ISSN 0002-9920 (print) ISSN 1088-9477 (online)

of the American Mathematical Society March 2017 Volume 64, Number 3

Women's History Month

Ad Honorem Sir Andrew J. Wiles

page 197

2018 Leroy P. Steele Prize: Call for Nominations page 195

Interview with New AMS President Kenneth A. Ribet page 229

New York Meeting page 291

Sir Andrew J. Wiles, 2016 Abel Laureate.

“The definition of a good mathematical problem is the mathematics it generates rather Notices than the problem itself.” of the American Mathematical Society March 2017 FEATURES

197 239229 26239 Ad Honorem Sir Andrew J. Interview with New The Graduate Student Wiles AMS President Kenneth Section Interview with Abel Laureate Sir A. Ribet Interview with Ryan Haskett Andrew J. Wiles by Martin Raussen and by Alexander Diaz-Lopez Allyn Jackson Christian Skau WHAT IS...an Elliptic Curve? Andrew Wiles's Marvelous Proof by by Harris B. Daniels and Álvaro Henri Darmon Lozano-Robledo The Mathematical Works of Andrew Wiles by Christopher Skinner

In this issue we honor Sir Andrew J. Wiles, prover of Fermat's Last Theorem, recipient of the 2016 Abel Prize, and star of the NOVA video The Proof. We've got the official interview, reprinted from the newsletter of our friends in the European Mathematical Society; "Andrew Wiles's Marvelous Proof" by Henri Darmon; and a collection of articles on "The Mathematical Works of Andrew Wiles" assembled by guest editor Christopher Skinner. We welcome the new AMS president, Ken Ribet (another star of The Proof). Marcelo Viana, Director of IMPA in Rio, describes "Math in Brazil" on the eve of the upcoming IMO and ICM. For Women's History Month we've got the story of Joan Clarke, an under-recognized English code-breaker during World War II, who like Alan Turing worked on the Enigma Project.

To fend off spring fever, read the Notices, submit an article or an item for the BackPage, participate in the commen- tary on the webpage www.ams.org/notices, and send in some Letters to the Editor. —Frank Morgan, Editor-in-Chief

FROM THE SECRETARY 252 100 Years Ago: Joan Clarke Allyn Jackson 195 2018 Leroy P. Steele Prizes: Call for Nominations 256 Commutative Algebra Provides a Big Surprise for 268 2017 Class of Fellows of the AMS Craig Huneke's Birthday Irena Swanson

260 Math PhD Careers: New Opportunities ALSO IN THIS ISSUE Emerging Amidst Crisis Yuliy Baryshnikov, Lee DeVille, Richard Laugesen 234 Math in Brazil: Sowing the Seeds Marcelo Viana 265 Book Review A Doubter's Almanac Sheldon Axler 245 Opinion The Common Vision Project: Four Reactions

IN EVERY ISSUE Notices AmericAn mAthemAticAl Society of the American Mathematical Society

272 Mathematics People

277 Mathematics Opportunities

278 New Publications Offered by the AMS 283 Mathematics Calendar Game Theory, Alive 286 Classified Advertising Anna R. Karlin, University of Washington, Seattle, and Yuval Peres, Microsoft Research, Redmond, WA 287 Meetings and Conferences of the AMS This beautifully written book introduces the reader to the rich and flourishing subject of game theory. From an exhaus- 304 The Back Page tive account of basics of the classical theory to a wide range of modern themes—including auctions, matching markets, sequential decision making—the book presents a great variety of topics in a rigorous yet entertaining manner. The authors guide the reader through the intricacies of game theory with an exquisite mathematical elegance. For anyone interested in the topic, experts and beginners alike, this book is a must. —Gábor Lugosi, Pompeu Fabra University, Barcelona, Spain 2017; approximately 396 pages; Hardcover; ISBN: 978-1-4704-1982-0; List US$75; AMS members US$60; Order code MBK/101

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Leroy P. Steele Prizes

The selection committee for these prizes requests nominations for con- sideration for the 2018 awards. Further information about the prizes can be found in the November 2015 Notices pp.1249–1255 (also available at www.ams.org/profession/prizes-awards/ams-prizes/steele-prize).

Three Leroy P. Steele Prizes are awarded each year in the following catego- ries: (1) the Steele Prize for Lifetime Achievement: for the cumulative influ- ence of the total mathematical work of the recipient, high level of research over a period of time, particular influence on the development of a field, and influence on mathematics through PhD students; (2) the Steele Prize for Mathematical Exposition: for a book or substantial survey or expos- itory-research paper; and (3) the Steele Prize for Seminal Contribution to Research: for a paper, whether recent or not, that has proved to be of fundamental or lasting importance in its field, or a model of important research. In 2018 the prize for Seminal Contribution to Research will be awarded for a paper in Discrete Mathematics/Logic.

Further information and instructions for submitting a nomination can be found at the Leroy P. Steele Prizes website:

www.ams.org/profession/prizes-awards/ams-prizes/steele-prize

Nominations for the Steele Prizes for Lifetime Achievement and for Mathematical Exposition will remain active and receive consideration for three consecutive years.

For questions contact the AMS Secretary at [email protected].

The nomination period is February 1, 2017 through March 31, 2017. Call for Nominations for Call Notices of the American Mathematical Society

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Wiles received the Abel Prize from Crown Prince Haakon of Norway.

Wiles ultimately established in a proof both surprising and profound. It is especially appropriate that Wiles’s groundbreaking work on elliptic curves was recognized by the awarding of the Abel Prize, elliptic curves being the natural domains of the elliptic functions introduced n May 24, 2016, Sir Andrew J. Wiles received by Abel. As the citation for the 2016 Abel Prize concludes: the Abel Prize in a ceremony held in the Aula of the University of Oslo in Oslo, Norway. Wiles, “Few results have as rich a mathematical history and as who received the prize from H.R.H. Crown dramatic a proof as Fermat’s Last Theorem.” Prince Haakon at the award ceremony, was The awarding of the Abel Prize was followed by the Othe fourteenth recipient of the 6 million NOK (about Abel Lectures on the next day, May 25. In his lecture, 750,000 USD) prize. A prize honoring the Norwegian Wiles explained how his proof of Fermat’s Last Theorem mathematician Niels Henrik Abel was first proposed by exemplified the movement of number theory from the the world-renowned mathematician Sophus Lie, also from abelian to the nonabelian. Henri Darmon, in his lecture Norway, and initially planned for the one-hundredth an- “Andrew Wiles’s Marvelous Proof,” described Wiles’s work niversary of Abel’s birth in 1902, but the establishment as “a centerpiece of the Langlands program” and ex- of the Abel Prize had to wait another hundred years. The plained its transformative impact, and Manjul Bhargava Abel Prize is administered by the Norwegian Academy of spoke about how Wiles’s work has implications for the Science and Letters. Birch–Swinnerton-Dyer Conjecture. The Abel Prize was awarded to Wiles for “his stunning proof of Fermat’s Last Theorem,” which opened a new era Photo Credits in number theory. The citation of the Abel Committee, Photos are courtesy of Audun Braastad. read by committee chair John Rognes on the occasion of the announcement of the 2016 Abel Prize, recounts the early history of Fermat’s Last Theorem—the assertion that The full text of the citation of the Abel Committee for any given integer 푛 ≥ 3, there are no integer solutions can be found at www.abelprize.no/c54154/binfil/ to 푥푛 + 푦푛 = 푧푛 with 푥푦푧 ≠ 0—and how it was eventually download.php?tid=67039. An expanded version of linked to the then-conjectural modularity of semistable Henri Darmon’s Abel Lecture is included in this issue. elliptic curves and that it was this modularity that Andrew DOI: http://dx.doi.org/10.1090/noti1486

March 2017 Notices of the AMS 197 Interview with Abel Laureate Sir Andrew J. Wiles Martin Raussen and Christian Skau

Raussen and Skau : Professor Wiles, please accept our con- gratulations for having been selected as the Abel Prize Laureate for 2016. To be honest, the two of us expected this interview to take place 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 Fermat’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 twenty years before it earned you the Abel Prize. Neverthe- less, 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 our best. Sir Andrew J. Wiles received the Abel Prize from Crown Prince Haakon of Norway. Fermat’s Last Theorem: A Historical Account We have to start at the very beginning, with a cita- this margin is too narrow to contain.” This remark was tion in Latin: “…nullam in infinitum ultra quadratum written in the year 1637 by the French lawyer and amateur potestatem in duos eiusdem nominis fas est dividere,” mathematician Pierre de Fermat [1601–1665] in the mar- which means: “it is impossible to separate any power gin of his copy of Diophantus’ Arithmetica. He certainly higher than the second into two like powers.” In mod- did not expect that it would keep mathematicians, profes- ern mathematical jargon, this can be written: “The sionals, and amateurs alike busy for centuries trying to equation 푥푛 + 푦푛 = 푧푛 has no solution in natural numbers unearth the proof. for 푛 greater that two.” And then it continues: “cuius rei Could you please give us a short account of some of demonstrationem mirabilem sane detexi. Hanc marginis the attempts towards proving Fermat’s Last Theorem up exiguitas non caperet,” which means: “I have discovered a until the time you embarked on your successful journey? truly marvellous proof of this, which 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? Wiles: The first serious attempt to solve it was pre- Martin Raussen is professor of mathematics at Aalborg Univer- sumably by Fermat himself. But, unfortunately, we know sity, Denmark. His e-mail address is [email protected]. nothing about it except for what he explained about his Christian Skau is professor of mathematics at the Norwegian proofs in the specific cases of 푛 = 3 and 푛 = 4.1 That is, University of Science and Technology, Trondheim, Norway. His he showed that you can’t have the sum of two cubes be e-mail address is [email protected]. another cube or the sum of two fourth powers being a For permission to reprint this article, please contact: [email protected]. 1Strictly speaking, Euler was the first to spell out a complete proof DOI: http://dx.doi.org/10.1090/noti1484 in the case 푛 = 3.

