It did not happen while trying to find a taxi or something similar. It was very hard work for a long time but then suddenly there was a moment where it became clear how the problem could be solved.

If you yourself made a list of the results that you are most proud of, what would it look like? I like all of them.

Mountaineering

You mentioned Arnold who died four years ago, an absolutely brilliant Rus- sian mathematician. Arnold is, among many other things, known for his contributions to the so-called KAM theory. You both followed Kolmogorov’s course and seminars in 1958. You told us that there was a close friendship already between your grandfathers. Both you and Arnold loved the outdoors and hiking. You once went to the Caucasus Mountains together and you have to tell us the story about what happened when you stayed in the tents with the shepherds. Yakov G. Sinai 217

That is a very funny story. The weather was very bad; there was a lot of rain. We came to the shepherds’ tent and they let us in and we could dry our clothes. We had lost our tent in the mountains so we decided to go back to try to find it. We started to walk back but these shepherds had some very big dogs – Caucasus dogs, a really big race. The shepherds wer- From left to right: Yakov Sinai, Martin Raussen and en’t there any longer and when Christian Skau. (Photo: Eirik Furu Baardsen) the dogs found out that we were leaving, they surrounded us and started to bark ferociously. Arnold began to yell back with all the obscenities he knew and the dogs did not touch him. But they attacked me. They didn’t touch my skin but they ripped my trousers apart. Finally, the shepherds came back and we were saved.

We would like to ask one final question that has nothing to do with mathematics: you have certainly focused on mathematics during your life but surely you have developed other interests also? I was interested, especially in former years, in many different sports. I was a volleyball player and I liked to ski, both downhill and cross-country. I also liked mountaineering but I cannot say I was a professional. I climbed often with a close friend of mine, Zakharov, who worked on integrable systems. We were climbing in the mountains together and once we were on a very difficult 300 metre long slope, which took us four hours to get down from! We had to use ropes and all sorts of gear. Nowadays, my possibilities are more limited.

Thank you very much for this most interesting conversation. We would like to thank you on behalf of the Norwegian, the Danish and the European Mathe- matical Societies. Thank you very much.

Abel Prize 2015: John F. Nash, Jr. and Louis Nirenberg

The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2015 to

John F. Nash, Jr. and Louis Nirenberg Princeton University, Courant Institute, New Jersey, USA New York University, USA

“for striking and seminal contributions to the theory of nonlinear partial differential equations and its applications to geometric analysis.”

Abel Laureate John F. Nash, Jr. Abel Laureate Louis Nirenberg. (Photo: Peter Badge) (Photo: Peter Badge)

* 13.6.1928, † 23.5.2015 * 28.2.1925

219 220 Abel Prize 2015

Partial differential equations are used to describe the basic laws of phenomena in physics, chemistry, biology, and other sciences. They are also useful in the analysis of geometric objects, as demonstrated by numerous successes in the past decades. John Nash and Louis Nirenberg have played a leading role in the development of this theory, by the solution of fundamental problems and the introduction of deep ideas. Their breakthroughs have developed into versatile and robust techniques, which have become essential tools for the study of nonlinear partial differential equations. Their impact can be felt in all branches of the theory, from fundamental existence results to the qualitative study of solutions, both in smooth and non- smooth settings. Their results are also of interest for the numerical analysis of partial differential equations. Isometric embedding theorems, showing the possibility of realizing an intrinsic geometry as a submanifold of Euclidean space, have motivated some of these developments. Nash’s embedding theorems stand among the most original results in geometric analysis of the twentieth century. By proving that any Riemannian geometry can be smoothly realized as a submanifold of Euclidean space, Nash’s smooth (C∞) theorem establishes the equivalence of Riemann’s intrinsic point of view with the older extrinsic approach. Nash’s non-smooth (C1) embedding theorem, improved by Kuiper, shows the possibility of realizing embeddings that at first seem to be forbidden by geometric invariants such as Gauss curvature; this theorem is at the core of Gromov’s whole theory of convex integration, and has also inspired recent spectacular advances in the understanding of the regularity of incompressible fluid flow. Nirenberg, with his fundamental embedding theorems for the sphere S2 in R3, having prescribed Gauss curvature or Riemannian metric, solved the classical problems of Minkowski and Weyl (the latter being also treated, simultaneously, by Pogorelov). These solutions were important, both because the problems were representative of a developing area, and because the methods created were the right ones for further applications. Nash’s work on realizing manifolds as real algebraic varieties and the Newlander–Nirenberg theorem on complex structures further illustrate the influence of both laureates in geometry. Regularity issues are a daily concern in the study of partial differential equations, sometimes for the sake of rigorous proofs and sometimes for the precious qualitative insights that they provide about the solutions. It was a breakthrough in the field when Nash proved, in parallel with De Giorgi, the first Hölder estimates for solutions of linear elliptic equations in general dimensions without any regularity assumption on the coefficients; among other consequences, this provided a solution to Hilbert’s 19th problem about the analyticity of minimizers of analytic elliptic integral functionals. A few years after Nash’s proof, Nirenberg, together with Agmon and Douglis, established several innovative regularity estimates for solutions of linear elliptic equations with Lp data, which extend the classical Schauder theory and are extremely useful in applications where such integrability conditions on the data are available. These works founded the modern theory of regularity, which has since grown immensely, with applications in analysis, geometry and probability, even in very rough, non-smooth situations. Symmetry properties also provide essential information about solutions of non- linear differential equations, both for their qualitative study and for the simplifica- John F. Nash, Jr. and Louis Nirenberg 221 tion of numerical computations. One of the most spectacular results in this area was achieved by Nirenberg in collaboration with Gidas and Ni: they showed that each positive solution to a large class of nonlinear elliptic equations will exhibit the same symmetries as those that are present in the equation itself. Far from being confined to the solutions of the problems for which they were devised, the results proved by Nash and Nirenberg have become very useful tools and have found tremendous applications in further contexts. Among the most popular of these tools are the interpolation inequalities due to Nirenberg, including the Gagliardo-Nirenberg inequalities and the John-Nirenberg inequality. The latter governs how far a function of bounded mean oscillation may deviate from its average, and expresses the unexpected duality of the BMO space with the Hardy space H1. The Nash–De Giorgi–Moser regularity theory and the Nash inequality (first proven by Stein) have become key tools in the study of probabilistic semigroups in all kinds of settings, from Euclidean spaces to smooth manifolds and metric spaces. The Nash–Moser inverse function theorem is a powerful method for solving perturbative nonlinear partial differential equations of all kinds. Though the widespread impact of both Nash and Nirenberg on the modern toolbox of nonlinear partial differential equations cannot be fully covered here, the Kohn–Nirenberg theory of pseudo-differential operators must also be mentioned. Besides being towering figures, as individuals, in the analysis of partial differential equations, Nash and Nirenberg influenced each other through their contributions and interactions. The consequences of their fruitful dialogue, which they initiated in the 1950s at the Courant Institute of Mathematical Sciences, are felt more strongly today than ever before.

From left to right: Louis Nirenberg and Jeanne Marie Aubin, King Harald, Alicia and John F. Nash Jr. (Photo: Håkon Mosvold Larsen) 222 Abel Prize 2015 Interview with John F. Nash, Jr.

This interview took place in Oslo on 18 May 2015, the day before the prize ceremony and only five days before the tragic accident that led to the death of John Nash and his wife Alicia. Nash’s untimely death made it impossible to follow the usual procedure for Abel interviews where interviewees are asked to proof-read and to edit first drafts. All possible misunderstandings are thus the sole responsibility of the interviewers.

Professor Nash, we would like to congratulate you as the Abel laureate in mathematics for 2015, a prize you share with Louis Nirenberg. What was your reaction when you first learned that you had won the Abel Prize? I did not learn about it like I did with the Nobel Prize. I got a telephone call late on the day before the announcement, which was confusing. However, I wasn’t entirely surprised. I had been thinking about the Abel Prize. It is an interesting example of a newer category of prizes that are quite large and yet not entirely predictable. I was given sort of a pre-notification. I was told on the telephone that the Abel Prize would be announced on the morning the next day. Just so I was prepared.

But it came unexpected? It was unexpected, yes. I didn’t even know when the Abel Prize decisions were announced. I had been reading about them in the newspapers but not following closely. I could see that there were quite respectable persons being selected.

Youth and Education

When did you realise that you had an exceptional talent for mathematics? Were there people that encouraged you to pursue mathematics in your formative years? Well, my mother had been a school teacher, but she taught English and Latin. My father was an electrical engineer. He was also a schoolteacher immediately before World War I. While at the grade school I was attending, I would typically do arithmetic – addition and multiplication – with multi-digit numbers instead of what was given at the school, namely multiplying two-digit numbers. So I got to work with four- and five-digit numbers. I just got pleasure in trying those out and finding the correct procedure. But the fact that I could figure this out was a sign, of course, of mathematical talent. Then there were other signs also. I had the book by E. T. Bell, Men of Mathemat- ics, at an early age. I could read that. I guess Abel is mentioned in that book?

Yes, he is. In 1948, when you were 20 years of age, you were admitted as a graduate student in mathematics at Princeton University, an elite institution that John F. Nash, Jr. and Louis Nirenberg 223 hand-picked their students. How did you like the atmosphere at Princeton? Was it very competitive? It was stimulating. Of course it was competitive also – a quiet competition of graduate students. They were not competing directly with each other like tennis players. They were all chasing the possibility of some special appreciation. Nobody said anything about that but it was sort of implicitly understood.

Games and game theory

You were interested in game theory from an early stage. In fact, you invented an ingenious game of a topological nature that was widely played, by both faculty members and students, in the Common Room at Fine Hall, the mathematics building at Princeton. The game was called “Nash” at Princeton but today it is commonly known as “Hex”. Actually, a Danish inventor and designer Piet Hein independently discovered this game. Why were you interested in games and game theory? Well, I studied economics at my previous institution, the Carnegie Institute of Technology in Pittsburgh (today Carnegie Mellon University). I observed people who were studying the linkage between games and mathematical programming at Princeton. I had some ideas: some related to economics, some related to games like you play as speculators at the stock market – which is really a game. I can’t pin it down exactly but it turned out that von Neumann [1903–1957] and Morgenstern [1902–1977] at Princeton had a proof of the solution to a two-person game that was a special case of a general theorem for the equilibrium of n-person games, which is what I found. I associated it with the natural idea of equilibrium and of the topological idea of the Brouwer fixed-point theorem, which is good material. Exactly when and why I started, or when von Neumann and Morgenstern thought of that, that is something I am uncertain of. Later on, I found out about the Kakutani fixed-point theorem, a generalisation of Brouwer’s theorem. I did not realise that von Neumann had inspired it and that he had influenced Kakutani [1911– 2004]. Kakutani was a student at Princeton, so von Neumann wasn’t surprised with the idea that a topological argument could yield equilibrium in general. I developed a theory to study a few other aspects of games at this time.

You are a little ahead of us now. A lot of people outside the mathematical community know that you won the Nobel Memorial Prize in Economic Sciences in 1994. That was much later.

Yes. Due to the film “A Beautiful Mind”, in which you were played by Rus- sell Crowe, it became known to a very wide audience that you received the Nobel Prize in economics. But not everyone is aware that the Nobel Prize idea was contained in your PhD thesis, which was submitted at Princeton in 224 Abel Prize 2015

1950, when you were 21-years-old. The title of the thesis was “Non-cooperative games”. Did you have any idea how revolutionary this would turn out to be? That it was going to have impact, not only in economics but also in fields as diverse as political science and evolutionary biology? It is hard to say. It is true that it can be used wherever there is some sort of equilibrium and there are competing or interacting parties. The idea of evolutionists is naturally parallel to some of this. I am getting off on a scientific track here.

But you realised that your thesis was good? Yes. I had a longer version of it but it was reduced by my thesis advisor. I also had material for cooperative games but that was published separately.

Did you find the topic yourself when you wrote your thesis or did your thesis advisor help to find it? Well, I had more or less found the topic myself and then the thesis advisor was selected by the nature of my topic.

Albert Tucker [1905–1995] was your thesis advisor, right? Yes. He had been collaborating with von Neumann and Morgenstern.

Princeton

We would like to ask you about your study and work habits. You rarely attended lectures at Princeton. Why? It is true. Princeton was quite liberal. They had introduced, not long before I arrived, the concept of an N-grade. So, for example, a professor giving a course would give a standard grade of N, which means “no grade”. But this changed the style of working. I think that Harvard was not operating on that basis at that time. I don’t know if they have operated like that since. Princeton has continued to work with the N-grade, so that the number of people actually taking the courses (formally taking courses where grades are given) is less in Princeton than might be the case at other schools.

Is it true that you took the attitude that learning too much second-hand would stifle creativity and originality? Well, it seems to make sense. But what is second-hand?

Yes, what does second-hand mean? Second-hand means, for example, that you do not learn from Abel but from someone who is a student of abelian integrals.

In fact, Abel wrote in his mathematical diary that one should study the masters and not their pupils. John F. Nash, Jr. and Louis Nirenberg 225

Yes, that’s somewhat the idea. Yes, that’s very parallel.

While at Princeton you contacted Albert Einstein [1879– 1955] and von Neumann, on separate occasions. They were at the Institute for Advanced Study in Princeton, which is located close to the campus of Princeton University. It was very audacious for a young student to contact such famous people, was it not? Well, it could be done. It fits into the idea of intellectual John F. Nash jr. and his wife Alicia were received by King functions. Concerning von Harald at the Royal Palace. (Photo: Håkon Mosvold Larsen) Neumann, I had achieved my proof of the equilibrium theorem for game theory using the Brouwer fixed-point theorem, while von Neumann and Morgenstern used other things in their book. But when I got to von Neumann, and I was at the blackboard, he asked: “Did you use the fixed-point theorem?” “Yes,” I said. “I used Brouwer’s fixed-point theorem.” I had already, for some time, realised that there was a proof version using Ka- kutani’s fixed-point theorem, which is convenient in applications in economics since the mapping is not required to be quite continuous. It has certain continuity properties, so-called generalised continuity properties, and there is a fixed-point theorem in that case as well. I did not realise that Kakutani proved that after being inspired by von Neumann, who was using a fixed-point theorem approach to an economic problem with interacting parties in an economy (however, he was not using it in game theory).

What was von Neumann’s reaction when you talked with him? Well, as I told you, I was in his office and he just mentioned some general things. I can imagine now what he may have thought, since he knew the Kakutani fixed- point theorem and I did not mention that (which I could have done). He said some general things, like: “Of course, this works.” He did not say too much about how wonderful it was.

When you met Einstein and talked with him, explaining some of your ideas in physics, how did Einstein react? He had one of his student assistants there with him. I was not quite expecting that. I talked about my idea, which related to photons losing energy on long travels through the Universe and as a result getting a red-shift. Other people have 226 Abel Prize 2015 had this idea. I saw much later that someone in Germany wrote a paper about it but I can’t give you a direct reference. If this phenomenon existed then the popular opinion at the time of the expanding Universe would be undermined because what would appear to be an effect of the expansion of the Universe (sort of a Doppler red-shift) could not be validly interpreted in that way because there could be a red-shift of another origin. I developed a mathematical theory about this later on. I will present this here as a possible interpretation, in my Abel lecture tomorrow. There is an interesting equation that could describe different types of space- times. There are some singularities that could be related to ideas about dark matter and dark energy. People who really promote it are promoting the idea that most of the mass in the Universe derives from dark energy. But maybe there is none. There could be alternative theories.

John Milnor, who was awarded the Abel Prize in 2011, entered Princeton as a freshman the same year as you became a graduate student. He made the observation that you were very much aware of unsolved problems, often cross-exam- ining people about these. Were you on the lookout for famous open problems while at Princeton? Well, I was. I have been in general. Milnor may have noticed at that time that I was looking at some particular problems to study. Milnor made various spectacular discoveries himself. For example, the non-standard differentiable structures on the seven-sphere. He also proved that any knot has a certain amount of curvature although this was not really a new theorem, since someone else1 had – unknown to Milnor – proved that.

A series of famous results

While you wrote your thesis on game theory at Princeton University, you were already working on problems of a very different nature, of a rather geometric flavour. And you continued this work while you were on the staff at MIT in Bos- ton, where you worked from 1951 to 1959. You came up with a range of really stunning results. In fact, the results that you obtained in this period are the main motivation for awarding you the Abel Prize this year. Before we get closer to your results from this period, we would like to give some perspective by quoting Mikhail Gromov, who received the Abel Prize in 2009. He told us, in the interview we had with him six years ago, that your methods showed “incredible originality”. And moreover: “What Nash has done in geometry is from my point of view incomparably greater than what he has done in economics, by many orders of magnitude.” Do you agree with Gromov’s assessment?

