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Home , Fields Medal, Jacques Tits, John Tate, Oslo Cathedral School, Pierre Deligne, Ragni Piene

The Award Ceremony May 21, 2013 The University Aula,

Procession accompanied by the “Abel Fanfare” (Klaus Sandvik) Performed by musicians from The Staff Band of the

His Majesty King Harald V enters the University Aula

Sicilienne (Gabriel Fauré) Aage Kvalbein, cello; Håvard Gimse, piano

Opening speech by Professor Kirsti Strøm Bull President of The Norwegian Academy of Science and Letters

Cafetin de Buenos Aires (Mariano Mores), Så skimrande var aldri havet (Evert Taube) and (Ary Barroso). Cello Quartet: Aage Kvalbein, ­ Jan Øyvind Grung Sture, Oda Dyrnes and Benedicte Årsland

The Abel Committee’s Citation by Professor Ragni Piene Chair of the Abel Committee

His Majesty King Harald V presents the Abel Prize to

Acceptance speech by Abel Laureate Pierre Deligne

Bachianas Brasileiras No. 5 Aria (Cantilena) (Heitor Villa-Lobos) Eir Inderhaug, soprano; 8 celli: Aage Kvalbein, Jan Øyvind Grung Sture, Oda Dyrnes, Benedicte Årsland, Erlend Vestby, Tone Østby, Elisabeth Teigen and Kjersti Støylen

His Majesty King Harald V leaves the University Aula

Procession leaves

The Prize Ceremony will be followed by a reception at Theatersalen, Hotel Continental. During the reception, Pierre Deligne will be interviewed by Tonje Steinsland. More info on page 13. mathématiques, in 1968. In 1972, Deligne received the doctorat d’État ès Sciences Professor Pierre Deligne Mathématiques from Université -Sud 11. Institute for Advanced Study, Princeton, New Jersey, USA Deligne went to the University of Brussels with the ambition of becoming a high- Abel Laureate 2013 school teacher, and of pursuing as a hobby for his own personal enjoyment. There, as a student of , Deligne was pleased to discover “for seminal contributions to and for their that, as he says, “one could earn one’s living by playing, i.e. by doing research in ­transformative impact on number theory, , and mathematics.” related fields” After a year at École normale supérieure in Paris as auditeur libre, Deligne was Pierre Deligne was born on 3 October 1944 in Etterbeek, Brussels, . He concurrently a junior at the Belgian National Fund for Scientific Research is Professor Emeritus in the School of Mathematics at the Institute for Advanced and a guest at the Institut des Hautes Études Scientifiques (IHÉS). Deligne was a Study in Princeton, New Jersey, USA. Deligne came to Princeton in 1984 from Insti- visiting member at IHÉS from 1968-70, at which time he was appointed a perma- tut des Hautes Études Scientifiques (IHÉS) at Bures-sur-Yvette near Paris, , nent member. where he was appointed its youngest ever permanent member in 1970. Concurrently, he was a Member (1972– 73, 1977) and Visitor (1981) in the School of Mathematics at the Institute for Advanced Study. He was appointed to a faculty When Deligne was around 12 years of position there in 1984. age, he started to read his brother’s university math books and to demand Pierre Deligne is a research who has excelled in making connections explanations. His interest prompted a between various fields of mathematics. His research has led to several important high-school math teacher, J. Nijs, to discoveries. One of his most famous contributions was his proof of the Weil conjec- lend him several volumes of “Elements tures in 1973. This earned him both the (1978) and the of Mathematics” by , (1988), the latter jointly with Alexandre Grothendieck. Deligne was awarded the the pseudonymous grey eminence Balzan Prize in 2004 and Wolf Prize in 2008. that called for a renovation of French mathematics. This was not the kind When Deligne was awarded the Fields Medal, and , both of reading matter that one would at the Harvard Mathematics Department, wrote in Science magazine that “There are normally dream of offering a young few mathematical subjects that Deligne’s questions and comments do not clarify, boy, but for Deligne it became a life for he combines powerful technique, broad knowledge, daring imagination, and changing experience. From then on unfailing instinct for the key idea.” Photo: Valérie Touchant-Landais he never looked back.