198 Notices of the AMS Volume 64, Number 3 fourth power. He did this by a beautiful method, which contradict another well-known conjecture called the Mod- we call infinite descent. It was a new method of proof, or ularity Conjecture. This conjecture had been proposed in at least a new way of presenting proofs in arithmetic. He a weak form by Taniyama [1927–1958] and was refined explained this method to his colleagues in letters, and he by Shimura, but the first real evidence for it came from also wrote about it in his famous margin, which was big André Weil [1906–1998], who made it possible to check enough for some of it at least. After the marginal notes this precise form of the Modularity Conjecture in some were published by Fermat’s son after his father’s death, detail. And a lot of evidence was amassed showing that it lay dormant for a while. Then it was picked up by Euler this should certainly be true. So, at that point, mathe- [1707–1783] and others who tried to find this truly mar- maticians could see that: “Yes, Fermat is going to be true. vellous proof. And they failed. It became quite dramatic Moreover, there has to be a proof of it.” What happened in the mid-nineteenth century—various people thought was that the Modularity Conjecture was a problem that they could solve it. There was a discussion concerning mathematics could not just put to one side and go on for this in the French Academy: Lamé [1795–1870] claiming he was just about to prove it, and Cauchy [1789–1857] another five hundred years. It was a roadblock right in the saying he thought he could too, and so on. middle of modern mathematics. It was a very, very central In fact, it transpired that the German mathematician problem. As for Fermat, you could just leave it aside and Kummer [1810–1893] had already written a paper where forget it almost forever. This Modularity Conjecture you he explained that the fundamental problem was what is could not forget. So, at the point when I heard that Ribet known now as the fundamental theorem of arithmetic. In had done this, I knew that this problem could be solved our normal number system, any number can be factorized and I was going to try. in essentially one way into prime factors. Take a number Raussen and Skau : Concerning speculations about Fer- like 12; it is 2 times 2 times 3. There is no other way of mat’s claimed proof, do you think he had the same idea as breaking it up. But in trying to solve the Fermat problem, Lamé, assuming, wrongly as it turned out, that the cyclo- you actually want to use systems of numbers where this tomic integers have unique factorization? uniqueness does not hold. Every attempt that was made Wiles: No, I don’t think so, though the idea might be to solve the Fermat problem had stalled because of this in there somewhere. It is very hard to understand. André failure of unique factorization. Kummer analyzed this in Weil wrote about this. All the other problems Fermat incredible detail. He came up with the most beautiful considered had to do with curves that were of genus results, and the end product was that he could solve zero or genus one. And suddenly he is writing down a it for many, many cases. For example, for 푛 ≤ 100 he curve that has higher genus. How is he going to think solved it for all primes except for 37, 59, and 67. But about it? When I was trying this myself as a teenager, he did not finally solve it. His method was based on the I put myself in Fermat’s frame of mind because there idea that Fermat had introduced—the method of infinite was hardly anything else I could do. I was capable of descent—but in these new number systems. The new number systems he was using spawned understanding his mathematics from the seventeenth algebraic number theory as we see it today. One tries to century but probably not much beyond that. It seemed to solve equations in these new systems of numbers instead me that everything he did came down to something about of solving them with ordinary integers and rational quadratic forms, and I thought that might be a way of numbers. Attempts in the style of Fermat carried on for a trying to think about it. Of course, I never succeeded, but while but somewhat petered out in the twentieth century. there is nothing else that suggests Fermat fell into this No one came up with a fundamentally new idea. In the trap with unique factorization. In fact, from the point of second half of the twentieth century, number theory view of quadratic forms, he understood that sometimes moved on and considered other questions. Fermat’s there was unique factorization and sometimes there was problem was all but forgotten by the professionals. not. So he understood that difference in his own context. Then, in 1985, Ger- I think it is unlikely that that was the mistake. hard Frey, a German Raussen and Skau : In the same book by André Weil mathematician, came up It was a that you referred to, titled Number Theory: An approach with a stunning new through History from Hammurapi to Legendre, it is men- idea where he took a roadblock right in tioned that Fermat looked at the equation of a cube minus hypothetical solution to the middle of a square equal to 2 [푥3 −푦2 = 2] and he showed that it has the Fermat problem and essentially only one solution, namely 푥 = 3 and 푦 = ±5. rewrote it so that it made André Weil speculates that Fermat at the time looked at what is called an elliptic modern the ring 푍[ − 2], which does have unique factorization. curve. And he showed, mathematics. √ or suggested, that this Wiles: Yes, he used unique factorization, but the way he elliptic curve had very did it was in terms of quadratic forms. And I think he also peculiar properties. He conjectured that you can’t really looked at quadratic forms corresponding to 푍[√ − 6] have such an elliptic curve. Building on this a year later, an where there is not unique factorization. So I think he American mathematician, Kenneth Ribet, demonstrated, understood. It was my impression when I thought about using this Frey curve, that any solution of Fermat would it that he understood the difference.

March 2017 Notices of the AMS 199 A Mathematical Education allowed to do number theory in your first year. And you Raussen and Skau : You were apparently already inter- could not really get down to it before your third year. ested in mathematical puzzles as quite a young boy. Have Raussen and Skau : But you were not interested in ge- you any thoughts about where this interest came from? ometry, not as much as in algebra and number theory, Were you influenced by anyone in particular? anyway? Wiles: I just enjoyed mathematics when I was very Wiles: No, I was primarily interested in algebra and young. At the age of ten, I was looking through library number theory. I was happy to learn these other things, shelves devoted to mathematics. I would pull out books but I really was most excited about number theory. My and at one point I pulled out a book of E. T. Bell teachers arranged for me to take extra classes in number [1883–1960] titled The Last Problem, which on its cover theory, but there was not that much to offer. describes the Fermat equation, the Wolfskehl Prize, and At one point, I decided that I should put all the years the romantic history of the problem. I was completely of Latin I had done at school to good use and try to captivated by it. read some of Fermat in the original, but I found that was Raussen and Skau : Were there other things that fasci- actually too hard. Even if you translated the Latin, the way nated you in this book by Eric Temple Bell? they wrote in those days wasn’t in the algebraic symbols Wiles: It is entirely about that one equation, really. And I was used to, so it was quite difficult. it is actually quite wordy. So there is less mathematics in Raussen and Skau : It must have been a relief when you were done and came to Cambridge to start studying some sense than you might think. I think it was more the number theory for real, with John Coates as your supervi- equation. Then, when I found this equation, I looked for sor. other elementary books on number theory and learned Wiles: That’s right. I had a year, a preliminary year, about congruences and solved congruences and so on, in which I just studied a range of subjects and then I and looked at other things that Fermat did. could do a special paper. John Coates was not yet at Raussen and Skau : You did this work besides your Cambridge, but I think he helped me, maybe over the ordinary schoolwork? summer. Anyway, that summer I met him and started Wiles: Yes, I don’t think my schoolwork was too taxing working with him right away, and that was just wonderful. from that point of view. The transition from undergraduate work, where you were Raussen and Skau : Was it already clear to you at that just reading and studying, to research—that was the real time that you had an extraordinary mathematical talent? break for me. It was just wonderful. Wiles: I certainly had a mathematical aptitude and obviously loved to do mathematics, but I don’t think I felt Elliptic Curves that I was unique. In fact, I don’t believe I was unique in Raussen and Skau : We assume it was John Coates who the school I attended. There were others who had just as initiated your work on elliptic curves and Iwasawa theory? strong a claim to be future mathematicians, and some of Wiles: Absolutely. He had some wonderful ideas and them have become mathematicians, too. was generous to share them with me. Raussen and Skau : Did you already plan to study math- Raussen and Skau : Did you tell John Coates that you ematics and to embark on a mathematical career at that were interested in the Fermat problem? age? Wiles: Perhaps I did. I don’t remember. It is really true Wiles: No, I don’t think I really understood you could that there hadn’t been any new ideas since the nineteenth spend your life doing mathematics. I think that only came century. People were trying to refine the old methods and, later. But I certainly wanted to study it as long as I could. yes, there were refinements. But it didn’t look like these I’m sure that as far as my horizon extended, it involved refinements and the solution were going to converge. It mathematics. was just too hard that way. Raussen and Skau : You started to study mathemat- Raussen and Skau : At the time you started to work ics as a student at Oxford in 1971. Can you tell us a little with John Coates, you had no idea that these elliptic curves bit about how that worked out? Were there any particu- were going to be crucial for the solution of Fermat’s Last lar teachers or any particular areas that were particularly Theorem? important for you? Wiles: No, it’s a wonderful coincidence. The strange Wiles: Before I went to college (actually in high school), thing is that, in a way, the two things that are most one of my teachers had a PhD in number theory. He prominent in Fermat that we remember today are his gave me a copy of Hardy and Wright’s An Introduction work on elliptic curves and his famous last theorem. For to the Theory of Numbers, and I also found a copy of example, this equation you mentioned, 푦2 + 2 = 푥3, is an Davenport’s The Higher Arithmetic. And these two books elliptic curve. And the two strands came together in the I found very, very inspiring in terms of number theory. proof. Raussen and Skau : So you were on track before you Raussen and Skau : Could you explain what an elliptic started studying? curve is and why elliptic curves are of interest in number Wiles: Yes, I was on track before. In fact, to some theory? extent, I felt college was a distraction because I had to Wiles: For a number theorist, the life of elliptic curves do all these other things: applied maths, logic, and so on, started with Fermat as equations of the form 푦2 equals and I just wanted to do number theory. You were not a cubic polynomial in 푥 with rational coefficients. Then,