1 István Fáry. John F. Nash, Jr. and Louis Nirenberg 227

It’s simply a question of taste, I say. It was quite a struggle. There was something I did in algebraic geometry, which is related to differential geometry with some subtleties in it. I made a breakthrough there. One could actually gain control of the geometric shape of an algebraic variety.

That will be the subject of our next question. You submitted a paper on real algebraic manifolds when you started at MIT, in October 1951. We would like to quote Michael Artin at MIT, who later made use of your result. He commented: “Just to conceive such a theorem was remarkable.” Could you tell us a little of what you dealt with and what you proved in that paper, and how you got started? I was really influenced by space-time and Einstein, and the idea of distributions of stars, and I thought: ‘Suppose some pattern of distributions of stars could be selected; could it be that there would be a manifold, something curving around and coming in on itself that would be in some equilibrium position with those distributions of stars?’ This is the idea I was considering. Ultimately, I developed some mathematical ideas so that the distribution of points (interesting points) could be chosen, and then there would be some manifold that would go around in a desired geometrical and topological way. So I did that and developed some additional general theory for doing that at the same time, and that was published. Later on, people began working on making the representation more precise because I think what I proved may have allowed some geometrically less beautiful things in the manifold that is represented, and it might come close to other things. It might not be strictly finite. There might be some part of it lying out at infinity. Ultimately, someone else, A. H. Wallace [1926–2008], appeared to have fixed it, but he hadn’t – he had a flaw. But later it was fixed by a mathematician in Italy, in Trento, named Alberto Tognoli [1937–2008].

We would like to ask you about another result, concerning the realisation of Riemannian manifolds. Riemannian manifolds are, loosely speaking, abstract smooth structures on which distances and angles are only locally defined in a quite abstract manner. You showed that these abstract entities can be realised very concretely as sub-manifolds in sufficiently high-dimensional Euclidean spaces. Yes, if the metric was given, as you say, in an abstract manner but was considered as sufficient to define a metric structure then that could also be achieved by an embedding, the metric being induced by the embedding. There I got on a side-track. I first proved it for manifolds with a lower level of smoothness, the 1C -case. Some other people have followed up on that. I published a paper on that. Then there was a Dutch mathematician, Nicolaas Kuiper [1920–1994], who managed to reduce the dimension of the embedding space by one.

Apart from the results you obtained, many people have told us that the methods you applied were ingenious. Let us, for example, quote Gromov and John 228 Abel Prize 2015

Conway. Gromov said, when he first read about your result: “I thought it was nonsense, it couldn’t be true. But it was true, it was incredible.” And later on: “He completely changed the perspective on partial differential equations.” And Conway said: “What he did was one of the most important pieces of mathematical analysis in the 20th century.” Well, that is quite something! Yes.

Is it true, as rumours have it, that you started to work on the embedding problem as a result of a bet? There was something like a bet. There was a discussion in the Common Room, which is the meeting place for faculty at MIT. I discussed the idea of an embedding with one of the senior faculty members in geometry, Professor Warren Ambrose [1914–1995]. I got from him the idea of the realisation of the metric by an embedding. At the time, this was a completely open problem; there was nothing there beforehand. I began to work on it. Then I got shifted onto the 1C -case. It turned out that one could do it in this case with very few excess dimensions of the embedding space compared with the manifold. I did it with two but then Kuiper did it with only one. But he did not do it smoothly, which seemed to be the right thing – since you are given something smooth, it should have a smooth answer. But a few years later, I made the generalisation to smooth. I published it in a paper with four parts. There is an error, I can confess now. Some 40 years after the paper was published, the logician Robert M. Solovay from the University of Califor- nia sent me a communication pointing out the error. I thought: “How could it be?” I started to look at it and finally I realised the error in that if you want to do a smooth embedding and you have an infinite manifold, you divide it up into portions and you have embeddings for a certain amount of metric on each portion. So you are dividing it up into a number of things: smaller, finite manifolds. But what I had done was a failure in logic. I had proved that – how can I express it? – that points local enough to any point where it was spread out and differentiated perfectly if you take points close enough to one point; but for two different points it could happen that they were mapped onto the same point. So the mapping, strictly speaking, wasn’t properly embedded; there was a chance it had self-intersections.

But the proof was fixed? The mistake was fixed? Well, it was many years from the publication that I learned about it. It may have been known without being officially noticed, or it may have been noticed but people may have kept the knowledge of it secret.2

2 The result in Nash’s paper is correct; it has been reproved by several researchers (notably Mikhail Gromov) using the general strategy devised by Nash. Nash gave his own account on this error in the case of embeddings of non-compact manifolds in the book The essential John Nash (eds. Harold W. Kuhn and Sylvia Nasar), Princeton University Press, 2002. John F. Nash, Jr. and Louis Nirenberg 229

May we interject the following to highlight how surprising your result was? One of your colleagues at MIT, Gian-Carlo Rota [1932–1999], professor of mathematics and also philosophy at MIT, said: “One of the great experts on the subject told me that if one of his graduate students had proposed such an out- landish idea, he would throw him out of his office.” That’s not a proper liberal, progressive attitude.

Partial differential equations

But nevertheless it seems that the result you proved was perceived as something that was out of the scope of the techniques that one had at the time. Yes, the techniques led to new methods to study PDEs in general.

Let us continue with work of yours purely within the theory of PDEs. If we are not mistaken, this came about as a result of a conversation you had with Louis Nirenberg, with whom you are sharing this year’s Abel Prize, at the Courant Institute in New York in 1956. He told you about a major unsolved problem within non-linear partial differential equations. He told me about this problem, yes. There was some work that had been done previously by a professor in California, C. B. Morrey [1907–1984], in two dimensions. The continuity property of the solution of a partial differential equation was found to be intrinsic in two dimensions by Morrey. The question was what happened beyond two dimensions. That was what I got to work on, and de Giorgi [1928– 1996], an Italian mathematician, got to work on it also.

But you didn’t know of each other’s work at that time? No, I didn’t know of de Giorgi’s work on this, but he did solve it first.

Only in the elliptic case though. Yes, well, it was really the elliptic case originally but I sort of generalised it to include parabolic equations, which turned out to be very favourable. With parabolic equations, the method of getting an argument relating to an entropy concept came up. I don’t know; I am not trying to argue about precedents but a similar entropy method was used by Professor Hamilton in New York and then by Perelman. They use an entropy which they can control in order to control various improvements that they need.

And that was what finally led to the proof of the Poincaré Conjecture? Their use of entropy is quite essential. Hamilton used it first and then Perelman took it up from there. Of course, it’s hard to foresee success. It’s a funny thing that Perelman hasn’t accepted any prizes. He rejected the Fields Prize and also the Clay Millennium Prize, which comes with a cash award of one million dollars. 230 Abel Prize 2015

Coming back to the time when you and de Giorgi worked more or less on the same problem. When you first found out that de Giorgi had solved the problem before you, were you very disappointed? Of course I was disappointed but one tends to find some other way to think about it. Like water building up and the lake flowing over, and then the outflow stream backing up, so it comes out another way.

Some people have been speculating that you might have received the Fields Medal if there had not been the coincidence with the work of de Giorgi. Yes, that seems likely; that seems a natural thing. De Giorgi did not get the Fields Medal either, though he did get some other recognition. But this is not mathematics, thinking about how some sort of selecting body may function. It is better to be thought about by people who are sure they are not in the category of possible targets of selection.

When you made your major and really stunning discoveries in the 1950s, did you have anybody that you could discuss with, who would act as some sort of sounding board for you? For the proofs? Well, for the proof in game theory there is not so much to discuss. Von Neumann knew that there could be such a proof as soon as the issue was raised.

What about the geometric results and also your other results? Did you have anyone you could discuss the proofs with? Well, there were people who were interested in geometry in general, like Professor Ambrose. But they were not so much help with the details of the proof.

What about Spencer [1912–2001] at Princeton? Did you discuss with him? He was at Princeton and he was on my General Exam committee. He seemed to appreciate me. He worked in complex analysis.

Were there any particular mathematicians that you met either at Princeton or MIT that you really admired, that you held in high esteem? Well, of course, there is Professor Levinson [1912–1975] at MIT. I admired him. I talked with Norman Steenrod [1910–1971] at Princeton and I knew Solomon Lefschetz [1884–1972], who was Department Chairman at Princeton. He was a good mathematician. I did not have such a good rapport with the algebra professor at Princeton, Emil Artin [1898–1962].

The Riemann Hypothesis

Let us move forward to a turning point in your life. You decided to attack arguably the most famous of all open problems in mathematics, the Riemann Hypothesis, which is still wide open. It is one of the Clay Millennium Prize John F. Nash, Jr. and Louis Nirenberg 231 problems that we talked about. Could you tell us how you experienced mental ex- haustion as a result of your endeavour? Well, I think it is sort of a rumour or a myth that I actually made a frontal attack on the hypothesis. I was cautious. I am a little cautious about my efforts when I try to attack some problem because the problem can attack back, so to From left to right: John F. Nash Jr., Christian Skau and say. Concerning the Riemann Martin Raussen. (Photo: Eirik Furu Baardsen, DNVA) Hypothesis, I don’t think of myself as an actual student but maybe some casual – whatever – where I could see some beautiful and interesting new aspect. Professor Selberg [1917–2007], a Norwegian mathematician who was at the Institute for Advanced Study, proved back in the time of World War II that there was at least some finite measure of these zeros that were actually on the critical line. They come as different types of zeros; it’s like a double zero that appears as a single zero. Selberg proved that a very small fraction of zeros were on the critical line. That was some years before he came to the Institute. He did some good work at that time. And then, later on, in 1974, Professor Levinson at MIT, where I had been, proved that a good fraction – around 1/3 – of the zeros were actually on the critical line. At that time he was suffering from brain cancer, which he died from. Such things can happen; your brain can be under attack and yet you can do some good reasoning for a while.

A very special mathematician?

Mathematicians who know you describe your attitude toward working on mathematical problems as very different from that of most other people. Can you tell us a little about your approach? What are your sources of inspiration? Well, I can’t argue that at the present time I am working in such and such a way, which is different from a more standard way. In other words, I try to think of what I can do with my mind and my experiences and connections. What might be favourable for me to try? So I don’t think of trying anything of the latest popular nonsense.

You have said in an interview (you may correct us) something like: “I wouldn’t have had good scientific ideas if I had thought more normally.” You had a different way of looking at things. 232 Abel Prize 2015

Well, it’s easy to think that. I think that is true for me just as a mathematician. It wouldn’t be worth it to think like a good student doing a thesis. Most mathematical theses are pretty routine. It’s a lot of work but sort of set up by the thesis advisor; you work until you have enough and then the thesis is recognised.

Interests and hobbies

Can we finally ask you a question that we have asked all the previous Abel Prize laureates? What are your main interests or hobbies outside of mathematics? Well, there are various things. Of course, I do watch the financial markets. This is not entirely outside of the proper range of the economics Nobel Prize but there is a lot there you can do if you think about things. Concerning the great depression, the crisis that came soon after Obama was elected, you can make one decision or another decision which will have quite different consequences. The economy started on a recovery in 2009, I think.

It is known that when you were a student at Princeton you were biking around campus whistling Bach’s “Little Fugue”. Do you like classical music? Yes, I do like Bach [1685–1750].

Other favourite composers than Bach? Well, there are lot of classical composers that can be quite pleasing to listen to, for instance when you hear a good piece by Mozart [1756–1791]. They are so much better than composers like Pachelbel [1653–1706] and others.

We would like to thank you very much for a very interesting interview. Apart from the two of us, this is on behalf of the Danish, Norwegian and European Mathematical Societies.

After the end of the interview proper, there was an informal chat about John Nash’s main current interests. He mentioned again his reflections about cosmology. Con- cerning publications, Nash told us about a book entitled Open Problems in Mathe- matics that he was editing with the young Greek mathematician Michael Th. Ras- sias, who was conducting postdoctoral research at Princeton University during that academic year. John F. Nash, Jr. and Louis Nirenberg 233 Interview with Louis Nirenberg

Partial differential equations (and geometry/physics)

Firstly, we want to congratulate you on being awarded (with Professor Nash) the Abel Prize for 2015. You will receive the Abel Prize from His Majesty the King of Norway in a ceremony tomorrow. Your first important achievement in mathematics was solving the so-called Weyl problem in your PhD thesis. Could you tell us what the Weyl problem is about? The problem was originally stated by Hermann Weyl. You have a two-dimensional sphere with a metric (that is, a way of measuring distance), and connected with the metric is its curvature. If this curvature is positive, the question is whether you can find a convex body in three dimensional space with a mapping to the sphere so that when you measure the distance in Euclidean space, it agrees with the metric? Weyl went quite far toward solving this problem but there were some estimates missing. My contribution was to fill in those missing estimates. When you express the problem mathematically, it involves partial differential equations. The equations were so-called non-linear partial differential equations and the problem was proving the existence of solutions of these equations. Much of my career has really been devoted to studying partial differential equations in general but also applying them to problems from geometry and complex analysis. I even wrote two papers with a friend in economics involving partial differential equations. In my mind, it is a wonderful field. A big part of the problem is proving that solutions exist because equations can be written down for which one knows there are no solutions.

Many of these problems come from physics so solutions would be expected to exist? Yes. But, for instance, for equations in fluid dynamics (the so-called Navier–Stokes equations that were introduced 150 years ago), mathematicians have not been able to prove that smooth solutions exist for all time. So that is still an open problem.

Is it true that the best result in that direction is your joint work with Caffarelli and Kohn from 1982? That result is not about the existence of solutions but about the dimension of singularities if they do occur. They cannot have a high dimension; for instance, they cannot fill a curve. They must fill a set of dimension less than one. You may wonder what the hell that is? It either has dimension zero or it has dimension one. But, no! There are concepts of dimension of any non-negative number. We proved that the one-dimensional measure had to be zero, so the set could not have dimension one. The paper is very technical. 234 Abel Prize 2015

But it is very important in connection with the Navier–Stokes equations? Well, it is a useful result mathematically whether engineers use it or not. Aeroplanes fly whether we solve the Navier–Stokes equations or not. But it is a big mathematical challenge to show that there are smooth solutions.

Do you often think about the Navier–Stokes equations? Once in a while. But I don’t really have any fresh ideas. I think it is up to younger people.

Start of a career in mathematics

May we ask how your mathematical life started? We were told that a certain teacher of Hebrew played an instrumental role. Is that true? My father tried to teach me Hebrew. I resisted, stupidly, and now I know no Hebrew at all. He hired a friend to give me lessons in Hebrew. This friend happened to like mathematical puzzles and half the lessons then consisted of these puzzles. I found them quite fascinating but, I must say now, at my age, I am no longer fascinated by puzzles. They are for young people. That started my interest in mathematics. I also went to a very good high school. This was during the depression and to be a high school teacher was considered a very good job. I had excellent teachers and I must say that the quality of the students was also very good. I particularly enjoyed the mathematics courses and especially geometry and physics. I then decided I would like to study physics.

Were there already clear signs that you had an exceptional talent for mathematics? The teachers considered me good but I think it became clearer in college that I had some talent in mathematics. When I graduated from university, I actually received a gold medal for my work in mathematics and physics.

You graduated from McGill University in Montreal. Perhaps you could tell us about your experience studying mathematics and physics at university? I finished high school and applied for a scholarship at McGill, which I didn’t get. The high school offered an additional year, equivalent to a first year at college. I did that, applied again to McGill and then got a scholarship. So I was at McGill for three years rather than the usual four. This was during World War II and I graduated in the Spring of 1945, just at the end of the war in Europe. It was a pleasure to study mathematics and physics. However, that was the only thing I studied. Because I missed the first year, I didn’t take any courses in other interesting subjects. I am sorry I didn’t.

How did you end up at the Courant Institute in New York? John F. Nash, Jr. and Louis Nirenberg 235

By pure luck! When I finished at McGill, I had a Summer job at the National Research Council where they did atomic research. A son of Courant [1888– 1972] had married a young woman from Montreal, whom I knew. They both worked there and one day she said they were going to New York to visit Courant. I asked her to ask him to suggest some place I could apply to do graduate work in physics. She came back and said that Courant suggested that I come and take a Master’s degree in mathematics. I could then go on to do physics, he said. I went down for an interview and was offered an assistantship in mathematics. I got a Master’s degree and I just stayed on. I never left New York University.