Although his father wanted him to become an engineer and to pursue a career that would afford him a good living, Deligne knew early on that he should do what he loved, and what he loved was mathematics. He studied mathematics at the Université Libre de Bruxelles (University of Brussels) and received his Licence en mathématiques, the equivalent of a B.A., in 1966 and his Ph.D., Doctorat en

4 5 ics, this teacher loaned him some of his mathematical works. For Deligne, this proved to Professor Kirsti Strøm Bull be of pivotal importance. President of The Norwegian Academy of Science and Letters A teacher played a crucial role in the development of Niels Henrik Abel as well. That is why we intend to emphasise the teacher’s important role in arousing and stimulating the inter- est of children and youth in mathematics by establishing a teacher’s prize in connection with the Abel Prize. This Prize is named after Abel’s teacher, . The Holmboe Prize will be awarded tomorrow at Niels Henrik Abel’s old school, Oslo . This year’s Prize-Winner is Anne-Mari Jensen. Your Majesty, Minister, Your Excellencies, Dear Prize Winner, honoured assembled guests, The Abel Board also supports several other initiatives to stimulate interest in mathematics On behalf of The Norwegian Academy of Science and Letters, I have the pleasure and among children and young people. honour of welcoming you all to the Abel Prize Award Ceremony for 2013. The Abel Prize was established by the Norwegian Government in 2002 in connection with the 200th an- Mathematical knowledge is absolutely necessary in order to deal with our age’s infrastruc- niversary of Niels Henrik Abel’s birth. The Abel Prize was first awarded in 2003 - ten years ture, buildings and communication systems, banking and insurance, the Internet, etc. ago. Today it is being awarded for the eleventh time. Mathematics plays an important role in modern society. Today’s society needs mathemati- cians. The Abel Prize is a prize for outstanding scientific work in mathematics. The Prize is a rec- ognition of scientific contributions of exceptional depth and significance for the discipline The Abel Prize also allows us to boost the fundamental importance of basic research in of mathematics. solving the challenges faced by society. In mathematics, we often find the clearest exam- ples that advanced solutions that are primarily a result of the desire to solve a theoretical Pierre Deligne is being awarded the Abel Prize “for seminal contributions to algebraic problem have had unintentional and unexpectedly great practical importance. geometry and for their transformative impact on number theory, representation theory, and related fields” to quote the Abel Committee’s explanation. Mathematics is a timeless, universal discipline, applied and developed throughout the world. A mathematical proof is true and can be appreciated in any culture or ideology. Deligne is a mathematician who has distinguished himself by finding connections between Mathematics links the past to the present. Even though mathematical research is in a pe- different fields of mathematics. His research has led to important discoveries. riod of rapid development, old ideas are not rejected as so often occurs in other branches of science. His discoveries have had enormous influence, and a number of mathematical concepts have been named after him. In just a short period of time, the Abel Prize has become one of the great international prizes in mathematics. With members from many different countries, who are nominated Like Niels Henrik Abel, Pierre Deligne had already made pioneering contributions to math- by the key international mathematics organisations, the Abel Committee deserves much ematics at a very young age. of the honour for the status that the Prize has attained. I would like to thank the Prize Committee, chaired by Professor Ragni Piene, for this important and demanding work. Pierre Deligne is best known for his impressive solution of the last of the , a solution he presented before he had turned 30. The Norwegian Academy of Science and Letters would also like to thank the other key participants involved in the Prize and its associated events - the Abel Board, the Norwe- Stimulating the interest of children and young people in mathematics is an important gian Government and the Norwegian Ministry of Education and Research, the Norwegian objective of the Abel Prize. Good, inspiring mathematics teachers play a very important Mathematics Council and the Norwegian mathematics community. role in this endeavour. Honoured guests, Abel Prize Winner Pierre Deligne, once again I wish you welcome to On many occasions, Pierre Deligne has emphasised the important effect that a mathemat- this year’s Abel Prize Ceremony. It is a great day for mathematics and for long-term basic ics teacher had on him when he was 14 years old. Noticing his keen interest in mathemat- research. 6 7 7 10