200 Notices of the AMS Volume 64, Number 3 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 푛 = 3 of Fermat’s Last Theorem, the equation being, in fact, an elliptic curve in disguise. Sometimes you 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 nineteenth 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 Interviewed in Oslo in May, Wiles told Martin understood in terms of doubly periodic functions in the Raussen and Christian Skau that he set out to prove early nineteenth century. But that is what underlies the the Modularity Conjecture with no idea from what complex solutions, solutions to the equation in complex branch of mathematics the answer would come. numbers. The solutions to the equation in rational numbers were studied by Poincaré [1854–1912]. What’s now known as Raussen and Skau : Was this the first general result the Mordell–Weil theorem was proved by Mordell [1888– concerning the Birch and Swinnerton-Dyer Conjecture? 1972] and then Weil in the 1920s, answering a question of Wiles: It was the first one that treated a family of cases Poincaré. In our setting, it says that the 퐾-rational points rather than individual cases. There was a lot of numerical on an elliptic curve over a number field 퐾, in particular for data for individual cases, but this was the first infinite 퐾 equal to the rationals, form a finitely generated abelian family of cases. group. That is, from Fermat’s language, you can start with Raussen and Skau : This was over the rational num- a finite number of solutions and, using those, generate bers? all the solutions by what he called the chord-and-tangent Wiles: Yes. process. Raussen and Skau : We should mention that the Birch By now you know the structure; it is a very beautiful and Swinnerton-Dyer Conjecture is one of the Clay Mil- algebraic structure, the structure of a group, but that lennium Prize Problems, which would earn a person who does not actually help you find the solutions. So, no one solves it one million dollars. really had any general methods for finding the solutions Wiles: That’s right. I think it’s appealing, partly be- until the conjectures of the 1960s, which emerged from cause it has its roots in Fermat’s work, just like the the Birch and Swinnerton-Dyer Conjecture. There are two Fermat problem. It is another “elementary-to-state” prob- aspects to it; one is somewhat analytic, and one is in terms lem concerned with equations—in this case of very low of what is called the Tate–Shafarevich group. Basically, degree—which we can’t master and which Fermat initiated. the Tate–Shafarevich group gives you the obstruction to I think it is a very appealing problem. an algorithm for finding the solutions. And the Birch Raussen and Skau : Do you think it is within reach? In and Swinnerton-Dyer Conjecture tells you that there is other words, do we have the necessary tools for somebody actually an analytic method for analyzing this so-called daring enough to attack it and succeed? Or do we have to Tate–Shafarevich group. If you combine all this together, wait for another three hundred years to see it solved? ultimately it should give you an algorithm for finding the Wiles: I don’t suppose it will take three hundred years, solutions. but I don’t think it is the easiest of the Millennium Problems. I think we are still lacking something. Whether Birch and Swinnerton-Dyer, Tate–Shafarevich, the tools are all here now, I am not sure. They may be. Selmer… There are always these speculations with these really Raussen and Skau : You worked on the Birch and difficult problems; it may be that the tools simply aren’t Swinnerton-Dyer Conjecture when you were a graduate there. I don’t believe that anyone in the nineteenth century student together with John Coates? could have solved Fermat’s Last Theorem, certainly not in Wiles: Yes, that is exactly what he proposed working the way it was eventually solved. There was just too big on. We got the first result in certain special families of a gap in mathematical history. You had to wait another elliptic curves on this analytic link between the solutions hundred years for the right pieces to be in place. You can and what is called the 퐿-function of the elliptic curve. never be quite sure about these problems, whether they Raussen and Skau : These were curves admitting com- are accessible to your time. That is really what makes plex multiplication? them so challenging; if you had the intuition for what can Wiles: Exactly; these were the elliptic curves with be done now and what can’t be done now, you would be complex multiplication. a long way towards a solution!

March 2017 Notices of the AMS 201 Raussen and Skau : You mentioned the Tate–Shafarevich The Modularity Conjecture group and in that connection the Selmer group appears. Raussen and Skau : What you proved in the end was a Selmer [1920–2006] was a Norwegian mathematician, special case of what is now called the Modularity Conjec- and it was Cassels [1922–2015] who was responsible for ture. In order to explain this, one has to start with modular naming this group the Selmer group. Could you say a few forms and how modular forms can be put in relation with words about the Selmer group and how it is related to the elliptic curves. Could you give us some explanations? Tate–Shafarevich group, even if it’s a little technical? Wiles: Yes; we have described an elliptic curve (over 2 3 Wiles: It is technical, but I can probably explain the the rationals) as an equation 푦 = 푥 +푎푥+푏, where the 푎 basic idea of what the Selmer group is. What you are and 푏 are assumed to be rational numbers. (There is also a condition that the discriminant should not vanish.) As I trying to do is to find the rational solutions on an said, at the beginning of the nineteenth century you could elliptic curve. The method is to take the rational points describe the complex solutions to this equation. You could on the elliptic curve—suppose you have got some—and describe these very nicely in terms of the Weierstrass ℘- you generate field extensions from these. So when I say function, in terms of a special elliptic function. But what generate extensions, I mean that you can take roots of we want is actually a completely different uniformization those points on the elliptic curve. Just like taking the 푛th of these elliptic curves which captures the fact that the 푎 root of 5 or the cube root of 2. You can do the same and 푏 are rational numbers. It is a parametrization just thing on an elliptic curve; you can take the 푛th root of a for the rational elliptic curves. And because it captures point. These are all points which added to themselves 푛 the fact that it is defined over the rationals, it gives you times give you the point you started with. They generate a much better hold on solutions over the rationals than certain extensions of the number field you started with, the elliptic functions do. The latter really only sees the in our case the rational number field Q. complex structure. You can put a lot of restrictions on those exten- And the place it comes from are modular forms or sions. And the Selmer group is basically the smallest modular curves. To describe modular functions first: we set of extensions you can get putting on all the obvious are used to functions which satisfy the relation of being restrictions. invariant under translation. Every time we write down Let me summarize this. You’ve got the group of points. a Fourier series, we have a function which is invariant under translation. Modular functions are ones which are They generate some extensions; that’s too big—you don’t invariant under the action of a much bigger group, usually want all extensions. You cut that down as much as you can a subgroup of 푆퐿 (ℤ). So, you would ask for a function 푓(푧) using local criteria, using 푝-adic numbers; that’s called 2 in one complex variable, usually on the upper-half-plane, the Selmer group. And the essential difference between which satisfies 푓(푧) is the same as 푓((푎푧 + 푏)/(푐푧 + 푑)) the group generated by the points and the Selmer group (or, more generally, is that times a power of 푐푧 + 푑). is the Tate–Shafarevich group. So the Tate–Shafarevich These are called group gives you the error term, if you like, in trying to modular functions, get at the points via the Selmer group. Modular functions and they were ex- Raussen and Skau : Selmer’s paper, which Cassels tensively studied in refers to, studied the Diophantine equation 3푥3 + 4푦3 + hold the key to the the nineteenth century. 5푧3 = 0 and similar ones. Selmer showed that it has just Surprisingly, they hold a trivial solution in the integers, while modulo 푛 it has arithmetic of the key to the arith- nontrivial solutions for all 푛. In particular, this curve has metic of elliptic curves. no rational points. Why did Cassels invoke Selmer’s name elliptic curves. Perhaps the simplest in naming the group? way to describe it is Wiles: Yes, there are quite subtle relationships between that because we have an action of 푆퐿2(ℤ) on the upper- these. What happens is you are actually looking at one half-plane 퐻—by the action z goes to (푎푧+푏)/(푐푧+푑)—we elliptic curve, which in this case would be 푥3 +푦3 +60푧3 = can look at the quotient 퐻 modulo this action. You can 0. That is an elliptic curve, in disguise, if you like, and the then give the quotient the structure of a curve. In fact, it naturally gets the structure of a curve over the rational Tate–Shafarevich group involves looking at other ones numbers. If you take a subgroup of 푆퐿 (ℤ), or more like it, for example, 3푥3 + 4푦3 + 5푧3 = 0, which is a genus 2 precisely what is called a congruence subgroup, defined one curve but which has no rational points. Its Jacobian by the 푐 value being divisible by 푁, then you call the curve is the original elliptic curve 푥3 + 푦3 + 60푧3 = 0. One way a modular curve of level 푁. The Modularity Conjecture of describing the Tate–Shafarevich group is in terms of asserts that every elliptic curve over the rationals is actu- these curves that have genus one but don’t have rational ally a quotient of one of these modular curves for some points. And by assembling these together you can make integer 푁. It gives you a uniformization of elliptic curves the Tate–Shafarevich group, and that is reflected in the by these other entities, these modular curves. On the face Selmer group. It is too intricate to explain in words, but of it, it might seem we are losing because this is a high it is another point of view. I gave it in more arithmetic genus curve—it is more complicated. But it actually has a terminology in terms of extensions. The more geometric lot more structure because it is a moduli space. terminology was in terms of these twisted forms. Raussen and Skau : And that is a very powerful tool?