Courant, Friedrichs and the CUNY

Courant was head of a very famous institute in Göttingen, Germany. He was kicked out when the Nazis came to power but he was offered a position at New York Uni- versity a year later to set up a graduate programme in the mathematics department. They only had undergraduate training at that time. He came to New York to do that but there were very few students in those first years. The number of students only increased after the war. When I came, just after the war, there were a number of very talented students. Some of them became well known mathematicians. I was part of a very good body of students and it was an exciting time. Usually, when you get a PhD at some university in America, you then leave. You go to another university for your first job. Courant was different. He kept the good people. If good people got PhDs, he simply offered them jobs.

Did it help being offered a job if you played an instrument? I didn’t play an instrument. But if I had, it may have helped even more. Of course, the rumour was that he hired people who played instruments (unless they played the piano, which he played himself).

Did you meet with him often? Oh, yes. He often invited the students socially to his home. His wife was completely devoted to music and played a number of instruments. She was the daughter of the mathematician Carl Runge [1856–1927], by the way. They had two daughters who were both very ardent musicians. One of them became a professional musician and is now married to Peter Lax. Courant was wonderful with young people – very encouraging and really exceptional.

Mathematically speaking, your mentor was Kurt Friedrichs [1901–1982]? Yes. Friedrichs was the person I regard as my Sensei (as the Japanese say). I really was most influenced by him. He worked mainly with partial differential equations but he also did other things. He wrote a book on quantum theory and a book, 236 Abel Prize 2015 together with Courant, on shock wave theory, which was widely used and translated into many languages.

You mentioned that there was a special atmosphere at the Courant Institute, in part because no distinction was made between pure and applied mathematics… That’s right. Courant insisted there was no difference between pure and applied mathematics. He did both and he encouraged people to do the same. It is just mathematics. When New York University hired him, they asked him what a mathematics department needs and he said: “A library and a coffee room.” So we have a very nice lounge that we use all the time.

It is remarkable that you are the fourth Abel Prize Laureate associated with the Courant Institute (after Peter Lax in 2005, Srinivasa Varadhan in 2007 and Mikhail Gromov in 2009). What has made this institute so successful? Well, partly it is just the warm atmosphere. I think graduate students are very happy there and there is a lot of interaction between the students and the faculty. It is, of course, much bigger now than it was when I was a student there. But the warm atmosphere has prevailed.

Who were your most important colleagues over your career? There is Friedrichs but also two other students of Courant: Fritz John [1910– 1994], a wonderfully talented mathematician who later became a faculty member (I had the fortune of writing one paper with him) and Hans Lewy [1904–1988] (I wrote several papers related to some of his work). Hans left Germany immediately after Hitler came to power. He came to the United States and had a career at Berkeley.

Partial differential equations and inequalities

Your name, often with various co-authors, is attached to many fundamental concepts and theorems in PDEs. If you just look at the citation list, your work has had a tremendous impact. Let’s start with Fritz John, with whom you authored a very influential paper about BMO functions (BMO standing for “Bounded Mean Oscillation”). That was his idea. He introduced BMO functions. It came from some work he had done in elasticity theory. He approached me saying: “I have a class of functions and I believe they should have such-and-such a property.” I worked on it and was able to prove that property. He then improved it so the final version is better than what I had done. It became a joint paper and I must say a lot of people have referred to it.

Absolutely! It became famous – if we may say so – because of the many applications. For instance, Charles Fefferman got the Fields Medal in 1978 and one John F. Nash, Jr. and Louis Nirenberg 237 of his main contributions was to show that the BMO space is dual to the Hardy space H1. Charles Fefferman did many things but in particular, he proved the duality result that you refer to.

Your paper with Fritz John contains the John–Nirenberg inequality. You love inequalities? I love inequalities. And what we proved in the paper was really an inequality.

Would you explain why inequalities are so important in the theory of PDEs? When you look at a partial differential equation, you may ask whether a solution exists. Now, you can’t write down the solution so you need to know some bounds. It cannot be too big, it cannot be too negative, its derivatives cannot be too big and so on. You try to get estimates of the size of the function and of its derivatives. All these estimates are inequalities. You are not saying that something is equal to something but that something is less than some constant. Thus, inequalities play an essential role in proving the existence of solutions. In addition, you want to prove properties of solutions and, again, inequalities play a central role. Hence, inequalities are absolutely fundamental to studying partial differential equations; for that matter, so are they for ordinary differential equations.

Let’s move on to your joint research with Shmuel Agmon and Avron Douglis [1918–1995]. There were two very important papers. Can you explain what they contained? What we did was to extend some classic work, by the Polish mathematician Schauder [1899–1943], to higher order equations. There is a fundamental paper of Schauder for second-order, so-called elliptic equations. We thought it would be useful for people to be able to deal with higher order equations and systems of equations so we proved the analogues of those results. In the other paper, we proved the results for systems and also for different norms, that is, for different ways of measuring the size of the solutions. We published several different kinds of inequalities and they have been used by many people.

You wrote a paper with Joseph Kohn introducing the important notion of pseudo-differential operators. You are one of the fathers of that concept. Can you explain why this concept is so important and how you came upon it? Joe Kohn had published a fundamental paper in complex analysis. It involved the regularity of solutions for a certain class of systems up to the boundary – a rather difficult paper! He suggested we should try to generalise this to more general systems of equations. We started to look at it and we had to consider so-called commutators of operators. You apply an operator and then you apply a second one. Then you take the difference of that result with the operator obtained by applying the second one and then the first. We needed properties of the commutator. We were using a certain space, called an Lp-space, and a theory due to Calderón 238 Abel Prize 2015

[1920–1998] and Zygmund [1900–1992] for certain singular integral operators. We needed to extend their result to commutators so we thought: “How do we extend these singular integral operators to make an algebra out of them?” That led to what we call pseudo-differential operators. The concept came from a very specific problem in systems of partial differential equations but it turned out to be a useful thing in itself. It grew out of Calderón and Zygmund theory. By the way, Calderón was a wonderful mathematician and he danced the tango, which I admired enormously.

You had a very bright student, August Newlander, with whom you wrote a very important joint paper in 1957. Can you tell us about the results you proved there? It was a problem I first heard of from André Weil [1906–1998]. He said: “Here’s a problem in complex analysis. Why don’t you people in partial differential equations work on this kind of problem?” I thought: “Why not? Let’s try.” I took a student who was very bright and I said: “Let’s look at the very simplest case, in the lowest dimension.” The student, Newlander, had the initial idea, which worked fine in low dimensions but, to our surprise and dismay, didn’t work in higher dimensions. We had to come up with a completely different proof in higher dimensions. It led from a linear problem to a non-linear problem. It was kind of strange but the non-linear problem was in some ways more accessible.

What was André Weil’s reaction when you solved the problem? He was very happy and so were other people in complex analysis. Many people have used the result. Some years later, Hörmander [1931–2012] found a linear proof of the same result. It was very technical but it was purely linear.

Are there any outstanding problems in the enormous field of partial differential equations, apart from the Navier–Stokes problem, that you would like to highlight? Well, I think almost nothing has been done in so-called over-determined systems, that is, where there are more equations than unknowns. You may have two unknowns and five equations so there have to be some compatibility relations. There’s almost no analytic theory of that. There is a theory developed by Cartan [1869–1951] and Kähler [1906–2000] but that assumes that everything is analytic. Outside analytic category, almost nothing is known about such systems. They often come up in geometry so I feel that this is a big gap in the theory of partial differential equations.

Mathematics and mathematicians all over the world

May we ask you some questions about international mathematics? We know that you travelled to post-war Europe very soon after your graduation. John F. Nash, Jr. and Louis Nirenberg 239

Yes. I had a fellowship and came to Zurich during the academic year 1951/52. I went mainly to be with Heinz Hopf [1888–1972], who was a geometer and a topologist. Heinz Hopf was a wonderful person – a lovely and extremely kind man. I also spent one month in Göttingen that year. That was arranged by Courant who felt I should go there. During that year, I didn’t actually carry out any research. What I did was to write up the things I had done before. I had been very slow at writing them up for publication because I somehow had a block against writing. So during that year I wrote several papers.

Did Courant ever return to Göttingen? Yes. After the war, he went back to Germany many times. He had many contacts and he wanted to help build up German mathematics again.

He must also have been very bitter? Well, he was bitter but, at the same time, he had friends and he wanted to encourage and help to develop mathematics in Germany.

You also went to the Soviet Union? Yes. The first time I went was in 1963. It was a joint Soviet-American sympo- sium on partial differential equations, arranged by Courant on one side and the Soviet mathematician Lavrentyev [1900–1980] on the other. There were about two dozen American mathematicians and about 120 Soviet mathematicians from all over the Soviet Union. It is one of the best meetings I have ever attended. It was in Novosibirsk, Siberia, which was the academic city that Lavrentyev had helped create. It was like being aboard a ship for two weeks with people you make friendships with immediately. I made friends with Russians that are still friends today. Some have died, unfortunately, but I have had very good friends in Rus- sia since then. I have never collaborated with any of them but they are still very warm friends; we would meet and talk about mathematics, politics and all kind of things.

How about China? I have been to China a number of times. The first visit was arranged by Chern [1911–2004], a Chinese mathematician who had settled in America. This was in 1975 and the Cultural Revolution was still going on, though I didn’t realise it at the time. For instance, I was visiting the Chinese Academy of Science but I was taken to Beijing University. I said I would like to meet the faculty but they said they were busy teaching – which was simply a lie. There was no teaching going on. They showed me the library and then they wanted to take me to some other university but I said: “There’s no point. Either I meet the faculty or I don’t go.” They had me give many lectures but I said I also wanted to hear what some of the people there were doing. So some young people spoke about some of their research. I learned later that they had to get permission to attend my lectures. I didn’t make close friends at that time. It was an interesting experience and, of course, things 240 Abel Prize 2015 have changed enormously since then. I did make friends with some who subsequently came and spent a year or two at Courant.

We should also mention that you were awarded the first Chern Medal of the International Mathematical Union. Yes. That’s true. That was in 2010.

You were also awarded the first Crafoord Prize in 1982, together with Arnol’d [1937–2010]. Perhaps it was a tongue-in-cheek comment but Arnol’d once said something like: “Mathematics is the part of physics in which experiments are cheap.” It wasn’t entirely tongue-in-cheek. He really felt that the contact of mathematics with physics and the real world was important. He didn’t get permission to go and get the Crafoord Prize. I visited Moscow just before I went to Sweden and had dinner with him in his home. He was waiting until the last minute to see if he would get permission, but he didn’t. When I went back to America, I got a call from a woman claiming to be Arnol’d’s sister. I thought: “How is that possible?” I had just seen Arnol’d a few weeks before and he never mentioned he had a sister in New Jersey. She came to my office and, indeed, it was Arnol’d’s sister. He never mentioned a word. It’s incredible!

Talking about Arnol’d, on some occasions he expressed frustration that results proved in the West had already been proved in Russia but, because of poor communication during the Cold War, these results were not known. Did he express these feelings to you? He tended to do that. I remember once he was visiting New York. Someone was giving a seminar talk and he was attending the lecture. During the talk, Arnol’d said: “Oh, that was already proved by such-and-such a Russian.” But the person giving the seminar talk then checked and the Russian had never proved it. So Arnol’d was not always correct. He tended to give more credit to Russians than was due. You may have heard the joke where the Russian says: “What you proved, I proved first. And anyway, the result is trivial.”

Problems, collaboration and “Sitzfleisch”

It is striking that 90% of your published papers describe joint work. Can you explain why this is so? It is just a pleasure! It is just an enormous pleasure talking mathematics with others and working with them. Of course, much of the work you do yourself. I mean, you discuss ideas and work with others but then you go home and think about what you have done. You get some ideas and you get together again and talk about the new ideas. You get reactions to your ideas and you react to their ideas. It is a wonderful experience. John F. Nash, Jr. and Louis Nirenberg 241

Do you usually start out with a goal in mind? Usually there is a goal. But somebody once used the expression: “There are those mathematicians who, when they come to a fork in the road, they take it.” I’m that kind of mathematician. So, I may be working on a problem with a colleague when we come to something that looks interesting, and we explore that and leave the original problem for a while.

Are you more of a problem solver? Yes, definitely. There are two kinds of mathematicians. There are those who develop theories and those who are primarily problem solvers. I am of the latter.

Do you come up with interesting problems through discussions with other mathematicians? What kinds of problems are you attracted to? Is there any pattern? It’s hard to say. A graduate student once asked me how I find good research problems. I said to him that I sometimes see a result but don’t like the proof. If the problem appeals to me, I start to think if there is a better proof. My ideas may lead to a better proof or may lead to something new. The student said he’d never seen a proof he didn’t like and I thought: “He is hopeless!”

May we ask you a question that we have asked several previous laureates? How does one find the proof of a mathematical result? Some people work with perseverance until a proof is complete but others tell us that insight appears in a sudden flash – like lightning. Do you have experiences of this sort? Both may happen. But most of the time you are stuck. Maybe you make a breakthrough with some problem as you get some insight and see something you didn’t see before. But the perseverance and all the work you carried out before seems to be necessary to have this insight. You need perseverance or, as the Germans say, you need “Sitzfleisch”.

Are you the kind of person that gets so involved in trying to solve a problem that you are, so to speak, lost to the world? Not all the time but it can happen for many hours. Sometimes, I wake in the middle of the night and start thinking about a problem for hours and cannot sleep. When you do that, it is very hard to fall asleep again! If I have an idea, I just follow it. I see if it leads to something. I still try to do that but in the last few years it has not led anywhere. I haven’t had any success.

Communicating mathematics

You have had 45 PhD students. That is an impressive number! Can you tell us what your philosophy is? How do you come up with problems for your students? It’s hard to say. Sometimes it is hard to think of a suitable problem. It is easier to think of problems that are too hard, and just not practical, than to think of a prob- 242 Abel Prize 2015 lem that is good and can be solved in reasonable time. I can’t really answer that question. I don’t know how I go about posing problems.

Were there occasions when you had to help students along? Oh, yes. I meet the students regularly, usually once a week. We discuss their progress and I might make suggestions. I may say: “Look at this paper, this may lead to something.”

How would you describe your love for mathematics? What is it about mathematics that is so appealing to you? Is it possible to communicate this love to people outside the mathematical community? Does one have to be a mathematician to appreciate the appeal of mathematics? Some people are very good at communicating to the general public. I am not so very good at that. But once you are in it, once you are hooked, it’s very exciting and fun. I have used the word “fun” before. But it is really fun to do mathematics. It is an enormous pleasure to think about mathematics even though you are stuck 90% of the time, perhaps even more.

That is what people outside mathematics cannot comprehend. Yes, it is hard to comprehend. You have to be in it and I think it does take some talent to be able to do mathematics. But it also takes, as I said, “Sitzfleisch”. You need to be stubborn and have perseverance, and you can’t give up. I have been stuck on some problems for years.

But you do think it’s important to try to communicate to the general audience? Yes, I do think that is important: (a) for the development of mathematics, and (b) to show them that it is a pleasure to do mathematics. Courant and Robbins [1915– 2001] wrote a very nice book: What is Mathematics? It is a lovely book. There is also a recent book by Edward Frenkel, a mathematician who came as an immigrant from Russia as a young man. It is called Love and Math. He makes a valiant attempt to get the general public interested in the branch of mathematics in which he works (which is also connected to physics). It is very hard to do. He tries but I think it is too hard for the general public. But he makes a real attempt to do it and I must say I admire him for that. I just recently read his book.

Music and movies

We have one final question that we have asked several laureates before. What are you interested in when you are not doing mathematics? I love music. I love movies. You won’t believe this but at the time when I lived in Mon- treal, in the province of Québec, you could not get into a movie before you were 16. Incredible! Now it’s hard to believe. So when I was 16, I went crazy and started to go to movies. When I moved to New York, there were suddenly all these foreign movies: John F. Nash, Jr. and Louis Nirenberg 243

Italian movies, Russian movies, French movies. I went crazy. I went almost every night to the movies. Since then, I have loved movies.

Have you seen “A Beautiful Mind”? Of course, and I have read the book.

What kind of music do you like? From left to right: Louis Nirenberg, Christian Skau and Mainly classical but I also Martin Raussen. (Photo: Eirik Furu Baardsen) listen to jazz. My grandson, who will be at the ceremony tomorrow, is a professional jazz drummer. And I love Argentinian tango. I have a large collection of records of Argentinian tango.

Not only on behalf of us but also on behalf of the Norwegian, Danish and Euro- pean Mathematical Societies, we would like to thank you for a very interesting interview.