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1. Endre Szemerédi receives the Abel Prize from H.M. King Harald V 7. Isadore M. Singer and Michael F. Atiyah receive the Abel Prize from H.M King Harald V 2. Mikhail Gromov and Minister Tora Aasland outside Akershus Castle at the Abel Banquet 8. Wreath-laying ceremony at the Abel Monument in Oslo 3. is received in audience by H.M. Queen Sonja at the Royal Palace 9. John Griggs Thompson and Jacques Tits 4. interviewed by Tonje Steinsland at Gamle Logen 10. Srinivasa S.R. Varadhan gives his Abel Lecture at the 5. The 2012 Abel Prize Award Ceremony at the University Aula 11. Jean-Pierre Serre, the first Abel Prize Laureate 6. John Torrence Tate inspires young KappAbel-winners at Oslo Cathedral School 12. Peter D. Lax gives his acceptance speech In Euclidean geometry the equation x² + y² =25 describes a circle with radius 5. As Professor Ragni Piene a Diophantine equation, it has the solution x=3, y=4 (and conversely) – (3,4,5) is a Chair of the Abel Committee Pythagorean triple. Suppose instead we work “modulo” a number p in the way we tell time “modulo” 12. “Modulo” 3 we have only three numbers: 0,1,2. So 25=1 modulo 3. The solutions are now x=0, y=1, and x=0, y=2 (and conversely). If we add a number “the square root of 2”, we get another solution. We can make larger and larger finite number systems in this way and ask how many solutions we get. The answers are encoded in a zeta-function, and the Weil conjectures concern the description of this Your Majesty, Minister, Your Excellencies, honored Laureate, dear colleagues and function. guests! André Weil envisioned that the proof of his conjectures would require methods from Geometric objects, such as lines, circles, and spheres, can be described by simple algebraic . In this spirit, Grothendieck and his school developed the theory of algebraic equations – for example, the points on a circle are the solutions to a quadratic ℓ-adic , which became a basic tool in Deligne’s proof. Deligne’s work shed equation. The resulting fundamental connection between geometry and algebra led to new light on the connection between the cohomology of algebraic varieties over the the development of algebraic geometry, in which geometric methods are used to study complex numbers and the Diophantine structure of algebraic varieties over finite fields. solutions of polynomial equations and, conversely, algebraic techniques are applied to The Weil conjectures have many important applications in number theory. analyze geometric objects. Among Deligne’s other important contributions are his generalization of classical Geometry and algebra are among the oldest and most fundamental branches of , his and Mumford’s compactification of the of stable curves mathematics. Geometry flourished in ancient Greece, and was already used at that as an algebraic , his work with Beilinson, Bernstein, and Gabber on perverse time to solve algebraic equations. Euclid’s Elements is the world’s all-time most sheaves, his clarification of the Riemann–Hilbert correspondence, and so on. successful textbook! Five hundred years after Euclid, Diophantus was the first to systematically use algebraic symbols as a precursor to polynomial equations. Deligne’s powerful concepts, ideas, results, and methods continue to influence the development of algebraic geometry as well as mathematics as a whole. The connection between algebra and geometry was developed and refined from the 16th century on by Viète, Descartes, and Fermat, and in the 19th century Abel, Riemann and others prepared the ground for the revolutionary transformations and extensions of algebraic geometry that took place in the 20th century, making it a central subject with deep connections to almost every area of mathematics. Pierre Deligne played a crucial role in many of those developments.

Whereas Fermat is best known for the Fermat Conjecture and Abel for proving that a fifth degree equation cannot be solved by radicals, Pierre Deligne is best known for his spectacular solution of the deepest of the Weil conjectures, namely the analogue of the for algebraic varieties over finite fields.

The Abel Committee: Gang Tian, , Ragni Piene, and Noga Alon. Photo: Eirik Furu Baardsen 10 11 Professor Pierre Deligne ­“motives”, which for me has been a guiding light. Acceptance speech I would also like to give thanks to the two extraordinary institutions at which I have spent my career: the IHÉS (Institut des Hautes Études Scientifiques) and the IAS (Institute for Advanced Study).