202 Notices of the AMS Volume 64, Number 3 Wiles: That is a very powerful tool, yes. You have But then, when your result is scrutinized by six referees function theory, you have deformation theory, geometric for a highly prestigious journal, it turns out that there is methods, etc. You have a lot of tools to study it. a subtle gap in one of your arguments and you are sent Raussen and Skau : Taniyama, the young Japanese back to the drawing board. After a while, you send for mathematician who first conjectured or suggested these your former student, Richard Taylor, to come to Princeton connections; his conjecture was more vague, right? to help you in your efforts. It takes a further ten months Wiles: His conjecture was more vague. He didn’t pin it of hard and frustrating work; we think we do not exagger- down to a function invariant under the modular group. I’ve ate by calling it a heroic effort under enormous pressure. forgotten exactly what he conjectured; it was invariant Then, in a sudden flash of insight, you realize that you can under some kind of group, but I forget exactly which combine some of your previous attempts with new results group he was predicting. But it was not as precise as the to circumvent the problem that had caused the gap. This congruence subgroups of the modular group. I think it turns out to be what you need in order to get the part of was originally written in Japanese, so it was not circulated the Modularity Conjecture that implies Fermat’s Last The- as widely as it might have been. I believe it was part of orem. What a relief that must have been! Would you like to notes compiled after a conference in Japan. give a few comments on this dramatic story? Raussen and Skau : It was an incredibly audacious con- Wiles: With regard jecture at that time, wasn’t it? to my own work, when Wiles: Apparently, yes. I became a profes- It is irresponsible Raussen and Skau : But then it gradually caught the at- sional mathematician tention of other mathematicians. You told us already about working with Coates, I to work on one Gerhard Frey, who came up with a conjecture relating realized I really had to Fermat’s Last Theorem with the Modularity Conjecture. stop working on Fer- problem to the Wiles: That’s right. Gerhard Frey showed that if you mat because it was take a solution to the Fermat problem, say 푎푝 + 푏푝 = 푐푝, time-consuming and I exclusion of and you create the elliptic curve 푦2 = 푥(푥 − 푎푝)(푥 + 푏푝), could see that in the everything else. then the discriminant of that curve would end up being last hundred years al- a perfect 푝th power. And if you think about what that most nothing had been means assuming the Modularity Conjecture—you have to done. And I saw others, even very distinguished mathe- assume something a bit stronger as well (the so-called maticians, had come to grief on it. When Frey came out epsilon conjecture of Serre)—then it forces this elliptic with this result, I was a bit skeptical that the Serre part curve to have the level 푁 that I spoke about to be equal of the conjecture was going to be true, but when Ribet to one, and hence the associated congruence subgroup is proved it, then, okay, this was it! equal to 푆퐿2(ℤ). But 퐻 modulo 푆퐿2(ℤ) is a curve of genus And it was a long, hard struggle. In some sense, it is zero. It has no elliptic curve quotient, so it wasn’t there irresponsible to work on one problem to the exclusion of after all, and hence there can’t be a solution to the Fermat everything else, but this is the way I tend to do things. problem. Whereas Fermat is very narrow (I mean, it is just this one equation, whose solution may or may not help with The Quest for a Proof anything else), the setting of the Modular Conjecture was Raussen and Skau : That was the point of departure for one of the big problems in number theory. It was a great your own work, with crucial further ingredients due to thing to work on anyway, so it was just a tremendous Serre and Ribet making this connection clear. May we opportunity. briefly summarize the story that then followed? It has When you are working on something like this, it takes been told by you many times, and it is the focus of a BBC many years to really build up the intuition to see the kinds documentary. of things you need and the kinds of things a solution will You had moved to the United States, first to Harvard, depend on. It’s something like discarding everything you then to Princeton University, becoming a professor there. can’t use and won’t work till your mind is so focused that When you heard of Ribet’s result, you devoted all your even making a mistake, you’ve seen enough that you’ll research time to proving the Modularity Conjecture for find another way to the end. semistable elliptic curves over the rationals. This work Funnily enough, concerning the mistake in the argu- went on for seven years of really hard work in isolation. ment that I originally gave, people have worked on that At the same time you were working as a professor in aspect of the argument and quite recently they have Princeton and you were raising small children. actually shown that you can produce arguments very like A proof seems to be accomplished in 1993, and the de- that. In fact, in every neighboring case, arguments similar velopment culminates in a series of three talks at the Isaac to the original method seem to work, but there is this Newton Institute in Cambridge back in England, announc- unique case that it doesn’t seem to work for, and there ing your proof of Fermat’s Last Theorem. You are cele- is not yet any real explanation for it. So the same kind of brated by your mathematical peers. Even the world press argument I was trying to use, using Euler systems and so takes an interest in your results, which happens very rarely on, has been made to work in every surrounding case, but for mathematical results. not the one I needed for Fermat. It’s really extraordinary.

March 2017 Notices of the AMS 203 Raussen and Skau : You once likened this quest for the Parallels to Abel’s Work proof of the Modularity Theorem to a journey through a Raussen and Skau : We want to read you a quotation: “The dark, unexplored mansion. Could you elaborate? ramparts are raised all around but, enclosed in its last re- Wiles: I started off really in the dark. I had no prior doubt, the problem defends itself desperately. Who will be insights of how the Modularity Conjecture might work or the fortunate genius who will lead the assault upon it or how you might approach it. One of the troubles with this force it to capitulate?” problem—it’s a little like the Riemann Hypothesis but Wiles: It must have been E. T. Bell, I suppose. Is it? perhaps even more so—was that you didn’t even know Raussen and Skau : No, it’s not. It is actually a what branch of mathematics the answer would be coming quote from the book Histoire des Mathématiques by from. Jean-Étienne Montucla [1725–1799], written in the late To start with, there are three ways of formulating eighteenth century. It is really the first book ever written the problem: one is geometric, one is arithmetic, and on the history of mathematics. The quotation refers to one is analytic. And there were analysts—I would not the solvability or unsolvability of the quintic equation understand their techniques at all well—who were trying by radicals. As you know, Abel [1802–1829] proved the unsolvability of the general quintic equation when he was to make headway on this problem. twenty-one years old. He worked in complete isolation, I think I was a little lucky because my natural instinct mathematically speaking, here in Oslo. Abel was obsessed, was with the arithmetic approach and I went straight or at least extremely attracted, to this problem. He also for the arithmetic route, but I could have been wrong. got a false start. He thought he could prove that one could The only previously known cases where the Modularity actually solve the quintic by radicals. Then he discovered Conjecture was known to hold were the cases of complex his mistake and he finally found the unsolvability proof. multiplication, and that proof is analytic, completely Well, this problem was, at that time, almost three hundred analytic. years old and very famous. If we move fast-forward two Partly out of necessity, I suppose, and partly because hundred years, the same quotation could be used about that’s what I knew, I went straight for an arithmetic the Fermat problem, which was around three hundred approach. I found it very useful to think about it in a fifty years old when you solved it. It is a very parallel story way that I had been studying in Iwasawa theory. With in many ways. Do you have any comments? John Coates, I had applied Iwasawa theory to elliptic Wiles: Yes. In some sense, I do feel that Abel, and then curves. When I went to Harvard, I learned about Barry Galois [1811–1832], were marking a transition in algebra Mazur’s work, where he had been studying the geometry from these equations, which were solvable in some very of modular curves using a lot of the modern machinery. simple way, to equations which can’t be solved by radicals. There were certain ideas and techniques I could draw on But this is an algebraic break that came with the quintic. from that. I realized after a while, I could actually use that In some ways, the whole trend in number theory now is to make a beginning—to find some kind of entry into the the transition from basically abelian and possibly solvable problem. extensions to insolvable extensions. How do we do the Raussen and Skau : Before you started on the Modu- arithmetic of insolvable extensions? larity Conjecture, you published a joint paper with Barry I believe the Modularity Conjecture was solved because Mazur, proving the main theorem of Iwasawa theory over we had moved on from this original abelian situation the rationals. Can you please tell us what Iwasawa theory to a nonabelian situation, and we were developing tools, is all about? modularity and so on which are fundamentally nonabelian Wiles: Iwasawa theory grew out of the work of Kummer tools. (I should say, though, that the proof got away mostly on cyclotomic fields and his approach to Fermat’s Last with using the solvable situation, not because it was more natural but because we have not solved the relevant Theorem. He studied the arithmetic, and in particular the problems in the general nonsolvable case.) ideal class groups, of prime cyclotomic fields. Iwasawa’s It is the same transition in number theory that he was idea was to consider the tower of cyclotomic fields making in algebra, which provides the tools for solving obtained by taking all 푝-power roots of unity at once. this equation. So I think it is very parallel. The main theorem of Iwasawa theory proves a relation Raussen and Skau : There is an ironic twist with Abel between the action of a generator of the Galois group on and the Fermat problem. When he was twenty-one years the 푝-primary class groups and the 푝-adic 퐿-functions. It is old, Abel came to Copenhagen to visit Professor Degen analogous to the construction used in the study of curves [1766–1825], who was the leading mathematician in Scan- over finite fields where the characteristic polynomial of dinavia at that time. Abel wrote a letter to his mentor in Frobenius is related to the zeta function. Oslo, Holmboe [1795–1850], stating three results about the Raussen and Skau : And these tools turned out to be Fermat equation without giving any proofs—one of them useful when you started to work on the Modularity Conjec- is not easy to prove, actually. This, of course, is just a cu- ture? riosity today. Wiles: They did; they gave me a starting point. It wasn’t But in the same letter, he gives vent to his frustration, obvious at the time, but when I thought about it for a intimating that he can’t understand why he gets an equa- while, I realized that there might be a way to start from tion of degree 푛2 and not 푛 when dividing the lemniscate there. arc in 푛 equal pieces. It was only after returning to Oslo