Abel Laureates John F. Nash Jr. and Louis Nirenberg at the award ceremony in the University Aula in Oslo. Back row: Kristian Ranestad, John Rognes and Kirsti Strøm Bull. (Photo: Berit Roald)

Abel Prize 2016: Sir Andrew J. Wiles

The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2016 to

Sir Andrew J. Wiles University of Oxford, UK

“for his stunning proof of Fermat’s Last Theorem by way of the modularity conjecture for semistable elliptic curves, opening a new era in number theory.”

Number theory, an old and beautiful branch of mathematics, is concerned with the study of arithmetic properties of the integers. In its modern form the subject is fundamentally connected to complex analysis, algebraic geometry, and representation theory. Num- ber theoretic results play an important role in our everyday lives through encryption algorithms for communications, financial transactions, and digital security. Fermat’s Last Theorem, first formulated by Pierre de Fermat in the 17th century, is the assertion that the equation xn + yn = zn has no solutions in positive integers for n > 2. Fermat proved his claim for n = 4, Leonhard Euler found a proof for n = 3, and Sophie Abel Laureate Sir Andrew J. Wiles. Germain proved the first general result that (Photo: Knut Falch) applies to infinitely many prime exponents. Ernst Kummer’s study of the problem unveiled several basic notions in algebraic number theory, such as ideal numbers and the subtleties of unique factorization. The complete proof found by Andrew Wiles relies on three further concepts in number theory, namely elliptic curves, modular forms, and Galois representations. Elliptic curves are defined by cubic equations in two variables. They are the natural domains of definition of the elliptic functions introduced by Niels Henrik Abel.

* 11.4.1953

245 246 Abel Prize 2016

Modular forms are highly symmetric analytic functions defined on the upper half of the complex plane, and naturally factor through shapes known as modular curves. An elliptic curve is said to be modular if it can be parametrized by a map from one of these modular curves. The modularity conjecture, proposed by Goro Shimura, Yutaka Taniyama, and André Weil in the 1950s and 60s, claims that every elliptic curve defined over the rational numbers is modular. In 1984, Gerhard Frey associated a semistable elliptic curve to any hypothetical counterexample to Fermat’s Last Theorem, and strongly suspected that this elliptic curve would not be modular. Frey’s non-modularity was proven via Jean-Pierre Serre’s epsilon conjecture by Kenneth Ribet in 1986. Hence, a proof of the Shimura- Taniyama-Weil modularity conjecture for semistable elliptic curves would also yield a proof of Fermat’s Last Theorem. However, at the time the modularity conjecture was widely believed to be completely inaccessible. It was therefore a stunning advance when Andrew Wiles, in a breakthrough paper published in 1995, introduced his modularity lifting technique and proved the semistable case of the modularity conjecture. The modularity lifting technique of Wiles concerns the Galois symmetries of the points of finite order in the abelian group structure on an elliptic curve. Building upon Barry Mazur’s deformation theory for such Galois representations, Wiles identified a numerical criterion which ensures that modularity for points of order p can be lifted to modularity for points of order any power of p, where p is an odd prime. This lifted modularity is then sufficient to prove that the elliptic curve is modular. The numerical criterion was confirmed in the semistable case by using an important companion paper written jointly with Richard Taylor. Theorems of Robert Langlands and Jerrold Tunnell show that in many cases the Galois representation given by the points of order three is modular. By an ingenious switch from one prime to another, Wiles showed that in the remaining cases the Galois representation given by the points of order five is modular. This completed his proof of the modularity conjecture, and thus also of Fermat’s Last Theorem. The new ideas introduced by Wiles were crucial to many subsequent developments, including the proof in 2001 of the general case of the modularity conjecture by Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor. As recently as 2015, Nuno Freitas, Bao V. Le Hung, and Samir Siksek proved the analogous modularity statement over real quadratic number fields. Few results have as rich a mathematical history Sir Andrew Wiles received the Abel Prize from and as dramatic a proof as Fermat’s Crown Prince Haakon. (Photo Audun Braastad) Last Theorem. Sir Andrew J. Wiles 247

Professor Wiles, please accept our congratulations for having been selected as the Abel Prize Laureate for 2016. To be honest, the two of us had expected this interview to take place already several years ago! You are famed not only among mathematicians, but also among the public at large for, and now we cite the Abel Committee: “the stunning proof of Fer- mat’s Last Theorem, by way of the Modularity Conjecture for elliptic curves, opening a new era in number theory”. This proof goes back to 1994, which means that you had to wait for more than 20 years before it earned you the Abel Prize. Nevertheless, you are the youngest Abel Prize Laureate so far. After you finished your proof of Fermat’s Last Theorem you had to undergo a deluge of interviews, which makes our task difficult. How on earth are we to come up with questions that you have not answered several times before? Well, we will try to do our best.

Fermat’s Last Theorem: A historical account

We have to start at the very beginning, with a citation in Latin: “…nullam in in- finitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere”, which means: “it is impossible to separate any power higher than the second into two like powers”. That is in modern mathematical jargon: The equation xn + yn = zn has no solution in natural numbers for n greater that two. And then it continues: “cujus rei demonstrationem mirabilem sane detexi. Hanc margin- is exiguitas non caperet”, which means: “I have discovered a truly marvellous proof of this, which this margin is too narrow to contain”. This remark was written in the year 1637 by the French lawyer and amateur mathematician Pierre de Fermat [1601–1665] in the margin of his copy of Diophantus’ Arithmetica. He certainly did not expect that it would keep mathematicians, professionals and amateurs alike, busy for centuries trying to unearth the proof. Could you please give us a short account of some of the attempts towards proving Fermat’s Last Theorem up until the time you embarked on your successful journey? Furthermore, why was such a simple-minded question so attractive and why were attempts to prove it so productive in the development of number theory? The first serious attempt to solve it was presumably by Fermat himself. But unfortunately we know nothing about it except for what he explained about his proofs in the specific cases of n = 3 and n = 4.1 That is, he showed that you can’t have the sum of two cubes be another cube, or the sum of two fourth powers being a fourth power. He did this by a beautiful method, which we call infinite descent. It was a new method of proof, or at least a new way of presenting proofs, in arithmetic. He explained this method to his colleagues in letters and he also wrote about it in his famous margin, which was big enough for some of it at least. After the marginal

1 Strictly speaking Euler was the first to spell out a complete proof in the case n = 3. 248 Abel Prize 2016 notes were published by Fermat’s son after his father’s death, it lay dormant for a while. Then it was picked up by Euler [1707–1783] and others who tried to find this truly marvellous proof. And they failed. It became quite dramatic in the mid-19th century – various people thought they could solve it. There was a discussion concerning this in the French Academy – Lamé [1795–1870] claiming he was just about to prove it – and Cauchy [1789–1857] said he thought he could too, and so on. In fact it transpired that the German mathematician Kummer [1810–1893] had already written a paper where he explained that the fundamental problem was what is known now as the fundamental theorem of arithmetic. In our normal number system any number can be factorized in essentially one way into prime factors. Take a number like 12; it is 2 times 2 times 3. There is no other way of breaking it up. But in trying to solve the Fermat problem you actually want to use systems of numbers where this uniqueness does not hold. Every attempt that was made to solve the Fermat problem had stalled because of this failure of unique factorization. Kummer analysed this in incredible detail. He came up with the most beautiful results, and the end product was that he could solve it for many, many cases. For example for n ≤ 100 he solved it for all primes except for 37, 59 and 67. But he did not finally solve it. His method was based on the idea that Fermat had introduced – the method of infinite descent – but in these new number systems. The new number systems he was using spawned algebraic number theory as we see it today. One tries to solve equations in these new systems of numbers instead of solving them with ordinary integers and rational numbers. Attempts in the style of Fermat carried on for a while but somewhat petered out in the twentieth century. No one came up with a fundamentally new idea. In the second half of the twentieth century number theory moved on and considered other questions. Fermat’s problem was all but forgotten by the professionals. Then in 1985, Gerhard Frey, a German mathematician, came up with a stunning new idea where he took a hypothetical solution to the Fermat problem and rewrote it so that it made what is called an elliptic curve. And he showed, or suggested, that this elliptic curve had very peculiar properties. He conjectured that you can’t really have such an elliptic curve. Building on this a year later an American mathematician, Kenneth Ribet, demonstrated, using this Frey curve, that any solution of Fermat would contradict another well-known conjecture called the Modular- ity Conjecture. This conjecture had been proposed in a weak form by Taniyama [1927–1958] and refined by Shimura, but the first real evidence for it came from André Weil [1906–1998] who made it possible to check this precise form of the Modularity Conjecture in some detail. And a lot of evidence was amassed showing that this should certainly be true. So at that point mathematicians could see: Yes, Fermat is going to be true. Moreover, there has to be a proof of it. What happened was that the Modularity Conjecture was a problem that mathematics could not just put to one side and go on for another five hundred years. It was a roadblock right in the middle of modern mathematics. It was a very, very central problem. As for Fermat you could just leave it aside and forget it almost forever. Sir Andrew J. Wiles 249

This Modularity Conjecture you could not forget. So at the point when I heard that Ribet had done this I knew that this problem could be solved and I was going to try.

Concerning speculations about Fermat’s claimed proof: Do you think he had the same idea as Lamé, assuming, wrongly it turned out, that the cyclotomic integers have unique factorization? No, I don’t think so, though the idea might be in there somewhere. It is very hard to understand. André Weil wrote about this. All the other problems Fermat considered had to do with curves that were of genus zero or genus one. And suddenly he is writing down a curve that has higher genus. How is he going to think about it? When I was trying this myself as a teenager, I put myself in Fermat’s frame of mind because there was hardly anything else I could do. I was capable of understanding his mathematics from the 17th century, but probably not much beyond that. It seemed to me that everything he did came down to something about quadratic forms, and I thought that might be a way of trying to think about it. Of course, I never succeeded, but there is nothing else that suggests Fermat fell into this trap with unique factorization. In fact, from the point of view of quadratic forms he understood that sometimes there was unique factorization and sometimes there was not. So he understood that difference in his own context. I think it is unlikely that that was the mistake.

In the same book by André Weil that you referred to, titled Number Theory: an approach through history from Hammurapi to Legendre, it is mentioned that Fermat looked at the equation a cube minus a square equal to 2 [x3 – y2 = 2], and he showed that it has essentially only one solution, namely x = 3 and y = ±5. André Weil speculates that Fermat at the time looked at the ring Z[–2], which does have unique factorization. Yes, he used unique factorization but the way he did it was in terms of quadratic forms. And I think he also looked at quadratic forms corresponding to Z[–6] where there is not unique factorization. So I think he understood. It was my impression when I thought about it that he understood the difference.

A mathematical education

You were apparently interested in mathematical puzzles already as a quite young boy. Have you any thoughts about where this interest came from? Were you influenced by anyone in particular? I just enjoyed mathematics when I was very young. At the age of ten I was looking through library shelves devoted to mathematics. I would pull out books and at one point I pulled out a book of E. T. Bell [1883–1960] titled The Last Problem, which on its cover describes the Fermat equation, the Wolfskehl Prize and the romantic history of the problem. I was completely captivated by it. 250 Abel Prize 2016

Were there other things that fascinated you in this book by Eric Temple Bell? It is entirely about that one equation, really. And it is actually quite wordy. So there is less mathematics in some sense than you might think. I think it was more the equation. Then, when I found this equation I looked for other elementary books on number theory and learned about congruences and solved congruences and so on, and looked at other things that Fermat did.

You did this work besides your ordinary school work? Yes, I don’t think my school work was too taxing from that point of view.

Was it already clear for you at that time that you had an extraordinary mathematical talent? I certainly had a mathematical aptitude and obviously loved to do mathematics, but I don’t think I felt that I was unique. In fact, I don’t believe I was in the school I attended. There were others who had just as strong a claim to be future mathematicians, and some of them have become mathematicians, too.

Did you plan to study mathematics and to embark on a mathematical career already at that age? No, I don’t think I really understood you could spend your life doing mathematics. I think that only came later. But I certainly wanted to study it as long as I could. I’m sure that as far as my horizon extended it involved mathematics.

You started to study mathematics as a student at Oxford in 1971. Can you tell us a little bit about how that worked out? Were there any particular teachers, any particular areas that were particularly important for you? Before I went to college, actually in high-school, one of my teachers had a PhD in number theory. He gave me a copy of Hardy and Wright’s An Introduction to the Theory of Numbers, and I also found a copy of Davenport’s The Higher Arithmetic. And these two books I found very, very inspiring in terms of number theory.

So you were on track before you started studying? Yes, I was on track before. In fact, to some extent I felt college was a distraction because I had to do all these other things, applied maths, logic and so on, and I just wanted to do number theory. You were not allowed to do number theory in your first year. And you could not really get down to it before your third year.

But you were not interested in geometry, not as much as in algebra and number theory, anyway? No, I was primarily interested in algebra and number theory. I was happy to learn these other things, but I really was most excited about number theory. My teachers arranged for me to take extra classes in number theory, but there was not that much on offer. Sir Andrew J. Wiles 251

At one point I decided that I should put all the years of Latin I had done at school to good use and try to read some of Fermat in the original, but I found that was actually too hard. Even if you translated the Latin, the way they wrote in those days wasn’t in the algebraic symbols I was used to; so it was quite difficult.

It must have been a relief when you were done and came to Cambridge to start studying number theory for real, with John Coates as your supervisor? That’s right. I had a year, a preliminary year, in which I just studied a range of subjects, and then I could do a special paper. John Coates was not yet at Cambridge, but I think he helped me – maybe over the summer. Anyway, that summer I met him and started working with him right away, and that was just wonderful. The transition from undergraduate work, where you were just reading and studying, to research, that was the real break for me. It was just wonderful.

Elliptic curves

We assume it was John Coates who initiated you to work on elliptic curves, and to Iwasawa theory? Absolutely. He had some wonderful ideas and was generous to share them with me.

Did you tell John Coates that you were interested in the Fermat problem? Perhaps I did. I don’t remember. It is really true that there hadn’t been any new ideas since the 19th century. People were trying to refine the old methods, and, yes, there were refinements. But it didn’t look like these refinements and the solution were going to converge. It was just too hard that way.

At the time you started to work with John Coates, you had no idea that these elliptic curves were going to be crucial for the solution of Fermat’s Last Theorem? No, it’s a wonderful coincidence. The strange thing is that, in a way, the two things that are most prominent in Fermat that we remember today are his work on elliptic curves and his famous last theorem. For example, this equation you mentioned, y2 + 2 = x3, is an elliptic curve. And the two strands came together in the proof.

Could you explain what an elliptic curve is and why elliptic curves are of interest in number theory? For a number theorist the life of elliptic curves started with Fermat as equations of the form y2 equals a cubic polynomial in x with rational coefficients. Then the problem is to find the rational solutions to such an equation. What Fermat noticed was the following: Sometimes you can start with one or even two rational solutions, and use them to generate infinitely many others. And yet sometimes there are no solutions. This latter situation occurs, for example, in the case n = 3 of Fermat’s Last Theorem, the equation being in fact an elliptic curve in disguise. Sometimes you 252 Abel Prize 2016 can show there are no rational solutions. You could have infinitely many and you could have none. This was already apparent to Fermat. In the early 19th century one studied these equations in complex numbers. Abel [1802–1829] himself came in at this point and studied elliptic functions and related these to elliptic curves, implying that elliptic curves have a group structure. They were very well understood in terms of doubly periodic functions in the early 19th century. But that is what underlies the complex solutions, solutions to the equation in complex numbers. The solutions to the equation in rational numbers were studied by Poincaré [1854–1912]. What’s now known as the Mordell–Weil theorem was proved by Mor- dell [1888–1972] and then Weil in the 1920s, answering a question of Poincaré. In our setting it says that the K-rational points on an elliptic curve over a number field K, in particular for K equal to the rationals, form a finitely generated abelian group. That is, from Fermat’s language you can start with a finite number of solutions and using those generate all the solutions by what he called the chord-and-tangent process.

Birch and Swinnerton–Dyer, Tate–Shafarevich, Selmer…

By now you know the structure, it is a very beautiful algebraic structure, the structure of a group, but that does not actually help you find the solutions. So no one really had any general methods for finding the solutions, until the conjectures of the 1960s, which emerged from the Birch and Swinnerton–Dyer Conjecture. There are two aspects to it; one is somewhat analytic and one is in terms of what is called the Tate–Shafarevich group. Basically the Tate–Shafarevich group gives you the obstruction to an algorithm for finding the solutions. And the Birch and Swinnerton–Dyer Conjecture tells you that there is actually an analytic method for analysing this so-called Tate–Shafarevich group. If you combine all this together, ultimately it should give you an algorithm for finding the solutions.