Established in 1930, the IAS was made possible by the generosity of Louis Bamberger and his sister Caroline Fuld and guided by the ideals of its first director, Abraham Flexner. Your Majesty, Minister, Excellencies, colleagues, family, friends and guests, As an aside, please allow me to mention that in a letter to the trustees of 1930, the founders insisted: “It is fundamental in our purpose, and our express desire, that, in the I am very honoured that this Abel prize associates me with the luminaries who received it appointments to the staff and faculty as well as in the admission of workers and stu- before me, amongst whom are my teachers and mentors Jacques Tits and Jean-Pierre dents, no account shall be taken, directly or indirectly, of race, religion, or sex.” This was Serre. far from common in the US of the 1930s, and I much prefer such a statement of princi- ples to the creeping quotas that clumsily attempt to enforce a similar ideal at present. The past century has been a golden century for mathematics. When I look back, I am amazed at all the questions that in my youth seemed inaccessible, but which have now The founding principle of the IAS is expounded in Flexner’s article of 1939, ”The Useful- been solved. The last half-century has also been a golden time for , but I ness of Useless Knowledge”*, which remain as relevant now as it was then. It explains worry that the prospects for young people are now far from being as good. that the current tendency of funding agencies to try to direct research is misguided, and that it is even worse to try to direct it toward directly applicable goals. Flexner explains, Throughout my life, I have received crucial help from many people and institutions. This by examples, that at the source of most of the important applications of sciences are for me is an occasion to give thanks. discoveries guided not by applications, but by curiosity. My first debt of gratitude is to Mr Jeff Nijs. Mr Nijs was a high school teacher. I first met When applying for a visit to IAS, visitors usually explain what they intend to do. One of him at age 12 as the father of a friend. He noted my interest in mathematics, protected the first things they are told upon arrival is that no matter what they said they were going it, nourished it by giving me my first serious mathematical books (a risky and felicitous to do, they are free to ignore it, and follow their curiosity. That indeed is how it should be. choice: Bourbaki’s set theory) and by arranging the possibility for me to borrow books from the Bibliothèque royale de Belgique. Later he introduced me to Jacques Tits. This freedom is a powerful incentive to do the best we can. As I can attest from exam- ples in my own work, it sometimes leads to dead ends, but that is a small price to pay. It was fortunate for me that, up to 1964, Tits was professor at the Free University of Brussels. I learned much from him, and when I was 20, he told me to go to Paris, where The IHÉS was founded in 1958 by Léon Motchane, in deliberate imitation of the IAS, but I benefited from Serre’s deep and luminous lectures at Collège de France and from without the security of an endowment and at a time when private support for mathemat- ­Grothendieck’s seminar at IHÉS. In due time, Serre also offered me suggestions that ics or physics was unheard of in France. This made the accomplishment of Motchane in proved crucial. One was to look at the work of Eichler and Shimura relating classical creating the IHÉS, wisely selecting the first permanent members and persuading them to automorphic forms to cohomology. Another was to pay attention to estimates proved by accept, and then keeping the IHÉS alive - a truly extraordinary accomplishment. Rankin. I am grateful to both the IHÉS and the IAS for their defense and illustration of curiosity- My main debt of gratitude is to . He did not mind my ignorance. driven research, and for enabling me to try to do my best. I am glad that the Abel com- He taught me my trade as well as ℓ-adic cohomology by asking me to write up, from mittee is guided by the same principles, and I hope this Abel prize will enable me to help his rough notes, the talks XVII and XVIII of SGA 4. He convinced his colleagues at IHÉS young mathematicians. to offer me a position. He exposed me to his , and especially to his idea of

* Harper’s magazine, October 1939, available at: library.ias.edu/files/UsefulnessHarpers.pdf 12 13 Aage Kvalbein Reception and interview Cellist Tonje Steinsland in conversation with Pierre Deligne

The Prize Award Ceremony will be followed by a reception at Theatersalen, Hotel Continental, Stortingsgaten 24/26, next to the National Theatre. During the reception, Pierre Deligne will be interviewed by journalist and TV personality Tonje Steinsland. Light refreshments will be served. The event is open for all guests attending the Award Ceremony.

Photo: Jan Ivar Vik Cellist Aage Kvalbein (born 1947 in Oslo) is one of ’s most prominent musicians, both as soloist, chamber musician and teacher. His debut recital in Oslo in 1973 received brilliant reviews and was followed by a hectic touring activity with up to 150 concerts per year. The tours included all the European countries, as well as USA, Mexico, Guatemala, Israel, Soviet, Hong Kong, Singapore, The Philippines and Korea. Kvalbein also took part in three competitions, in Essen, Munich and , and he won all of them. Here in Norway he has been awarded several prizes: The Debutant Prize, The Norwegian Music Critics’ Award, The Grieg Prize and three times the Norwegian “Grammy” Award. At the age of 33 years he became Norway’s first professor of cello at the Norwegian State Academy of Music, a position he still holds. Over the years he has made more than 40 recordings of solo and chamber music for cello. A large number of works have been dedicated to him by the most prominent Norwegian composers, and from composers in , Ireland, Belgium, Italy, USA and Finland.

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