204 Notices of the AMS Volume 64, Number 3 that he discovered the double periodicity of the lemniscate of that? This is something the mind does for you. It is the integral and also of the general elliptic integral of the first flash of insight. kind. I remember (this is a trivial example in a non- If one thinks about it, what he did on the Fermat equa- mathematical setting) someone once showed me some tion turned out to be just a curiosity. But what he achieved script—it was some Gothic script—and I couldn’t make on elliptic functions, and implicitly on elliptic curves, head nor tail of it. I was trying to understand a few letters, turned out later to be a relevant tool for solving it. Of and I gave up. Then I came back half an hour later and I course, Abel had no idea that this would have anything to could read the whole thing. The mind somehow does this do with arithmetic. So this story tells us that mathematics for you and we don’t quite know how, but we do know sometimes develops in mysterious ways. what we have to do to set up the conditions where it will Wiles: It certainly does, yes. happen. Raussen and Skau : This is reminiscent of a story about Work Styles Abel. While in Berlin, he shared an apartment with some Raussen and Skau : May we ask for some comments about Norwegian friends who were not mathematicians. One of work styles of mathematicians in general and also about his friends said that Abel typically woke up during the your own? Freeman Dyson, a famous physicist and mathe- night, lit a candle, and wrote down ideas that he woke up matician at IAS in Princeton, said in his Einstein Lecture in with. Apparently his mind was working while asleep. 2008: “Some mathematicians are birds, others are frogs. Wiles: Yes, I do that, except I don’t feel the need to Birds fly high in the air and survey broad vistas of math- write them down when I wake up with it because I know ematics out to the horizon. They delight in concepts that I will not forget it. But if I have an idea when I am about unify our thinking and bring together diverse problems to go to sleep, I am terrified that I will not wake up with from different parts of the landscape. Frogs live in the mud that idea, so then I have to write it down. below and see only the flowers that grow nearby. They Raussen and Skau : Are you thinking in terms of for- delight in the details of particular objects and they solve mulas or in terms of geometric pictures or what? problems one at a time.” Wiles: It is not really Freeman Dyson didn’t say that birds were better than geometric. I think it is frogs or the other way around. He considered himself a patterns and I think I often feel that frog rather than a bird. it is just parallels be- When we are looking at your work, for us it seems rather tween situations I have doing difficult to decide where to place you in his classification seen elsewhere and the scheme: among the birds (those who create theories) or one I am facing now. In mathematics is among the frogs (those who solve problems). What is your a perfect world, what own perception? is it all pointing to? like being a Wiles: Well, I don’t feel like either. I’m certainly not a What are the ingredi- squirrel. bird—unifying different fields. I think of frogs as jumping ents that ought to go a lot. I think I’m very, very focused. I don’t know what into this proof? What the animal analogy is, but I think I’m not a frog in the am I not using that I still have in my pocket? Sometimes sense of enjoying the nearby landscape. I’m very, very it is just desperation. I assemble every piece of evidence concentrated on the problem I happen to work on and I I have and that’s all I’ve got. I have got to work with that am very selective. And I find it very hard to even take my and there is nothing else. mind off it enough to look at any of the flowers around, I often feel that doing mathematics is like being a so I don’t think that either of the descriptions quite fit. squirrel and there are some nuts at the top of a very tall Raussen and Skau : Based on your own experience, tree. But there are several trees and you don’t know which could you describe the interplay between hard, concen- one. What you do is that you run up one and you think, no, trated, and persevering work on the one side and, on the it does not look good on this one, and you go down and other side, these sudden flashes of insight that seemingly up another one, and you spend your whole life just going come out of nowhere, often appearing in a more relaxed up and down these trees. But you’ve only got up to thirty setting. Your mind must have worked unconsciously on feet. Now, if someone told you the rest of the trees—it’s the problem at hand, right? not in them, you have only one tree left—then you would Wiles: I think what you do is that you get to a situation just keep going until you found it. In some sense, it is where you know a theory so well, and maybe even more ruling out the wrong things—that is really crucial. And than one theory, so that you have seen every angle and if you just believe in your intuition and your intuition is tried a lot of different routes. correct and you stick with your one tree, then you will There is this tremendous amount of work in the find it. preparatory stage where you have to understand all the details and maybe some examples—that is your essential Problems in Mathematics launch pad. When you have developed all this, you let Raussen and Skau : Felix Klein [1849–1925] once said: the mind relax and then at some point—maybe when you “Mathematics develops as old results are being understood go away and do something else for a little bit—you come and illuminated by new methods and insights. Proportion- back and suddenly it is all clear. Why did you not think ally with a better and deeper understanding new problems