You worked already on the Birch and Swinnerton–Dyer Conjecture when you were a graduate student together with John Coates? Yes, that is exactly what he proposed working on. We got the first result in certain special families of elliptic curves on this analytic link between the solutions and what is called the L-function of the elliptic curve.

These were curves admitting complex multiplication? Exactly, these were the elliptic curves with complex multiplication.

Was this the first general result concerning the Birch and Swinnerton–Dyer Conjecture? It was the first one that treated a family of cases rather than individual cases. There was a lot of numerical data for individual cases, but this was the first infinite family of cases. Sir Andrew J. Wiles 253

This was over the rational numbers? Yes.

We should mention that the Birch and Swinnerton–Dyer Conjecture is one of the Clay Millennium Prize Problems which would earn a person who solves it one million dollars. That’s right. I think it’s appealing, partly because it has its roots in Fermat’s work, just like the Fermat problem. It is another ‘elementary to state’ problem, concerned with equations – in this case of very low degree – which we can’t master and which Fermat initiated. I think it is a very appealing problem.

Do you think it is within reach? In other words, do we have the necessary tools for somebody daring enough to attack it and succeed? Or do we have to wait for another three hundred years to see it solved? I don’t suppose it will take three hundred years, but I don’t think it is the easiest of the Millennium Problems. I think we are still lacking something. Whether the tools are all here now, I am not sure. They may be. There are always these speculations with these really difficult problems; it may be that the tools simply aren’t there. I don’t believe that anyone in the 19th century could have solved Fermat’s Last Theorem, certainly not in the way it was eventually solved. There was just too big a gap in mathematical history. You had to wait another hundred years for the right pieces to be in place. You can never be quite sure about these problems whether they are accessible to your time. That is really what makes them so challenging; if you had the intuition for what can be done now and what can’t be done now you would be a long way towards a solution!

You mentioned the Tate–Shafarevich group and in that connection the Selmer group appears. Selmer [1920–2006] was a Norwegian mathematician, and it was Cassels [1922–2015] who is responsible for naming this group the Selmer group. Could you say a few words about the Selmer group and how it is related to the Tate–Shafarevich group, even if it’s a little technical? It is technical, but I can probably explain the basic idea of what the Selmer group is. What you are trying to do is to find the rational solutions on an elliptic curve. The method is to take the rational points on the elliptic curve – suppose you have got some – and you generate field extensions from these. So when I say generate extensions, I mean that you can take roots of those points on the elliptic curve. Just like taking the n’th root of 5 or the cube root of 2. You can do the same thing on an elliptic curve, you can take the n’th root of a point. These are all points which added to themselves n times gives you the point you started with. They generate certain extensions of the number field you started with, so in our case of the rational number field Q. You can put a lot of restrictions on those extensions. And the Selmer group is basically the smallest set of extensions you can get putting on all the obvious restrictions. 254 Abel Prize 2016

Let me summarize this. You’ve got the group of points. They generate some extensions; that’s too big, you don’t want all extensions. You cut that down as much as you can using local criteria, using p-adic numbers; that’s called the Selmer group. And the difference essentially between the group generated by the points and the Selmer group is the Tate–Shafarevich group. So the Tate–Shafarevich group gives you the error term if you like, in trying to get at the points via the Selmer group.

Selmer’s paper, which Cassels refers to, studied the Diophantine equation, 3x3 + 4y3 + 5z3 = 0 and similar ones. Selmer showed that it has just a trivial solution in the integers, while modulo n it has non-trivial solutions for all n. In particular, this curve has no rational points. Why did Cassels invoke Selmer’s name in naming the group? Yes, there are quite subtle relationships between these. What happens is you are actually looking at one elliptic curve, which in this case would be x3 + y3 + 60z3 = 0. Thatis an elliptic curve, in disguise, if you like, and the Tate–Shafarevich group involves looking at other ones like it, for example 3x3 + 4y3 + 5z3 = 0, which is a genus one curve, but which has no rational points. Its Jacobian is the original elliptic curve, x3 + y3 + 60z3 = 0. One way of describing the Tate–Shafarevich group is in terms of these curves that have genus one but don’t have rational points. And by assembling these together you can make the Tate–Shafarevich group, and that is reflected in the Selmer group. It is too intricate to explain in words but it is another point of view. I gave it in a more arithmetic terminology in terms of extensions. The more geometric terminology was in terms of these twisted forms.

The Modularity Conjecture

What you proved in the end was a special case of what is now called the Modu- larity Conjecture. In order to explain this one has to start with modular forms, and how modular forms can be put in relation with elliptic curves. Could you give us some explanations? Yes; an elliptic curve (over the rationals) we have described as an equation y2 = x3 + ax + b, where the a and b are assumed to be rational numbers. (There is also a condition that the discriminant should not vanish). As I said, at the beginning of the 19th century you could describe the complex solutions to this equation. You could describe these very nicely in terms of the Weierstrass ℘-function, in terms of a special elliptic function. But what we want is actually a completely different uniformization of these elliptic curves which captures the fact that the a and b are rational numbers. It is a parametrization just for the rational elliptic curves. And because it captures the fact that it is defined over the rationals it gives you a much better hold on solutions over the rationals than the elliptic functions do. The latter really only sees the complex structure. And the place it comes from are modular forms or modular curves. To describe modular functions first: we are used to functions which satisfy the relation of being Sir Andrew J. Wiles 255 invariant under translation. Every time we write down a Fourier series we have a function which is invariant under translation. Modular functions are ones which Z are invariant under the action of a much bigger group, usually a subgroup of SL2( ). So, you would ask for a function f(z) in one complex variable, usually on the upper half plan, which satisfies f(z) is the same as f((az + b) / (cz + d)); or more generally, is that times a power of cz + d. These are called modular functions and were extensively studied in the 19th century. Surprisingly they hold the key to the arithmetic of elliptic curves. Perhaps the Z simplest way to describe it is that because we have an action of SL2( ) on the upper half plane H – by the action z goes to (az + b) / (cz + d) – we can look at the quotient H modulo this action. You can then give the quotient the structure of a curve. In fact, it naturally gets the structure of a curve over the rational numbers. Z If you take a subgroup of SL2( ), or more precisely what is called a congruence subgroup, defined by the c value being divisible by N, then you call the curve a modular curve of level N. The Modularity Conjecture asserts that every elliptic curve over the rationals is actually a quotient of one of these modular curves for some integer N. It gives you a uniformization of elliptic curves by these other entities, these modular curves. On the face of it, it might seem we are losing because this is a high genus curve, it is more complicated. But it actually has a lot more structure because it is a moduli space.

And that is a very powerful tool? That is a very powerful tool, yes. You have function theory, you have deformation theory, geometric methods etc. You have a lot of tools to study it.

Taniyama, the young Japanese mathematician who first conjectured or suggested these connections, his conjecture was more vague, right? His conjecture was more vague. He didn’t pin it down to a function invariant under the modular group. I’ve forgotten exactly what he conjectured; it was invariant under some kind of group, but I forget exactly which group he was predicting. But it was not as precise as the congruence subgroups of the modular group. I think it was originally written in Japanese so it was not circulated as widely as it might have been. I believe it was part of notes compiled after a conference in Japan.

It was an incredibly audacious conjecture at that time, wasn’t it? Apparently, yes.

But then it gradually caught the attention of other mathematicians. You told us already about Gerhard Frey, who came up with a conjecture relating Fer- mat’s Last Theorem with the Modularity Conjecture. That’s right. Gerhard Frey showed that if you take a solution to the Fermat problem, say ap + bp = cp, and you create the elliptic curve y2 = x (x – ap)(x + bp), then the discriminant of that curve would end up being a perfect p’th power. And if you think 256 Abel Prize 2016 about what that means assuming the Modularity Conjecture – you have to assume something a bit stronger as well (the so called epsilon conjecture of Serre) – then it forces this elliptic curve to have the level N that I spoke about to be equal to one, Z and hence the associated congruence subgroup is equal to SL2( ). But H modulo Z SL2( ) is a curve of genus zero. It has no elliptic curve quotient so it wasn’t there after all, and hence there can’t be a solution to the Fermat problem.

The quest for a proof

That was the point of departure for your own work, with crucial further ingredients due to Serre and Ribet making this connection clear. May we briefly summarize the story that then follows? It has been told by you many times, and it is the focus of a BBC-documentary. You had moved to the United States, first to Harvard, then to Princeton University, becoming a professor there. When you heard of Ribet’s result you devoted all your research time to prove the Modularity Conjecture for semistable elliptic curves over the rationals. This work went on for seven years of really hard work in isolation. At the same time you were working as a professor in Princeton and you were raising small kids. A proof seems to be accomplished in 1993, and the development culminates in a series of three talks at the Isaac Newton Institute in Cambridge back in England, announcing your proof of Fermat’s Last Theorem. You are celebrated by your peer mathematicians. Even the world press takes an interest in your results, which happens very rarely for mathematical results. But then when your result is scrutinized by six referees for a highly prestigious journal, it turns out that there is a subtle gap in one of your arguments, and you are sent back to the drawing board. After a while you send for your former student, Richard Taylor, to come to Princeton to help you in your efforts. It takes a further ten months of hard and frustrating work; we think we do not exaggerate by calling it a heroic effort under enormous pressure. Then in a sudden flash of insight you realize that you can combine some of your previous attempts with new results to circumvent the problem that had caused the gap. This turns out to be what you need in order to get the part of the Modularity Conjecture that implied Fermat’s Last Theorem. What a relief that must have been! Would you like to give a few comments to this dramatic story? With regard to my own work when I became a professional mathematician working with Coates I realized I really had to stop working on Fermat because it was time-consuming and I could see that in the last hundred years almost nothing had been done. And I saw others, even very distinguished mathematicians, had come to grief on it. When Frey came out with this result, I was a bit sceptical that the Serre part of the conjecture was going to be true, but when Ribet proved it then, okay, this was it! Sir Andrew J. Wiles 257

And it was a long hard struggle. In some sense it is irresponsible to work on one problem to the exclusion of everything else, but this is the way I tend to do things. Whereas Fermat is very narrow, I mean it is just this one equation, whose solution may or may not help with anything else, yet the setting of the modular conjecture was one of the big problems in number theory. It was a great thing to work on anyway, so it was just a tremendous opportunity. When you are working on something like this it takes many years to really build up the intuition to see what kinds of things you need and what kinds of things a solution will depend on. It’s something like discarding everything you can’t use and won’t work till your mind is so focused that even making a mistake, you’ve seen enough that you’ll find another way to the end. Funnily enough, concerning the mistake in the argument that I originally gave, people have worked on that aspect of the argument and quite recently they have actually shown that you can produce arguments very like that. In fact, in every neighbouring case arguments similar to the original method seem to work but there is this unique case that it doesn’t seem to work for, and there is not yet any real explanation for it. So the same kind of argument I was trying to use, using Euler systems and so on, has been made to work in every surrounding case but not the one I needed for Fermat. It’s really extraordinary.

You once likened this quest for the proof of the Modularity Theorem in terms of a journey through a dark unexplored mansion. Could you elaborate? I started off really in the dark. I had no prior insights how the Modularity Con- jecture might work or how you might approach it. One of the troubles with this problem – it’s a little like the Riemann Hypothesis but perhaps even more so with this one – is you didn’t even know what branch of mathematics the answer would be coming from. To start with, there are three ways of formulating the problem, one is geometric, one is arithmetic and one is analytic. And there were analysts – I would not understand their techniques at all well – who were trying to make headway on this problem. I think I was a little lucky because my natural instinct was with the arithmetic approach and I went straight for the arithmetic route, but I could have been wrong. The only previously known cases where the Modularity Conjecture were known to hold were the cases of complex multiplication, and that proof is analytic, completely analytic. Partly out of necessity, I suppose, and partly because that’s what I knew, I went straight for an arithmetic approach. I found it very useful to think about it in a way that I had been studying in Iwasawa theory. With John Coates I had applied Iwasa- wa theory to elliptic curves. When I went to Harvard I learned about Barry Mazur’s work, where he had been studying the geometry of modular curves using a lot of the modern machinery. There were certain ideas and techniques I could draw on from that. I realized after a while I could actually use that to make a beginning – to find some kind of entry into the problem. 258 Abel Prize 2016

Before you started on the Modularity Conjecture, you published a joint paper with Barry Mazur, proving the main theorem of Iwasawa Theory over the rationals. Can you please tell us what Iwasawa Theory is all about? Iwasawa theory grew out of the work of Kummer on cyclotomic fields and his approach to Fermat’s Last Theorem. He studied the arithmetic, and in particular the ideal class groups, of prime cyclotomic fields. Iwasawa’s idea was to consider the tower of cyclotomic fields obtained by taking all p-power roots of unity at once. The main theorem of Iwasawa theory proves a relation between the action of a generator of the Galois group on the p-primary class groups and the p-adic L-functions. It is analogous to the construction used in the study of curves over finite fields where the characteristic polynomial of Frobenius is related to the zeta function.

And these tools turned out to be useful when you started to work on the Mod- ularity Conjecture? They did, they gave me a starting point. It wasn’t obvious at the time, but when I thought about it for a while I realized that there might be a way to start from there.

Parallels to Abel’s work

We want to read you a quotation: “The ramparts are raised all around but, enclosed in its last redoubt, the problem defends itself desperately. Who will be the fortunate genius who will lead the assault upon it or force it to capitulate?” It must been E. T. Bell, I suppose? Is it?

No, it’s not. It is actually a quote from the book “Histoire des Mathématiques” by Jean-Étienne Montucla [1725–1799], written in the late 18th century. It is really the first book ever written on the history of mathematics. The quotation refers to the solvability or unsolvability of the quintic equation by radicals. As you know Abel proved the unsolvability of the general quintic equation when he was 21 years old. He worked in complete isolation, mathematically speaking, here in Oslo. Abel was obsessed, or at least extremely attracted, to this problem. He also got a false start. He thought he could prove that one could actually solve the quintic by radicals. Then he discovered his mistake and he finally found the unsolvability proof. Well, this problem was at that time almost 300 years old and very famous. If we move fast forward 200 years the same quotation could be used about the Fermat problem, which was around 350 years old when you solved it. It is a very parallel story in many ways. Do you have comments? Yes. In some sense I do feel that Abel, and then Galois [1811–1832], were marking a transition in algebra from these equations which were solvable in some very simple way to equations which can’t be solved by radicals. But this is an algebraic break that came with the quintic. In some ways the whole trend in number theory now is Sir Andrew J. Wiles 259 the transition from basically abelian and possibly solvable extensions to insolvable extensions. How do we do the arithmetic of insolvable extensions? I believe the Modularity Conjecture was solved because we had moved on from this original abelian situation to a non-abelian situation, and we were developing tools, modularity and so on, which are fundamentally non-abelian tools. (I should say though that the proof got away mostly with using the solvable situation, not because it was more natural but because we have not solved the relevant problems in the general non-solvable case). It is the same transition in number theory that he was making in algebra, which provides the tools for solving this equation. So I think it is very parallel.

There is an ironic twist with Abel and the Fermat Problem. When he was 21 years old, Abel came to Copenhagen to visit Professor Degen [1766–1825], who was the leading mathematician in Scandinavia at that time. Abel wrote a letter to his mentor in Oslo, Holmboe [1795–1850], stating three results about the Fermat equation without giving any proofs – one of them is not easy to prove, actually. This, of course, is just a curiosity today. But in the same letter he gives vent to his frustration, intimating that he can’t understand why he gets an equation of degree n2, and not n, when dividing the lemniscate arc in n equal pieces. It was only after returning to Oslo that he discovered the double periodicity associated with the lemniscate integral, and also with the general elliptic integral of the first kind. If one thinks about it, what he did on the Fermat equation turned out to be just a curiosity. But what he achieved on elliptic functions, and implicitly on elliptic curves, turned out later to be a relevant tool for solving it. Of course, Abel had no idea that this would have anything to do with arithmetic. So this story tells us that mathematics sometimes develops in mysterious ways. It certainly does, yes.