March 2017 Notices of the AMS 205 naturally arise.” And David Hilbert [1862–1943] stressed nowadays often depends on mathematical modelling that “problems are the lifeblood of mathematics.” Do you and optimization methods. Science and industry propose agree? challenges to the mathematical world. Wiles: I certainly agree with Hilbert, yes. Good problems In a sense, mathematics has become more applied than are the lifeblood of mathematics. I think you can see this it ever was. One may ask whether this is a problem for clearly in number theory in the second half of the pure mathematics. It appears that pure mathematics some- last century. For me personally, there is obviously the times is put to the sidelines, at least from the point of view Modularity Conjecture but also the whole Langlands of the funding agencies. Do you perceive this as a serious program and the Birch and Swinnerton-Dyer Conjecture. problem? These problems give you a very clear focus on what we Wiles: Well, I think in comparison with the past, one should try to achieve. We also have the Weil Conjectures feels that mathematicians two, three hundred years ago on curves and varieties over finite fields and the Mordell were able to handle a much broader spectrum of mathe- Conjecture and so on. matics, and a lot more of it touched applied mathematics These problems somehow concentrate the mind and than a typical pure mathematician would do nowadays. also simplify our goals in mathematics. Otherwise, we can On the other hand, that might be because we only remem- get very, very spread out and not sure what’s of value and ber the very best and most versatile mathematicians from what’s not of value. the past. Raussen and Skau : Do we have as good problems today I think it is always going to be a problem if funding as when Hilbert formulated his twenty-three problems in agencies are short-sighted. If they want to see a result 1900? in three years, then it is not going to work. It is hard to Wiles: I think so, yes. imagine a pure development and then the application all Raussen and Skau : Which one do you think is the most happening within three to five years. It is probably not important problem today? And how does the Langlands going to happen. program fit in? On the other hand, I don’t believe you can have a happily Wiles: Well, I think the Langlands program is the functioning applied maths world without the pure maths broadest spectrum of problems related to my field. I to back it up, providing the future and keeping them on think that the Riemann Hypothesis is the single greatest the straight and narrow. So it would be very foolish not to problem from the areas I understand. It is sometimes invest in pure mathematics. It is a bit like only investing hard to say exactly why that is, but I do believe that in energy resources that you can see now. You have to solving it would actually help solve some of these other invest in the future; you have to invest in fusion power problems. And then, of course, I have a very personal or solar power or these other things. You don’t just use attachment to the Birch and Swinnerton-Dyer Conjecture. up what is there and then start worrying about it when it Raussen and Skau : Intuition can lead us astray some- is gone. It is the same with mathematics; you can’t just times. For example, Hilbert thought that the Riemann Hy- use up the pure mathematics we have now and then start pothesis would be solved in his lifetime. There was another worrying about it when you need a pure result to generate problem on his list, the seventh, that he never thought your applications. would be solved in his lifetime, but which was solved by Gel- fond [1906–1968] in 1934. So our intuition can be wrong. Mathematical Awards Wiles: That is right. I’m not surprised that Hilbert Raussen and Skau : You have already won a lot of prizes felt that way. The Riemann Hypothesis has such a clear as a result of your achievements, culminating in proving statement, and we have the analogue in the function field Fermat’s Last Theorem. You have won the Rolf Schock setting. We understand why it is true there and we feel we Prize, given by the Swedish Academy; the Ostrowski Prize, ought to be able to translate it. Of course, many people which was given to you in Denmark; the Fermat Prize in have tried and failed. But I would still expect it to be France; the Wolf Prize in Israel; the Shaw Prize in Hong solved before the Birch and Swinnerton-Dyer Conjecture. Kong (the prize that has been named the Nobel Prize of the East), and the list goes on, culminating with the Abel Investing in Mathematics Prize tomorrow. May we ask you whether you enjoy these Raussen and Skau : Let’s hope we’ll find out in our life- awards and the accompanying celebrations? times! Wiles: I certainly love them, I have to say. I think they Classical mathematics has, roughly speaking, two are a celebration of mathematics. I think with something sources: one of them coming from the physical sciences like Fermat, it is something people are happy to see in and the other one from—let’s for simplicity call it num- their lifetime. I would obviously be very happy to see the ber theoretical speculations, with number theory not Riemann Hypothesis solved. It is just exciting to see how associated to applications. it finally gets resolved and just to understand the end That has changed. For example, your own field of ellip- of the story—because a lot of these stories we won’t live tic curves has been applied to cryptography and security. to see the end of. Each time we do see the end of such People are making money with elliptic curves nowadays! a story, it is something we will naturally celebrate. For On the other hand, many sciences apart from physics me, I learned about the Fermat problem from this book really take advantage and profit from mathematical of E. T. Bell and about the Wolfskehl Prize attached to thinking and mathematical results. Progress in industry it. The Wolfskehl Prize was still there—only just, I may

206 Notices of the AMS Volume 64, Number 3 say; I only had a few years left before the deadline for it I like to read and I like various kinds of literature: expired. novels, some biographies—it is fairly balanced. I don’t Raussen and Skau : This gives us the lead to talk a little have any other focused obsessions. When I was in school, about that prize. The Wolfskehl Prize was founded in 1906 I played on chess teams and bridge teams, but when by Paul Wolfskehl [1856–1906], who was a German physi- I started to do serious mathematics, I completely lost cian with an interest in mathematics. He bequeathed one interest in those. hundred thousand Reichmarks (equivalent to more than Raussen and Skau : What about music; are you fond of one million dollars in today’s money) to the first person to music? prove Fermat’s Last Theorem. The prize was, according to Wiles: I go and listen to concerts, but I am not the testament, valid until 13 September 2007 and you re- myself actively playing anything. I enjoy listening to ceived it in 1997. By then, due in part to the hyperinflation music—classical, preferably. Germany suffered after World War I, the prize money had Raussen and Skau : Are you interested in other sciences dwindled a lot. apart from mathematics? Wiles: For me, the amount of money was unimportant. Wiles: I would say somewhat. These are things I It was the sentimental feeling attached to the Wolfskehl do to relax, so I don’t like them to be too close to Prize that was important for me. mathematics. If it is something like animal behavior or astrophysics or something from a qualitative point of Graduate Students view—I certainly enjoy learning about those—likewise Raussen and Skau : You have had altogether twenty-one about what machines are capable of, and many other PhD students and you have attracted very gifted students. kinds of popular science, but I’m not going to spend my Some of them are really outstanding. One of them, Man- time learning the details of string theory. I’m too focused jul Bhargava, won the Fields Medal in 2014. It must be a to be willing to do that. Not that I would not be interested, pleasure to be advisor to such students. but this is my choice. Wiles: Yes, I don’t want to take too much credit for Raussen and Skau : We would like to thank you very it. In the case of Manjul, I suggested a problem to him, much for this wonderful interview, first of all on behalf but after that I had nothing much more to do. He was of the two of us but also on behalf of the Norwegian, the coming up with these absolutely marvelous discoveries. Danish, and the European Mathematical Societies. Thank In some sense, you get more credit if you have very gifted you so much! students, but the truth is that very gifted students don’t Wiles: Thank you very much! really require that much help. Raussen and Skau : What is the typical way for you of interacting with graduate students? Photo Credits Wiles: Well, I think the hardest thing to learn as a Photo of Sir Andrew J. Wiles with Crown Prince Haakon is graduate student is that afterwards you need to carry on courtesy of Audun Braastad. with the rest of your professional life; it’s hard to pick Photo of Sir Andrew J. Wiles with Martin Raussen, and problems. And if you just assign a problem and they do it, Christian Skau is courtesy of Erik F. Baardsen, DNVA. 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 mathe- matics to look at, having not quite focused on the problem. Then, as they start working and become experts, they can Sidebar 1. Abel Prize Winners see a better way of pinning down what the right question is. And then they are part of the process of choosing 2016: Andrew Wiles the problem. I think that is a much better investment for 2015: John Forbes Nash Jr. and Louis Nirenberg their future. It doesn’t always work out that way, and 2014: Yakov Sinai sometimes the problem you give them turns out to be the 2013: Pierre Deligne right thing. But usually it is not that way and usually it’s 2012: Endre Szemerédi a process to find the right problem. 2011: John Milnor 2010: John Tate Hobbies and Interests 2009: Mikhail Leonidovich Gromov Raussen and Skau : We always end the Abel interviews by 2008: John G. Thompson and Jacques Tits asking the laureate what they enjoy doing when they are 2007: S. R. Srinivasa Varadhan not working with mathematics. What are your hobbies and interests outside mathematics? 2006: Lennart Carleson Wiles: Well, it varies at different times. When I was 2005: Peter Lax doing Fermat and being a father with young children, that 2004: Michael Atiyah and Isadore Singer combination was all-consuming. 2003: Jean-Pierre Serre

March 2017 Notices of the AMS 207 Sidebar 2. Notices articles on Wiles

July/August 1993: Wiles Proves Taniyama’s Conjecture; Fermat’s Last Theorem Follows, by Kenneth A. Ribet, math.berkeley.edu/~ribet/Articles/notices.pdf October 1993: Fermat Fest Draws a Crowd, by Allyn Jackson October 1994: Another Step Toward Fermat, by Allyn Jackson, www.ams.org/notices/199501/ rubin.pdf July 1995: The Proof of Fermat’s Last Theorem by R. Taylor and A. Wiles, by Gerd Faltings, www. ams.org/notices/199507/faltings.pdf July 1996: Wiles Receives NAS Award in Mathematics, by John Coates, www.ams.org/notices/ 199607/comm-wiles.pdf January 1997: Review of BBC’s Horizon Program, ”Fermat’s Last Theorem,” reviewed by Andrew Granville, www.ams.org/notices/199701/comm-granville.pdf March 1997: Announcement: 1997 Cole Prize, www.ams.org/notices/199703/comm-cole.pdf November 1997: Paul Wolfskehl and the Wolfskehl Prize, by Klaus Barner, www.ams.org/ notices/199710/barner.pdf November 1997: Book Review: Fermat’s Enigma by Simon Singh, reviewed by Allyn Jackson, www.ams.org/notices/199710/comm-fermat.pdf December 1999: Research News: A Proof of the Full Shimura-Taniyama-Weil Conjecture Is Announced, by Henri Darmon, www.ams.org/notices/199911/comm-darmon.pdf December 2001: Theater Review: Fermat’s Last Tango, reviewed by Robert Osserman, www.ams. org/notices/200111/rev-osserman.pdf September 2005: Wiles Receives 2005 Shaw Prize, by Allyn Jackson, www.ams.org/notices/ 200508/comm-shaw.pdf June/July 2016: Sir Andrew J. Wiles Awarded Abel Prize, www.ams.org/publications/ journals/notices/201606/rnoti-p608.pdf

208 Notices of the AMS Volume 64, Number 3 Andrew Wiles’s Marvelous Proof Henri Darmon

ermat famously claimed to have discovered “a truly marvelous proof” of his Last Theorem, which the margin of his copy of Diophantus’s Arithmetica was too narrow to contain. While this proof (if it ever existed) is lost to posterity, AndrewF Wiles’s marvelous proof has been public for over two decades and has now earned him the Abel Prize. According to the prize citation, Wiles merits this recogni- tion “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.” Few can remain insensitive to the allure of Fermat’s Last Theorem, a riddle with roots in the mathematics of ancient Greece, simple enough to be understood It is also a centerpiece and appreciated Wiles giving his first lecture in Princeton about his by a novice of the “Langlands approach to proving the Modularity Conjecture in (like the ten- early 1994. year-old Andrew program,” the Wiles browsing of Theorem 2 of Karl Rubin’s contribution in this volume). the shelves of imposing, ambitious It is also a centerpiece of the “Langlands program,” the his local pub- imposing, ambitious edifice of results and conjectures lic library), yet edifice of results and which has come to dominate the number theorist’s view eluding the con- conjectures which of the world. This program has been described as a “grand certed efforts of unified theory” of mathematics. Taking a Norwegian per- the most brilliant has come to dominate spective, it connects the objects that occur in the works minds for well of Niels Hendrik Abel, such as elliptic curves and their over three cen- the number theorist’s associated Abelian integrals and Galois representations, turies, becoming with (frequently infinite-dimensional) linear representa- over its long his- view of the world. tions of the continuous transformation groups whose tory the object of study was pioneered by Sophus Lie. This report focuses lucrative awards on the role of Wiles’s theorem and its “marvelous proof” like the Wolfskehl Prize and, more importantly, motivating in the Langlands program in order to justify the closing a cascade of fundamental discoveries: Fermat’s method of phrase in the prize citation: how Wiles’s proof has opened infinite descent, Kummer’s theory of ideals, the ABC con- “a new era in number theory” and continues to have a jecture, Frey’s approach to ternary diophantine equations, profound and lasting impact on mathematics. Serre’s conjecture on mod 푝 Galois representations,…. Our “beginner’s tour” of the Langlands program will Even without its seemingly serendipitous connection only give a partial and undoubtedly biased glimpse of to Fermat’s Last Theorem, Wiles’s modularity theorem the full panorama, reflecting the author’s shortcomings is a fundamental statement about elliptic curves (as evi- as well as the inherent limitations of a treatment aimed denced, for instance, by the key role it plays in the proof at a general readership. We will motivate the Langlands program by starting with a discussion of diophantine Henri Darmon is James McGill Professor of Mathematics at McGill equations: for the purposes of this exposition, they are University and a member of CICMA (Centre Interuniversitaire en equations of the form Calcul Mathématique Algébrique) and CRM (Centre de Recherches Mathématiques). His e-mail address is [email protected]. (1) 풳 ∶ 푃(푥1, … , 푥푛+1) = 0,

This report is a very slightly expanded transcript of the Abel Prize where 푃 is a polynomial in the variables 푥1, … , 푥푛+1 lecture delivered by the author on May 25, 2016, at the University with integer (or sometimes rational) coefficients. One can of Oslo. examine the set, denoted 풳(퐹), of solutions of (1) with For permission to reprint this article, please contact: coordinates in any ring 퐹. As we shall see, the subject [email protected]. draws much of its fascination from the deep and subtle DOI: http://dx.doi.org/10.1090/noti1487 ways in which the behaviours of different solution sets

March 2017 Notices of the AMS 209 can resonate with each other, even if the sets 풳(ℤ) or (2) the reciprocal roots of 푄(푇) and 푅(푇) are complex 풳(ℚ) of integer and rational solutions are foremost in numbers of absolute value 푝푖/2 with 푖 an integer our minds. Examples of diophantine equations include in the interval 0 ≤ 푖 ≤ 2푛. 푑 푑 푑 Fermat’s equation 푥 + 푦 = 푧 , the Brahmagupta–Pell The first statement—the rationality of the zeta function, 2 2 equation 푥 −퐷푦 = 1 with 퐷 > 0, as well as elliptic curve which was proved by Bernard Dwork in the early 1960s— 2 3 equations of the form 푦 = 푥 + 푎푥 + 푏, in which 푎 and 푏 is a key part of the Weil conjectures, whose formulation are rational parameters, the solutions (푥, 푦) with rational in the 1940s unleashed a revolution in arithmetic ge- coordinates being the object of interest in the latter case. ometry, driving the development of étale cohomology It can be instructive to approach a diophantine equation by Grothendieck and his school. The second statement, −푠 by first studying its solutions over simpler rings, such as which asserts that the complex function 휁푝(풳; 푝 ) has the complete fields of real or complex numbers. The set its roots on the real lines ℜ(푠) = 푖/2 with 푖 as above, (2) ℤ/푛ℤ ∶= {0, 1, … , 푛 − 1} is known as the Riemann hypothesis for the zeta func- tions of diophantine equations over finite fields. It was of remainders after division by an integer 푛 ≥ 2, equipped proved by Pierre Deligne in 1974 and is one of the major with its natural laws of addition, subtraction, and mul- achievements for which he was awarded the Abel Prize in tiplication, is another particularly simple collection of 2013. numbers of finite cardinality. If 푛 = 푝 is prime, this ring That the asymptotic behaviour of 푁 can lead to deep is even a field: it comes equipped with an operation of 푝 insights into the behaviour of the associated diophantine division by nonzero elements, just like the more familiar collections of rational, real, or complex numbers. The fact equations is one of the key ideas behind the Birch and Swinnerton-Dyer conjecture. Understanding the patterns that 픽푝 ∶= ℤ/푝ℤ is a field is an algebraic characterisation of the primes that forms the basis for most known ef- satisfied by the function ficient primality tests and factorisation algorithms. One (9) 푝 ↦ 푁푝 or 푝 ↦ 휁푝(풳; 푇) of the great contributions of Evariste Galois, in addition as the prime 푝 varies will also serve as our motivating to the eponymous theory which plays such a crucial role question for the Langlands program. in Wiles’s work, is his discovery of a field of cardinality 푟 푟 It turns out to be fruitful to package the zeta functions 푝 for any prime power 푝 . This field, denoted 픽 푟 and 푝 over all the finite fields into a single function of a complex sometimes referred to as the Galois field with 푝푟 elements, variable 푠 by taking the infinite product is even unique up to isomorphism. −푠 For a diophantine equation 풳 as in (1), the most basic (10) 휁(풳; 푠) = ∏ 휁푝(풳; 푝 ) invariant of the set 푝 푛+1 as 푝 ranges over all the prime numbers. In the case of the (푥1, … , 푥푛+1) ∈ 픽푝푟 such that (3) 풳(픽푝푟 ) ∶= { } simplest nontrivial diophantine equation 푥 = 0, whose 푃(푥1, … , 푥푛+1) = 0 solution set over 픽푝푟 consists of a single point, one has 푟 of solutions over 픽푝 is of course its cardinality 푁푝푟 = 1 for all 푝, and therefore (4) 푁푝푟 ∶= #풳(픽푝푟 ). 푇푟 (11) 휁 (푥 = 0; 푇) = exp ( ) = (1 − 푇)−1. What patterns (if any) are satisfied by the sequence 푝 ∑ 푟≥1 푟 (5) 푁푝, 푁푝2 , 푁푝3 , … , 푁푝푟 ,…? It follows that This sequence can be packaged into a generating series 1 −1 like (12) 휁(푥 = 0; 푠) = ∏ (1 − ) 푝푠 ∞ ∞ 푝 푟 푁푝푟 푟 (6) ∑ 푁푝푟 푇 or ∑ 푇 . 1 1 1 푟=1 푟=1 푟 (13) = ∏ (1 + + + + ⋯) 푝푠 푝2푠 푝3푠 For technical reasons it is best to consider the exponential 푝 ∞ of the latter: 1 (14) = ∑ = 휁(푠). ∞ 푛푠 푁푝푟 푟 푛=1 (7) 휁푝(풳; 푇) ∶= exp ( ∑ 푇 ) . 푟=1 푟 The zeta function of even the humblest diophantine This power series in 푇 is known as the zeta function equation is thus a central object of mathematics: the celebrated Riemann zeta function, which is tied to some of 풳 over 픽푝. It has integer coefficients and enjoys the following remarkable properties: of the deepest questions concerning the distribution of prime numbers. In his great memoir of 1860, Riemann (1) It is a rational function in 푇: proved that, even though (13) and (14) only converge 푄(푇) absolutely on the right half-plane ℜ(푠) > 1, the function (8) 휁푝(풳; 푇) = , 푅(푇) 휁(푠) extends to a meromorphic function of 푠 ∈ ℂ (with a where 푄(푇) and 푅(푇) are polynomials in 푇 whose single pole at 푠 = 1) and possesses an elegant functional degrees (for all but finitely many 푝) are inde- equation relating its values at 푠 and 1 − 푠. The zeta pendent of 푝 and determined by the shape—the functions of linear equations 풳 in 푛 + 1 variables are just complex topology—of the set 풳(ℂ) of complex shifts of the Riemann zeta function, since 푁푝푟 is equal to solutions; 푝푛푟, and therefore 휁(풳; 푠) = 휁(푠 − 푛).