Work styles

May we ask for some comments about work styles of mathematicians in general and also about your own? Freeman Dyson, a famous physicist and mathematician at IAS in Princeton, said in his Einstein lecture in 2008: “Some mathematicians are birds, others are frogs. Birds fly high in the air and survey broad vistas of mathematics out to the horizon. They delight in concepts that unify our thinking and bring together diverse problems from different parts of the landscape. Frogs live in the mud below and see only the flowers that grow nearby. They delight in the details of particular objects and they solve problems one at a time”. Freeman Dyson didn’t say that birds were better than frogs, or the other way around. He considered himself a frog rather than a bird. 260 Abel Prize 2016

When we are looking at your work, it seems rather difficult for us to decide where to place you in his classification scheme: among the birds, those who create theories, or among the frogs, those who solve problems. What is our own perception? Well, I don’t feel like either. I’m certainly not a bird – unifying different fields. I think of frogs as jumping a lot. I think I’m very, very focused. I don’t know what the animal analogy is, but I think I’m not a frog in the sense that I enjoy the nearby landscape. I’m very, very concentrated on the problem I happen to work on and I am very selective. And I find it very hard to even take my mind off it enough to look at any of the flowers around, so I don’t think that either of the descriptions quite fits.

Based on your own experience could you describe the interplay between hard, concentrated and persevering work on the one side, and on the other side these sudden flashes of insights that seemingly come out of nowhere, often appearing in a more relaxed setting. Your mind must have worked unconsciously on the problem at hand, right? I think what you do is that you get to a situation where you know a theory so well, and maybe even more than one theory, so that you have seen every angle and tried a lots of different routes. It is this tremendous amount of work in the preparatory stage where you have to understand all the details, and maybe some examples, that is your essential launch pad. When you have developed all this, then you let the mind relax and then at some point – maybe when you go away and do something else for a little bit – you come back and suddenly it is all clear. Why did you not think of that? This is something the mind does for you. It is the flash of insight. I remember – this is a trivial example in a non-mathematical setting – once someone showed me some script, it was some gothic script, and I couldn’t make head or tail of it. I was trying to understand a few letters, and I gave up. Then I came back half an hour later and I could read the whole thing. The mind somehow does this for you and we don’t quite know how, but we do know what we have to do to set up the conditions where it will happen.

This is reminiscent of a story about Abel. While in Berlin he shared an apartment with some Norwegian friends, who were not mathematicians. One of his friends said that Abel typically woke up during the night, lighted a candle and wrote down ideas that he woke up with. Apparently his mind was working while asleep. Yes, I do that except I don’t feel the need to write them down when I wake up with it because I know I will not forget it. But I am terrified if I have an idea when I am about to go to sleep that I would not wake up with that idea, so then I have to write it down.

Are you thinking in terms of formulas or in terms of geometric pictures, or what? Sir Andrew J. Wiles 261

It is not really geometric. I think it is patterns, and I think it is just parallels between situations I have seen elsewhere and the one I am facing now. In a perfect world, what is it all pointing to, what are the ingredients that ought to go into this proof, what am I not using that I have still in my pocket? Sometimes it is just desperation. I assemble every piece of evidence I have and that’s all I’ve got. I have got to work with that and there is nothing else. I often feel that doing mathematics is like being a squirrel and there are some nuts at the top of a very tall tree. But there are several trees and you don’t know which one. What you do is that you run up one and you think, no, it does not look good on this one, and you go down and up another one, and you spend your whole life just going up and down these trees but you’ve only got up to thirty feet. Now if someone told you the rest of the trees – it’s not in them, you have only one tree left, then you would just keep going until you found it. In some sense it is ruling out the wrong things that is really crucial. And if you just believe in your intuition, and your intuition is correct, and you stick with your one tree then you will find it.

Problems in mathematics

Felix Klein [1849–1925] once said: “Mathematics develops as old results are being understood and illuminated by new methods and insights. Proportion- ally with a better and deeper understanding new problems naturally arise.” And David Hilbert [1862–1943] stressed that “problems are the lifeblood of mathematics”. Do you agree? I certainly agree with Hilbert, yes. Good problems are the lifeblood of mathematics. I think you can see this clearly in number theory in the second half of the last century. For me personally obviously the Modularity Conjecture, but also the whole Langlands Program and the Birch and Swinnerton–Dyer Conjecture: These problems give you a very clear focus on what we should try to achieve. We also have the Weil Conjectures on curves and varieties over finite fields and the Mordell Conjec- ture and so on. These problems somehow concentrate the mind and also simplify our goals in mathematics. Otherwise we can get very, very spread out and not sure what’s of value and what’s not of value.

Do we have as good problems today as when Hilbert formulated his twenty-three problems in 1900? I think so, yes.

Which one do you think is the most important problem today? And how does the Langlands program fit in? Well, I think the Langlands program is the broadest spectrum of problems related to my field. I think that the Riemann Hypothesis is the single greatest problem from the areas I understand. It is sometimes hard to say exactly why that is, but I do 262 Abel Prize 2016 believe that solving it would actually help solve some of these other problems. And then of course I have a very personal attachment to the Birch and Swinnerton–Dyer Conjecture.

Intuition can lead us astray sometimes. For example, Hilbert thought that the Riemann Hypothesis would be solved in his lifetime. There was another problem on his list, the 7th, that he never thought would be solved in his lifetime, but which was solved by Gelfond [1906–1968] in 1934. So our intuition can be wrong. That is right. I’m not surprised that Hilbert felt that way. The Riemann Hypothesis has such a clear statement and we have the analogue in the function field setting. We understand why it is true there, and we feel we ought to be able to translate it. Of course, many people have tried and failed. But I would still myself expect it to be solved before the Birch and Swinnerton–Dyer Conjecture.

Investing in mathematics

Let’s hope we’ll find out in our lifetimes! Classical mathematics has, roughly speaking, two sources: one of them coming from the physical sciences and the other one from, let’s for simplicity call it number theoretical speculations, with number theory not associated to applications. That has changed. For example, your own field of elliptic curves has been applied to cryptography and security. People are making money with elliptic curves nowadays! On the other hand, many sciences apart from physics really take advantage and profit from mathematical thinking and mathematical results. Progress in industry nowadays often depends on mathematical model- ling and optimization methods. Science and industry propose challenges to the mathematical world. In a sense, mathematics has become more applied than it ever was. One may ask whether this is a problem for pure mathematics. It appears that pure mathematics sometimes is put to the side lines, at least from the point of view of the funding agencies. Do you perceive this as a serious problem? Well, I think in comparison with the past one feels that mathematicians two, three hundred years ago were able to handle a much broader spectrum of mathematics, and a lot more of it touched applied mathematics than would a typical pure mathematician do nowadays. On the other hand that might be because we only remember the very best and most versatile mathematicians from the past. I think it is always going to be a problem if funding agencies are short-sighted. If they want to see a result in three years then it is not going to work. It is hard to imagine a pure development and then the application all happening within three to five years. It is probably not going to happen. Sir Andrew J. Wiles 263

On the other hand, I don’t believe you can have a happily functioning applied maths world without the pure maths to back it up, providing the future and keeping them on the straight and narrow. So it would be very foolish not to invest in pure mathematics. It is a bit like only investing in energy resources that you can see now. You have to invest in the future; you have to invest in fusion power or solar power or these other things. You don’t just use up what is there and then start worrying about it when it is gone. It is the same with mathematics, you can’t just use up the pure mathematics we have now and then start worrying about it when you need a pure result to generate your applications.

Mathematical awards

You have already won a lot of prizes as a result of your achievements, culmi- nating in proving Fermat’s Last Theorem. You have won the Rolf Schock Prize, given by the Swedish Academy, the Ostrowski Prize, which was given to you in Denmark, the Fermat Prize in France, the Wolf Prize in Israel, the Shaw Prize in Hong Kong – the prize that has been named the Nobel Prize of the East; and the list goes on, ending with the Abel Prize tomorrow. May we ask you whether you enjoy these awards and the accompanying celebrations? I certainly love them, I have to say. I think they are a celebration of mathematics. I think with something like Fermat it is something people are happy to see in their lifetime. I would obviously be very happy to see the Riemann Hypothesis solved. It is just exciting to see how it finally gets resolved and just to understand the end of the story. Because a lot of these stories we won’t live to see the end of. Each time we do see the end of such a story it is something we naturally will celebrate. For me I learned about the Fermat problem from this book of E. T. Bell and about the Wolfskehl Prize attached to it. The Wolfskehl Prize was still there – only just I may say – I only had a few years left before the deadline for it expired.

This gives us the lead to talk a little about that prize. The Wolfskehl Prize was founded in 1906 by Paul Wolfskehl [1856–1906], who was a German physician with an interest in mathematics. He bequeathed one hundred thousand Reichmarks (equivalent to more than one million dollars in today’s money) to the first person to prove Fermat’s Last Theorem. The prize was, according to the testament, valid until September 13, 2007, and you received it in 1997. By then, due in part to hyperinflation Germany suffered after World War I, the prize money had dwindled a lot. For me the amount of money was unimportant. It was the sentimental feeling attached to the Wolfskehl Prize that was important for me. 264 Abel Prize 2016

Graduate students

You have had altogether twenty-one PhD-students and you have attracted very gifted students. Some of them are really outstanding. One of them, Manjul Bhargava, won the Fields medal in 2014. It must have been a pleasure to be advisor to such students? Yes, I don’t want to take too much credit for it. In the case of Manjul I suggested a problem to him but after that I had nothing much more to do. He was coming up with these absolutely marvellous discoveries. In some sense you get more credit if you have very gifted students, but the truth is that very gifted students don’t really require that much help.

What is the typical way for you of interacting with graduate students? Well, I think the hardest thing to learn as a graduate student is that afterwards you need to carry on with the rest of your professional life; it’s hard to pick problems. And if you just assign a problem and they do it, in some sense that hasn’t given them terribly much. Okay, they solved that problem, but the hard thing is then to have to go off and find other problems! So I prefer it if we come to a decision on the problem together. I give them some initial idea and which area of mathematics to look at, having not quite focused on the problem. Then as they start working and become experts they can see a better way of pinning down what the right question is. And then they are part of the process of choosing the problem. I think that is a much better investment for their future. It doesn’t always work out that way, and sometimes the initial problem you give them turns out to be the right thing. But usually it is not that way, and usually it’s a process to find the right problem.

Hobbies and interests

We always end the Abel interviews by asking the laureate what he enjoys doing when he doesn’t work with mathematics. What are your hobbies and interests outside of mathematics? Well, it varies at different times. When I was doing Fermat, and being a father with young children, that combination was all-consuming. I like to read and I like various kinds of literature, novels, some biographies, it is fairly balanced. I don’t have any other focused obsessions. When I was in school I played on chess teams and bridge teams, but when I started to do serious mathematics I completely lost interest in those.

What about music; are you fond of music? I go and listen to concerts, but I am not myself actively playing anything. I enjoy listening to music, classical, preferably. Sir Andrew J. Wiles 265

Are you interested in other sciences apart from mathematics? I would say somewhat. These are things I do to relax, so I don’t like them to be too close to mathematics. If it is something like animal behaviour or astrophysics or something from a qualitative point of view, I certainly enjoy learning about those. Likewise about what machines are capable of, From left to right: Sir Andrew J. Wiles, Christian Skau and and many other kinds of pop- Martin Raussen. (Photo: Eirik Furu Baardsen, DNVA) ular science, but I’m not going to spend my time learning the details of string theory. I’m too focused to be willing to do that. Not that I would not be interested, but this is my choice.

We would like to thank you very much for this wonderful interview. That is first of all on behalf of the two of us, but also on the behalf of the Norwegian, the Danish and the European Mathematical Society. Thank you very much!

Abel Laureate Sir Andrew Wiles at the award ceremony. (Photo: Audun Braastad)

An ImaginaryImaginary Interview Interview with NielsNiels HenrikHenrik Abel Abel

This interview with Niels Henrik Abel is entirely imaginary, but it is not inconceiv- able – based upon what we know of Abel’s life – that an actual interview with Abel at the end of his life would overlap to a large degree with our imaginary interview. It follows the pattern that has been established in previous interviews with Abel Prize recipients.e have chosen 6 February 1829 as time for the interview and “Bukkerommet” at Froland Verk in Aust-Agder as location. At that time, Abel had celebrated Christmas and had taken part in the New Year’s festivities at Froland Verk, where his ﬁancée was governess. At the time of the interview, he had been bedridden for about a month after being struck down with pneumonia, the underlying tuberculosis becoming fatal in the end. However, by the beginning of February, he had recuperated from the pneumonia and the fever was gone.e know from reliable sources (those that looked after him) that Abel at the time, though weak, was optimistic about the future and thought that he would recover and get well again. (Alas, we know that this was an illusion – the tuberculosis killed him and he died exactly two months after our imaginary interview.) In the interview, you will hear the authentic voice of Abel.e have been careful not to ascribe anything to Abel that he himself had not written either in his letters or his mathematical notebooks, or which cannot be found in his mathematical manuscripts, published or not.e have expanded on some of his answers but only if we have a strong indication that this is what he meant or had in mind. To enliven the interview, we have used geometric ﬁgures and/or formulas for illustration in those parts of the interview that deal with Abel’s mathematical discoveries. As we do when we prepare for interviews with actual Abel Prize recipients, we have ‘contacted’ people who knew Abel or who expressed opinions about him and his achievements, notably Holmboe, Degen, Crelle, Legendre, Jacobi, Bessel and Gauss. (However, as part of our imaginary set-up, we have assumed that these people could be contacted with 21st century speed and not by the slow mail of the 1820s!)

Tell us about your ﬁrst encounter with mathematics. My father sent me and my older brother Hans Mathias to Christiania in 1815, when I was 13 years old, to enrol as pupils at the Cathedral School. This was the best secondary school in Norway at the time and this was the ﬁrst time I was exposed to mathematics beyond elementary arithmetical computations. I liked mathematics

267 268 Imaginary Interview

Born Aug. 5 1802

1815

1816

1817

1818 Pupil at the Cathedral School in Christiania (later Oslo) New math teacher 1819 (Bernt Michael Holmboe)

1820 Death of Abel’s father

1821 “Proof” of solvability (sic!) of the quintic Examen Artium

1822

1822 Entrance exam to the university 1823 Visits professor Degen in Copenhagen

“Anni mirabiles” (The miraculous years) 1824 Discoveries: 1) Abel’s integral equation Preparation for the travel 2) Unsolvability of the quintic to Göttingen and Paris – in 3) Elliptic functions isolation in Christiania 1825 4) The addition theorem

Berlin and Crelle: “Journal für die reine und 1826 Angewandte Mathematik”. Abel’s foreign trip to Berlin Paris and the disappearance of the Paris Memoir. and Paris 1827 Avoids travelling to Göttingen and Gauss.

1828 Development of the theory of elliptic functions. Hectic work period in Chris- (Abel-Jacobi “competition”.) tiania Theory of equations.

Last 31/2 months at 1829 Froland Verk Interview February 6 January 6: Last manuscript – proof of April 6 the addition theorem in its most general form.

A chronology of Abel's life. A chronology of Abel’s life. Niels Henrik Abel 269

Main building at Froland Verk (located 200 kilometers south-west of Oslo) where Abel stayed from De- cember 19, 1828 to his death on April 6, 1829.

and was pretty good at it. However, we had a maths teacher named Hans Peter Bader who, like my co-pupils, I hated. I would describe him as a bully bordering on a sadist. Even though Bader was a competently trained mathematics teacher, his behaviour dampened my interest in mathematics.

Could you be more specific? Bader physically thrashed and beat pupils in his class if he was not satisfied with their answers or solutions – sometimes even for no apparent reason. One day – this happened in November 1817 – he beat a pupil named Henrik Stoltenberg, who fell to the floor while Bader continued to kick him. Henrik was bedridden and died a week later. hether that was directly related to Bader’s mistreatment or not I do not know but the result was that Bader was fired, which made us pupils very happy. I should mention that Henrik Stoltenberg’s father, Carl Peter Stoltenberg, was a member of our parliament (Storting) and was therefore an influential person. hat happened then? For me, it was a watershed – a before and after – in my life.e got a new maths teacher in early 1818, Bernt Michael Holmboe. He was only 23 years old and he exhibited an enthusiasm for mathematics that was contagious. More importantly, 270 Imaginary Interview

Abel stayed in the “Buck room” (“Bukkerommet” in Norwegian), named after the eponymous goat on the wallpaper.

he opened, ﬁguratively speaking, the door for me into mathematics at a deeper level by introducing me to the great masters of mathematics, especially Euler, Lagrange and Gauss. In the beginning, Holmboe gave me private tutoring but after a while I studied the works of these masters by myself. You could say that I devoured the books of these giants of mathematics. Initially, I borrowed books from Holmboe’s rather impressive book collection and then later from the university library.

Is it fair to say that Euler, Lagrange and Gauss became your teachers of mathematics, so to say? That is absolutely true. My life was changed when I studied these masters. I holdthe ﬁrm conviction that if one wants to make progress in mathematics, one should study the masters, not their pupils. I decided that I would devote my life to mathematics. During all this time, Holmboe encouraged me and goaded me on, selﬂessly admitting that he could not keep up with me.

This may be an improper question to ask but your father was a public ﬁgure very much in the public spotlight so it would be unnatural to skirt around this matter. Your father was elected as a member of our parliament (Storting) in 1818 but he was involved in a scandal that almost resulted in impeachment and Niels Henrik Abel 271 dismissal from parliament. Anyway, he was heavily criticised and viciously ridiculed in the newspapers at the time. How did this aﬀect you living in Chris- tiania and experiencing this close up? It is painful for me to talk about this. I felt that my father was unfairly attacked. He returned to his vicarage at Gjerstad in Aust-Agder a broken man, his reputation ir- reparably damaged. He died two years later, in 1820, the early death certainly being hastened by the condemnation he had been exposed to. Initially, at the Cathedral School, there were a few pupils that teased me about my father but fortunately both Holmboe and Rosted, the rector, protected me and shielded me from being taunted. My main consolation during this painful period was mathematics. hen I was immersed in a mathematical problem or a mathematical theory, I was completely lost to the world and the surrounding noise and din.

This leads naturally to the next question. Tell us about your ﬁrst mathematical discovery. Let me start by saying that I was fascinated, early on, by the theory of equations and, in particular, solvability problems. The ﬁrst result of some importance that I obtained occurred during my last year at the Cathedral School when I was 18 years old. I proved, or so I thought, that the general quintic equation

5 4 3 2 x + a4x + a3x + a2x + a1x + a0 = 0 can be solved by radicals, thereby settling a problem that mathematicians had struggled with for more than 300 years. However, the proof was wrong – I found the mistake myself – which of course was a let-down. But it had the beneﬁcial eﬀect that I became convinced that it was impossible to solve the general equation of degree n ≥ 5 by radicals. Two and a half years later, in late Autumn 1823, I succeeded in proving the unsolvability.

Your answer brings to mind what the French mathematician Jean Étienne Mon- tucla (1725–1799) metaphorically says in his widely read “Histoire des Math- ématiques” – you may perhaps have read it yourself? – describing the allure of solving the quintic:

“The ramparts are raised all around but, enclosed in its last redoubt, the problem defends itself desperately. ho will be the fortunate genius who will lead the assault upon it or force it to capitulate?”

So you were the fortunate genius! Is it possible to give an indication of how you proved your impossibility result so that non-experts can get a ﬂavour of what is involved? I can try. First, I will say that I am indebted to Lagrange for his analysis of the solutions of the general cubic and quartic equations, which were obtained by the Italians in the early 1500s. Lagrange gave a methodical explanation of why solutions by radicals were possible in these cases. hen he applied the same analysis to the general 272 Imaginary Interview quintic, the ensuing so-called resolvent equation became of degree six, so higher than ﬁve. This was in contrast to the cubic and quartic cases, where the resolvent equations were of lower degree – two and three respectively. All this indicated that something dramatically new happened when the degree of the general equation was higher than four. Now, let us look at the general equation of degree two, whose solution was known in antiquity, 2 x + a1x + a0 = 0,

a1 1 2 a1 1 2 x1 = + a 4a0, x2 = a 4a0, − 2 2 1 − − 2 − 2 1 − √ √ 2 = . a1 − 4a0 x1 − x2 √ hat we should note are two things: (i) The radical that occurs in the solution is a square root. (ii) The radical can be expressed as a polynomial of the roots. The general cubic equation, 3 2 x + a2x + a1x + a0 = 0, which, by a simple substitution, can be written in the form x3 + px + q = 0, has roots given by the so-called Cardano’s formulas:

2 3 2 3 3 q q p 3 q q p x1 = + + + + , √−2 √ 4 27 √−2 − √ 4 27

2 3 2 3 2 3 q q p 3 q q p x2 = ω + + + ω + , √−2 √ 4 27 √−2 − √ 4 27

2 3 2 3 3 q q p 2 3 q q p x3 = ω + + + ω + , √−2 √ 4 27 √−2 − √ 4 27

2πi where ω = e 3 and where q2 p3 1 1 + = (ω + )(x1 x2)(x1 x3)(x2 x3), √ 4 27 9 2 − − − 2 3 3 q q p 1 2 + + = (x1 + ωx2 + ω x3), √−2 √ 4 27 3

2 3 3 q q p 1 2 + = (x1 + ω x2 + ωx3). √−2 − √ 4 27 3 Niels Henrik Abel 273

hat we should note is that the first (“inner”) radical is a square root, while the next is a cubic root. Furthermore, all the radicals that occur in the solution can be expressed as polynomials of the roots x1, x2 and x3 (as well as the roots of unity – in this case a cubic root of unity). A similar result is true for the general quartic equation. I realised that the key to proving unsolvability by radicals of the quintic and higher degrees was, firstly, to prove that the radicals occurring in a putative solution could be expressed as polynomials in the roots (with certain roots of unity appearing in the coefficients). Then, secondly, by permuting the roots, one would get a number of different values, which would not be compatible with the restricted number of values that the radicals can obtain. By slightly modifying the second part of the proof, it can be summarised as follows. In a putative solution by radicals, the first (“inner”) radical would have to be a square root. If the degree n of the general equation is larger or equal to three then the second radical has to be a cubic root. However, if n ≥ 5 then it followed from my analysis that the second radical has to be a quintic radical. Hence, we get a contradiction to the assumption that there exists a radical solution if n ≥ 5. There is a peculiar antisymmetry, so to say, between the two parts that the proof consists of. In the first part, one starts with the last (“outermost”) radical and works successively inwards to the first (“innermost”) radical; in the second part, one starts with the innermost radical and works outwards. I should mention that when I came to Paris, I learned that there was an Italian – his name was Paolo Ruffini – who had written several papers claiming to have proved the unsolvability result. However, his writing is so obscure that it is difficult to judge the validity of his proof.

You wrote up your proof of the unsolvability of the general quintic and higher degree equations as a pamphlet in 1824 and later published a more detailed proof in the first volume of “Journal für die Reine und angewandte Mathema- tik”,the journal founded by Crelle in Berlin in 1826. However, you made some other important discoveries before you completed the proof of this spectacular result. Could you tell us about some of these? I would like to mention two results, the first of which concerns indefinite integrals (or anti-derivatives). It is well known that the derivative of an elementary function is again an elementary function. By an elementary function, I mean: (i) Rational functions. (ii) Algebraic functions y = ϕ(x), i.e. y is implicitly defined by n n 1 y + an 1(x)y − + + a1(x)y + a0(x)=0, − ···

where the ai(x)’s are rational functions. (iii) The exponential function ex and its inverse, log x. (iv) The trigonometric functions and their inverse arc-functions. (v) All functions that can be deﬁned by means of any ﬁnite combination of these preceding classes of functions. 274 Imaginary Interview

Of course, (i) is subsumed under (ii). For example, let f(x) be the elementary function x 1 f(x)= x2 1 + log(x + x2 1). 2 − 2 − √ √ Then, the derivative is x2 f ′(x)= , √ 2 x − 1 again an elementary function. Is the converse true, namely, is the (indeﬁnite) integral of an elementary function again an elementary function? This question had been investigated by many mathematicians, among them Laplace, but no one had been able to prove the conventional conjecture, namely, that this is not true in general. I developed a theory around this problem and gave some criteria for when the integral of an elementary function is again elementary. I also proved that log x + dx ∫ x a = = log x = 1 2 + is not an elementary function if a ̸ 0. (If a 0, we get x dx 2 log x C.) This discovery, which I made during theinter of 1822-23, I deemed∫ so important that I wrote it up in French – the lingua franca of mathematics. However, the manuscript disappeared when it was sent between various departments as an appendix to my application for a travel grant.

But haven’t you written up this result again? It seems to us that the result is of considerable importance? No, I have not done that, partly because I was so busy making even more important discoveries. However, I have not forgotten it. In the memoir I worked on this last Autumn, “Précis d’une théorie des fonctions elliptiques”, which is half-finished, I announce in a footnote that I am in possession of a comprehensive theory on the integration of elementary functions, in particular algebraic functions, and I will get back to it later. This is one of my many projects that I will pursue when I get well and am able to work again. e interrupted you with our last question. So let’s get back to the other discovery you mentioned, which you made during the period we are talking about. In the first year of publication of the Norwegian language scientific journal “Magazin for Naturvidenskapen”, which Hansteen, professor in applied mathematics at the university in Christiania, founded in 1823, there appeared an article of mine titled “Opløsning af et Par Opgaver ved Hjelp af bestemte Integraler” (“The solution of some problems by means of definite integrals”). One of the problems I solved was a mechanical problem that involved an integral equation. As far as I know, no one had looked at similar equations before. Let me describe the mechanical problem: Niels Henrik Abel 275

A C

D E

M P

s B

A monotonically increasing function ψ(x) is given. Find the shape of the curve ADMB such that a mass released at rest at M will reach B at time ψ(x) if it glides (without friction) along the curve under the inﬂuence of gravity. The relevant equation turns out to be x ( ) g t = ψ( ). ⋆ √ dt x ( ) ∫0 x − t The problem is to ﬁnd the function g(x), where ds = g(x)dx. I solved this by introducing fractional derivation. Then, equation (⋆) can be written as / d1 2s ψ(x)=√π , dx1/2 where s is the arc length between A and M. The solution then becomes 1/2ψ( ) x ψ( ) = 1 d− x = 1 t s √π 1/2 π √ dt dx− ∫0 x − t and then ds g(x)= . dx One gets to fractional derivation this way, starting with the Taylor series of a function F(x):

∞ n F(x)= anx . = ∑n 0 The k’th derivative is k d F(x) ( ) ∞ n! ∞ Γ(n + 1) = k ( )= n−k = n−k, k F x an ( )!x an Γ( + )x dx n k = n 1 k ∑n=k − ∑n 0 − 276 Imaginary Interview

Γ Γ( )= = , , , where denotes the gamma function. (Recall that m ∞ if m 0 −1 −2 ···.) Now, substitute k with any α. (α) I deﬁne the α-derivative F (x) of F(x) by

α d F(x) (α) ∞ Γ(n + 1) α = ( )= n− . α F x an Γ( + α)x dx = n 1 ∑n 0 − (α+β) α β ( ) ( ) x = 0 = −1 ( )= ( ) One can show that F F ◦ F , F F and F x 0 F t dt. By using the equality ∫

Γ(x)Γ(y) 1 = x−1( )y−1 , Γ( + ) t 1 − t dt x y ∫0 which is due to Euler, one gets the following identity by interchanging summation and integration:

x (α) 1 α ( )= ( )( )− −1 . F x Γ( α) F t x − t dt − ∫0 Here, α should be less than one. α = 1 Setting − 2 , one gets

1/2 x x d− F(x) ( 1 ) 1 F(t) 1 F(t) = − 2 ( )= = , / F x 1 dt dt −1 2 Γ √ √π √ dx 2 ∫0 x − t ∫0 x − t ( ) which is exactly the form we want. So,

1/2ψ( ) x ψ( ) ( )= 1 d− x = 1 t . s x √π 1/2 π √ dt dx− ∫0 x − t In particular, if ψ(x)=constant, that is, if the time of descent of the mass is the same regardless of the height from which it starts, we get the solution

s = k√x, where k is a constant. This is a well known equation for the cycloid and so we get, as a very special case of my solution, that the cycloid is the isochrone curve, as was shown by Christian Huygens in 1673.

You got a small travel grant, generously given to you by Søren Rasmussen, professor of pure mathematics at the university, so that you could travel to Copen- hagen in the Summer of 1823 to visit Professor Ferdinand Degen at Copenhagen Niels Henrik Abel 277

University. Degen was arguably the strongest mathematician in Scandinavia at the time. Can you tell us the most important thing that came out of this visit for you? ell, first, on a personal note, during my stay in Copenhagen, I met Christine Kemp, who later become my fiancée and who is governess here at Froland Verk. She is tirelessly looking after me while I am recuperating. Mathematically, the most important thing for me was that Degen encouraged me to study Legendre’s books treating elliptic integrals, which I did right after returning to Christiania. This led, in turn, to what I will characterise as a major discovery, namely, that by inverting the elliptic integral (of the first kind), which in Legendre’s normal form is

x dx ( 2)( 2 2) ∫0 1 − x 1 − k x √ = x dx (note that k i gives 0 √ 4 ), one gets a function with two periods. I had actually 1−x met a roadblock before∫ this discovery that had held me up for some time.

This sounds interesting! Is this the same roadblock to which you refer inthe letter that Holmboe told us he received from you during your sojourn in Copen- hagen?e quote from the letter:

“... You remember the little paper which treated the inverse functions of the elliptic transcendentals; I asked Degen to read it, but he could not discover any erroneous conclusions or where the mistake may be hidden. God knows how I can pull out of it! ...”

You are right. Let me explain. Reading Gauss’ “Disquisitiones Arithmeticae”, it was especially Chapter VII that intrigued me. There, Gauss studies cyclotomic equations (or circle division equations). He proves that dividing a circle into n equal parts leads to a cyclotomic equation that can be solved by radicals. He also proves that a regular n-gon can be constructed by ruler and compass if and only if n is a product of distinct Fermat primes (multiplied by 2m for some m, of course). Then, he says that his theory can be applied not only to circular functions (i.e. sine/cosine functions) but also to other transcendental functions, for example those that depend on the integral

dx . √ 4 ∫ 1 − x This points to the lemniscate: 278 Imaginary Interview

r = ( )= dr s arc AP √1 r4 0 − r = ϕ(s) ∫ y s P = √ θ r r cos 2 A x a r1 a 1 − r2

= 2 ( 2 = ) ( 2 + 2)2 = 2 2 r1 · r2 a 2a 1 x y x − y

Let us ﬁrst consider the circle. The equation to ﬁnd the n points that divide the unit n = = circle in n equal parts is of course x − 1 0 or, removing the trivial root x 1, n−1 + n−2 + + + = . x x ··· x 1 0 One can also express this in terms of an integral:

y x = ( )= dx = s s arc AP √1 x2 arcsinx A 0 − P ∫ x = sin s 2π = x dx = ( 2π ) n 0 √ 2 , x sin n 1−x x ∫ x 1 (n-division of the circumference.)

x2 + y2 = 1

I found the equation for the n points dividing the lemniscate into n equal parts and the equation is of degree n2, with n2 distinct roots! But there are only n division 2 points so how should one interpret the other n − n roots? This was what prompted my question to Degen. Niels Henrik Abel 279

= 2 ( 2 = ) r1 · r2 a 2a 1

a 1 −a r1 r2 1 ω = dr 4 √1 r4 0 − r ∫ = dr s 4 π √1 r ω = 2 = . ... 0 − ( ,√ ) 5 2411 (Gauss) ∫ M 1 2 = ϕ( ) r s n-division of the circumference: r ϕ( )=ϕ( + ω)=ϕ( + ω) ω = dr = ϕ ω s s s i n √1 r4 r n 0 − ∫ ( )

Could you explain to us why the existence of two independent periods of the inverse function x = ϕ(s) of the lemniscate integral,

x dx s = , √ 4 ∫0 1 − x leads to an equation of degree n2 – and not degree n – for the division of the circumference into n equal parts? This is easy to explain. In fact, the equation for dividing the circumference into n ϕ( kω+miω ) equal pieces boils down to ﬁnding the equation that has roots n , where , ω ω ϕ ϕ 0 ≤ k m ≤ n − 1 and and i are the two periods of ( being the inverse lemniscate function). So, the equation in question must be of degree n2 since the number of roots is n2. In the circle case, the inverse function is the sine function, which only has one period, namely 2π. So, in this case, the division equation is of degree n.

So, you discovered the double periodicity of the inverse function of an elliptic integral (of the ﬁrst kind) in 1824, naming this inverse an elliptic function. You were in Christiania at the time, preparing for your travels abroad to Göttingen and Paris to visit Gauss and leading French mathematicians. However, your ﬁrst publication on elliptic functions, “Recherches sur les fonctions elliptiques”, which was published in two instalments in Crelle’s journal, was not written up until you returned to Christiania after your travels abroad. hat was the cause of this delay? During my stays in Berlin and Paris (especially in Paris), I worked intermittently on elliptic functions, uncovering their amazing properties. The “Recherches” memoir 280 Imaginary Interview

gives a comprehensive theory of these functions. I thought I was the only mathematician in possession of this theory but I was mistaken. If I had known that Jacobi was on my heels, I would not have postponed writing up my discoveries.

Holmboe has told us about a letter you sent to him from Paris in December 1826, which is related to what you just said. In this letter you write, among other things:

“... I have lifted the mystery which rested over Gauss’ theory of division of the circle. ... I am preparing a memoir on elliptic functions in which there are many queer things which I ﬂatter myself will startle someone; among other things it is about the division of arcs of the lemniscate. ... All I have described about the lemniscate is the fruit of my eﬀorts in the theory of equations, my favourite topic. You will not believe how many delightful theorems I have discovered.”

In the same letter, you said you would visit Göttingen and Gauss on your forthcoming journey back to Berlin. hy did you change your mind? The reason I did not make a detour to Göttingen on my way back to Berlin from Paris when I left on 29 December was simply that I was nearly broke! I could not aﬀord to visit Göttingen. I hoped that when I got to Berlin, money would have arrived for me from Norway (this happened but not immediately). Unfortunately, an occasion never arose later for me to meet Gauss. I admired Gauss enormously but I wished his mathematical writing had been more expansive and explanatory (more like Euler and Lagrange perhaps). Gauss was like a fox erasing his tracks with his tail. In fact, Gauss usually declined to present the intuition behind his very elegant proofs and erased all traces of how he discovered them.

Maybe it was a good thing you did not meet Gauss! In fact, we have been in contact with Bessel, a conﬁdant of Gauss. After the ﬁrst part of your “Recherches” memoir was published in Crelle’s journal in September 1827, Bessel wrote to Gauss urging him to publish his results on elliptic functions. Here is a quotation from Gauss’ letter replying to Bessel:

“I shall most likely not soon prepare my investigation on the transcendental functions which I have had for many years – since 1798 – because I have many other matters which must be cleared up. Herr Abel has now, as I see, anticipated me and relieved me of the burden in regard to about one third of these matters, particularly since he has executed all developments with great stringency and elegance. He has followed exactly the same road which I travelled in 1798; it is no wonder that our results are so similar. To my surprise this extended also to the form and even, in part, to the choice of notations, so several of his formulas appeared as if they were copied from mine. But to avoid every misunderstanding, I Niels Henrik Abel 281

must observe that I cannot recall ever having communicated any of these investigations to others.”

Do you have any comments? ell, maybe it was a good thing after all that I did not meet Gauss. If we had met, I would certainly have told him about my discoveries on elliptic functions, only to be told, I am sure, that he had made the same discoveries almost 30 years earlier. That would certainly have been a disappointment for me. On the other hand, Ifeel proud and very honoured to hear Gauss’ nice words and praise of the ﬁrst part of my “Recherches” memoir, especially since Gauss is not known to lavish praise on the works of other mathematicians. It is likely he would also have been pleased with the second part of “Recherches”,which treats transformation theory, including examples of complex multiplication.

Finally, we come to your greatest discovery: the addition theorem. The genesis of that discovery, we suspect, is alluded to in a letter dated 2 March 1824 that you sent from Christiania to Degen in Copenhagen. As you know, we have been in contact with Degen about this. Here is a quotation from your letter to him:

“... I have come across a remarkable discovery: I can express a property of all transcendental functions of the form ϕ(z)dz, where ϕ(z) is an arbitrary algebraic function of z, by an equation∫ of the following form (denoting ϕ(z)dz = ψ(z)): ∫ ψ( )+ψ( )+ + ψ( )=ψ(α )+ψ(α )+ + ψ(α )+ , z1 z2 ··· zn 1 2 ··· n p

where z1, z2,... are algebraic functions of an arbitrary number of variables (n depends on this number and is in general much larger; α1,α2,... are constant entries and p is an algebraic/logarithmic function, which in many cases is zero). This theorem and a memoir based on it I expect to send to the French Institute, for I believe it will throw light over the whole theory of transcendental functions...”

Is this an early version of the addition theorem? You are absolutely right. I should say, though, that, in the letter to Degen, the theorem is stated somewhat awkwardly and in a rudimentary form (but it contains the gist of the matter). I consider the addition theorem as my most important discovery. hen I arrived in Paris in the Summer of 1826, I wrote up a memoir on transcendental functions that is based upon the addition theorem and presented it to l’Académie des Sciences on 30 October 1826. Cauchy and Legendre were appointed as referees and I was eagerly awaiting their judgment. 282 Imaginary Interview

And what was their judgement? I have not heard a word and, since more than two years have elapsed, I fear that my Paris memoir is lost. hy have you not made enquiries about this?e have been in contact with Legendre, to whom you wrote a long letter last November telling him about your mathematical discoveries. But you do not mention anything about the Paris memoir, even though Legendre is one of the referees. It seems to us very strange that you did not bring this up in your correspondence with him. I guess I did not want to embarrass him by reminding him about my memoir. I did publish a short article, which appeared in Crelle’s journal last December, where I treated hyperelliptic integrals using a special case of the addition theorem. In a footnote, I said that I had presented a memoir to l’Académie des Sciences in 1826 treating a much more general situation. Finally, exactly a month ago, on 6 January, in this very room, I wrote a four-page letter to Crelle stating and proving the addition theorem in its most general form. This is the fundamental theorem upon which the Paris memoir is based. I asked Crelle to publish it, which I am sure he will. So, at least the main theorem of the memoir will be saved for posterity. e have some news for you about all this. In correspondence with Jacobi, he tells us that after he read your footnote in your hyperelliptic paper, he sent a letter to Legendre expressing his astonishment that the memoir you submitted to the academy in Paris had not been published. Jacobi informs us that Crelle has shown him your four-page letter and, upon reading it and putting it into context with your hyperelliptic paper, he writes:

“... die grösste mathematische Entdeckung unserer Zeit, obgleich erst eine künftige grosse Arbeit ihre ganze Bedeutung aufweise könne...” [“... the greatest mathematical discovery of our time, even though only a great work in the future will reveal its full signiﬁcance...”]

This is very good news! Thank you for conveying it. Jacobi’s praise makes mefeel humble.

Could you tell us in broad terms what the addition theorem is all about? Instead of giving you the general statement, let me state it for the very special case of hyperelliptic integrals, on which I published the paper in Crelle’s journal that I mentioned earlier. Let n be any natural number and let x1, x2,...,xn be any complex numbers. Then,

x1 x2 xn g zk dx + dx + + dx = dx , ··· 0 f(x) 0 f(x) 0 f(x) 0 f(x) ∫ ∫ ∫ ∑k=1 ∫ √ √ √ √ Niels Henrik Abel 283 where f(x) is a polynomial of degree 2g + 2 or 2g + 1 with no multiple roots and z1, z2,...,zg are algebraic functions of x1, x2,...,xn. This is a generalisation of Euler’s addition theorem for elliptic integrals:

x1 x2 dx + dx ( 2)( 2 2) ( 2)( 2 2) ∫0 1 − x 1 − k x ∫0 1 − x 1 − k x z √ √ = dx , ( 2)( 2 2) ∫0 1 − x 1 − k x where √ ( 2)( 2 2)+ ( 2)( 2 2) = x1 1 − x2 1 − k x2 x2 1 − x1 1 − k x1 . z 2 2 2 √ 1 − k x1x√2

There are several other areas of mathematics that we have not touched upon where you have made fundamental contributions. Your work on inﬁnite series and convergence criteria is one area that comes to mind. But time is running out for our interview. However, we cannot end the interview without some words about the theory of equations, which you said was your favourite topic. Let us start with the aforementioned letter you sent to Legendre in November. You ended the letter with these words:

“I have had the good fortune in ﬁnding a deﬁnite rule with whose help one can recognise whether any given equation can be solved by radicals or not. A corollary of my theory is that it is impossible to solve the general equation of degree greater than four.”

Could you elaborate? A week ago, I received a reply letter from Legendre, which ends with these words:

“You announce to me, Sir, a very beautiful work on algebraic equations that has as object to resolve for each given numerical equation whether it can be solved by radicals, and to declare unsolvable those that do not satisfy the required conditions. A necessary consequence of this theory is that the general equation of degree higher than four cannot be solved. I urge you to let this new theory appear in print as quickly as you are able. It will be of great honour to you, and will universally be considered the greatest discovery which remains to be made in mathematics.”

You can just imagine the eagerness with which I want to send him the memoir I am writing up – of which a rough sketch exists – presenting my theory. But I do not have the strength to work now so I have to be patient till I regain my strength. Besides my memoir on the unsolvability of the general equation of degree n ≥ 5, I have only published one other paper entirely devoted to the theory of equations, 284 Imaginary Interview titled “Mémoire sur une classe particulière d’équations résolubles algébriquement”. The transformation theory of elliptic functions (of which the division of the lemniscate arc into equal parts is a very special case) gives an abundance of examples of algebraic equations that satisfy the conditions of the memoir, and thus can be solved by radicals. In fact, my various papers on elliptic functions are interspersed with examples of equations that can be solved by radicals. As I said above, I am working on a memoir that will attack the general problem of the algebraic solution (i.e. by radicals) of numerical equations.

Crelle told us about the letter you sent him last year dated 25 September. Here is a quote from the letter:

“… It pleases me greatly that you will print my “Précis d’une théorie des fontions elliptiques”. I shall exert myself to make it as clear and good as possible, and hope I shall succeed. But do you not think that it would be better to commence with this paper instead of the one on the equations? I ask you urgently.”

“Firstly, I believe that the elliptic functions will be of greater interest; secondly, my health will hardly permit me to occupy myself with the equations for a while. I have been ill for a considerable period of time, and compelled to stay in bed. Even if I am now recovered, the physician has warned me that any strong exertion can be very harmful.”

“Now the situation is this: the equations will require a disproportion- ately greater eﬀort on my part than the elliptic functions. Therefore, I should prefer, if you do not absolutely insist on the article on equations – in that case you shall have it – to begin with the elliptic functions. The equations will follow soon afterward …”

Do you have any comments? ell, the letter is self-explanatory, I think. There was an agreement between Crelle and myself that I should write a memoir on the theory of equations, settling once and for all the question of whether an arbitrary numerical equation can be solved by radicals or not. This is a problem I have been thinking about for a long time and, during my sojourn in Paris, I made an important discovery that will be crucial for the solution. As I told you, I did start writing the memoir in question but it was interrupted by my sickness in September so only a rough sketch exists. Crelle wrote back to me admonishing me to follow my doctor’s advice. Therefore, I applied all my strength last Autumn to write up my “Précis” memoir – of which half is ﬁn- ished – which treats the theory of elliptic functions from a purely algebraic point of view.

So you have a lot of things on your mathematical agenda when you get well and are able to work again? Niels Henrik Abel 285

Absolutely! The ﬁrst order of business is to complete the “Précis” memoir. Then, I will write up the memoir on equations. Another project is the theory of integrating elementary functions, especially algebraic functions, that I mentioned earlier. But perhaps the most interesting project is to use the addition theorem to invert integrals of algebraic functions, similarly to what was done with elliptic integrals. This last problem seems to me to be of the utmost importance because the solution will throw new light on the higher transcendental functions. In fact, I do not think Professor Degen was exaggerating when, in a letter to Professor Hansteen in 1821, he poetically predicted that:

“… The serious investigator … would discover a Strait of Magellan leading into wide expanses of a tremendous analytic ocean.”

You received a travel grant in early 1824 from the university in Christiania to travel abroad for two years to visit Göttingen to meet Gauss and to visit Paris, “the focus of all my mathematical desires”, as you described it. How- ever, there was a catch: the collegium at the university decided that, before you started your travels, you should stay in Christiania for two years to study, among other things, foreign languages, primarily French and German, in order to be more prepared for your journey abroad. They presumably thought you were too young to start your travels right away. Anyway, it took about one and a half years before you embarked on your journey. hat was your reaction to all this? I was very unhappy that I could not start my travels right away. I was so eager to meet my peers abroad. However, in hindsight, I deem it a “blessing in disguise” that I was held back. It was during the “waiting period” of one and half years before my journey, in total mathematical isolation, so to say, in Christiania, that I made my most important discoveries. During my travels, I spent most of my time working out and developing the ideas that came to me then. The only really new things I learned after I left Norway were when I came across Cauchy’s “Cours d’Analyse” while in Berlin. This spurred me to rigorously examine infinite series and related convergence questions. I published a memoir in the first volume of Crelle’s journal (in 1826) on this material. hat you are telling us does remind us about what happened to Isaac Newton when, at almost the same age as you, he was forced to spend the years 1665– 67 in total scientific isolation at his birthplaceoolsthorpe in Lincolnshire because of the plague that befell Cambridge. It was during these two years that he made his most fundamental discoveries. I don’t like to be compared with Newton. He is in a class all by himself. However, I see the analogy with his isolation and mine.

Can you say something about your working style? For instance, have you had moments of epiphany, where, all of a sudden, you see the light, and the solution 286 Imaginary Interview of a problem you have been struggling with appears to you completely trans- parent? In this connection, we mention that your friend Christian Boeck, who was one of the Norwegians that you shared an apartment with in Berlin, tells us that you habitually woke up during the night, lit a candle and wrote down ideas that you had woken up to. Many times, I have experienced the solution to a problem I have been working on appearing to me in a sudden ﬂash of insight. This only happens after working on the problem intensely over a long period, though.

Could we ask you a somewhat vague question? Do you have what we may call, for lack of a more precise term, a “philosophy of mathematics”? To put it in a more down-to-earth way, what is your thinking about mathematics – how it ought to be pursued with respect to formulating problems and how to go about solving these problems? Let me answer your question this way. One should give a problem such a form that it is always possible to solve it, something one can always do with any problem. In presenting a problem in this manner, the actual wording of it contains the germ of its solution and shows the route one should take. The reason why this method has been so little used in mathematics is the extreme complication it appears to be subject to in the plurality of problems, especially if these are of a certain general nature. However, in many of these cases, the complication is only an illusion and vanishes at first sight. I have treated several topics in analysis and algebra in this manner and, although I have often posed myself problems that surpass my powers, I have nevertheless attained a great number of general results that have shed a broad light on the nature of these quantities, the knowledge of which is the object of mathematics. e cannot end this interview without hearing your opinions on the various mathematicians you met during your sojourns in Berlin and Paris. Holmboe showed us the letter you sent him from Paris in October 1826, where you gave some rather biting and harsh characterisations of some of the French mathematicians. First of all, August Leopold Crelle, who I met and befriended in Berlin, is the most kind and honourable man one can imagine. Crelle, though a construction engineer and not a mathematician, founded “Journal für die reine und angewandte Mathema- tik” in 1826 and I have published almost all my papers there. Crelle has, and still is, working indefatigably to secure a professorship for me in Berlin. Now … Paris and French mathematicians. I made only a fleeting and superficial contact with them during my sojourn in Paris. The only one I found amiable was Legendre. The French are extremely reserved toward strangers. Everybody works for themselves without concern for others. All want to instruct and nobody wants to learn. The most absolute egotism reigns everywhere. Let me exemplify bymen- tioning two episodes, where I met Cauchy and Laplace, respectively. Niels Henrik Abel 287

I showed Cauchy (who is a ﬁrst class mathematician, no question about that) my memoir on transcendental functions, which I dare say is good (without any boasting). He would hardly cast a glance at it. The memoir was presented to l’Académie des Sciences and Cauchy was appointed as one of the referees. More than two years have elapsed and I have not heard a word. He has presumably lost it. Laplace, who is a small, lively man, suﬀers from the “disease” that he interferes with the speech of others. I have tried to talk with him, only to be abruptly interrupted, so I was not able to communicate with him. e end this interview by asking about your interests aside from mathematics, like music, literature or art? I love the theatre and this is, by far, my greatest interest aside from mathematics. Everywhere I have travelled, I have gone to the theatre to see plays. From Paris, I remember with fondness a play by Molière where the eminent actress Mademoiselle Mars played the leading role. As for music, I am simply tone-deaf. Crelle used to have musical soirées that I attended. I listened politely but the music did not move me at all. I should also mention one other interest of mine: playing cards with my friends. The main thing is the social aspect of card playing. I love being among friends and if I am alone for a long time, I get depressed.

Thank you for this most interesting interview! Martin aussen and Christian Skau, Editors Interviews with the Abel Prize Laureates 2003–2016

Interviews with the Abel Prize Laureates 2003–2016

Martin Raussen Christian Skau Editors

ISBN 978-3-03719-177-4

www.ems-ph.org

Raussen/Skau | Fonts: Helvetica Neue | Farben: Pantone 116, Pantone 287, Cyan | RB 15.1 mm