210 Notices of the AMS Volume 64, Number 3 Moving on to equations of degree two, the general involves the same basic constituents, Dirichlet series, quadratic equation in one variable is of the form 푎푥2 +푏푥+ as in the one-variable case. This means that quadratic 푐 = 0, and its behaviour is governed by its discriminant diophantine equations in any number of variables are well understood, at least as far as their zeta functions are (15) Δ ∶= 푏2 − 4푎푐. concerned. This purely algebraic fact remains true over the finite The plot thickens when equations of higher degree fields, and for primes 푝 ∤ 2푎Δ one has are considered. Consider for instance the cubic equation 3 0 if Δ is a nonsquare modulo 푝, 푥 − 푥 − 1 of discriminant Δ = −23. For all 푝 ≠ 23, (16) 푁 = { 푝 2 if Δ is a square modulo 푝. this cubic equation has no multiple roots over 픽푝푟 , and therefore 푁푝 = 0, 1, or 3. A simple expression for 푁푝 in A priori, the criterion for whether 푁푝 = 2 or 0—whether this case is given by the following theorem of Hecke: the integer Δ is or is not a quadratic residue modulo 푝—seems like a subtle condition on the prime 푝. To get Theorem (Hecke). The following hold for all primes 푝 ≠ a better feeling for this condition, consider the example 23: 2 of the equation 푥 − 푥 − 1, for which Δ = 5. Calculating (1) If 푝 is not a square modulo 23, then 푁푝 = 1. whether 5 is a square or not modulo 푝 for the first few (2) If 푝 is a square modulo 23, then primes 푝 ≤ 101 leads to the following list: 0 if 푝 = 2푎2 + 푎푏 + 3푏2, (17) (21) 푁 = { 푝 3 if 푝 = 푎2 + 푎푏 + 6푏2, 2 for 푝=11, 19, 29, 31, 41, 59, 61, 71, 79, 89, 101, … 푁 ={ 푝 0 for 푝=2, 3, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, … for some 푎, 푏 ∈ ℤ. A clear pattern emerges from this experiment: whether Hecke’s theorem implies that 푁 = 0 or 2 seems to depend only on the rightmost digit ∞ 푝 3 −푠 of 푝, i.e., on what the remainder of 푝 is modulo 10. One (22) 휁(푥 − 푥 − 1; 푠) = 휁(푠) × ∑ 푎푛푛 , is led to surmise that 푛=1 where the generating series 2 if 푝 ≡ 1, 4 (mod 5), (18) 푁푝 = { (23) 0 if 푝 ≡ 2, 3 (mod 5), 푛 2 3 6 8 13 16 23 퐹(푞)∶=∑ 푎푛푞 = 푞−푞 −푞 +푞 +푞 −푞 −푞 +푞 +⋯ a formula that represents a dramatic improvement over is given by the explicit formula (16), allowing a much more efficient calculation of 푁푝 for example. The guess in (18) is in fact a consequence of 1 2 2 2 2 (24) 퐹(푞) = ⎛ ∑ 푞푎 +푎푏+6푏 − 푞2푎 +푎푏+3푏 ⎞ . Gauss’s celebrated law of quadratic reciprocity: 2 ⎝푎,푏∈ℤ ⎠ 2 Theorem (Quadratic reciprocity). For any equation 푎푥 + The function 푓(푧) = 퐹(푒2휋푖푧) that arises by setting 푞 = 2 푏푥+푐, with Δ ∶= 푏 −4푎푐, the value of the function 푝 ↦ 푁푝 푒2휋푖푧 in (24) is a prototypical example of a modular form: (for 푝 ∤ 푎Δ) depends only on the residue class of 푝 modulo namely, it satisfies the transformation rule 4Δ and hence is periodic with period length dividing 4|Δ|. (25) 푎푧 + 푏 푎, 푏, 푐, 푑 ∈ ℤ, 푎푑 − 푏푐 = 1 The repeating pattern satisfied by the 푁푝’s as 푝 varies 푓 ( ) = (푐푧+푑)푓(푧), { 푎 greatly facilitates the manipulation of the zeta functions 푐푧 + 푑 23|푐, ( 23 ) = 1. of quadratic equations. For example, the zeta function of under so-called modular substitutions of the form 푧 ↦ 푎푧+푏 . 풳 ∶ 푥2 − 푥 − 1 = 0 is equal to 푐푧+푑 This property follows from the Poisson summation for- 1 1 1 1 1 1 1 휁(풳; 푠) = 휁(푠)×{(1− − + )+( − − + ) mula applied to the expression in (24). Thanks to (25), the 2푠 3푠 4푠 6푠 7푠 8푠 9푠 zeta function of 풳 can be manipulated with the same ease 1 1 1 1 as the zeta functions of Riemann and Dirichlet. Indeed, (19) + (1 − − + )+⋯}. 푠 푠 푠 푠 ∞ −푠 11 12 13 14 Hecke showed that the 퐿-series ∑푛=1 푎푛푛 attached to ∞ 2휋푖푛푧 The series that occurs on the right-hand side is a prototyp- a modular form ∑푛=1 푎푛푒 possesses very similar an- ical example of a Dirichlet 퐿-series. These 퐿-series, which alytic properties, notably an analytic continuation and a are the key actors in the proof of Dirichlet’s theorem on Riemann-style functional equation. the infinitude of primes in arithmetic progressions, enjoy The generating series 퐹(푞) can also be expressed as an many of the same analytic properties as the Riemann zeta infinite product: function: an analytic continuation to the entire complex (26) ∞ plane and a functional equation relating their values at 푠 1 2 2 2 2 ⎛ 푞푎 +푎푏+6푏 −푞2푎 +푎푏+3푏 ⎞=푞 (1−푞푛)(1−푞23푛). and 1 − 푠. They are also expected to satisfy a Riemann hy- ∑ ∏ 2 ⎝푎,푏∈ℤ ⎠ 푛=1 pothesis which generalises Riemann’s original statement and is just as deep and elusive. The first few terms of this power series identity can readily It is a (not completely trivial) fact that the zeta function be verified numerically, but its proof is highly nonobvious of the general quadratic equation in 푛 variables and indirect. It exploits the circumstance that the space of holomorphic functions of 푧 satisfying the transformation 푛 푛 rules (25) together with suitable growth properties is a one- (20) ∑ 푎푖푗푥푖푥푗 + ∑ 푏푖푥푖 + 푐 = 0 푖,푗=1 푖=1 dimensional complex vector space which also contains

March 2017 Notices of the AMS 211 the infinite product above. Indeed, the latter is equal to after a suitable change of variables and which are non- 휂(푞)휂(푞23), where singular, a condition equivalent to the assertion that the 3 2 ∞ discriminant Δ ∶= −16(4푎 + 27푏 ) is nonzero. (27) 휂(푞) = 푞1/24 ∏(1 − 푞푛) To illustrate Wiles’s theorem with a concrete example, 푛=1 consider the equation is the Dedekind eta function whose logarithmic derivative (32) 퐸 ∶ 푦2 = 푥3 − 푥, (after viewing 휂 as a function of 푧 through the change of 2휋푖푧 of discriminant Δ = 64. After setting variables 푞 = 푒 ) is given by (33) ∞ −푠 −푠 −푠 −1 휂′(푧) −1 휁(퐸; 푠) = 휁(푠−1)×(푎1 + 푎22 + 푎33 + 푎44 + ⋯) , (28) = −휋푖( + 2 ∑ ( ∑ 푑)푒2휋푖푛푧) 휂(푧) 12 푛=1 푑|푛 the associated generating series satisfies the following identities reminiscent of (24) and (26): 푖 ∞ ∞ 1 (29) = , (34) ∑ ∑ 2 4휋 푚=−∞ 푛=−∞ (푚푧 + 푛) 푛 5 9 13 17 25 where the term attached to (푚, 푛) = (0, 0) is excluded 퐹(푞) = ∑ 푎푛푞 = 푞 − 2푞 − 3푞 + 6푞 + 2푞 − 푞 + ⋯ 2 2 from the last sum. The Dedekind 휂-function is also (35) = ∑ 푎 ⋅ 푞(푎 +푏 ) connected to the generating series for the partition 푎,푏 function 푝(푛) describing the number of ways in which ∞ 푛 can be expressed as a sum of positive integers via the (36) = 푞 ∏(1 − 푞4푛)2(1 − 푞8푛)2, identity 푛=1 2 ∞ where the sum in (35) runs over the (푎, 푏) ∈ ℤ for which (30) 휂−1(푞) = 푞−1/24 ∑ 푝(푛)푞푛, the Gaussian integer 푎 + 푏푖 is congruent to 1 modulo 푛=0 (1+푖)3. (This identity follows from Deuring’s study of zeta which plays a starring functions of elliptic curves with complex multiplication role alongside Jeremy and may even have been known earlier.) Once again, the 2휋푖푧 “There are five Irons and Dev Patel in holomorphic function 푓(푧) ∶= 퐹(푒 ) is a modular form a recent film about satisfying the slightly different transformation rule elementary the life of Srinivasa (37) 푎푧 + 푏 푎, 푏, 푐, 푑 ∈ ℤ, 푎푑 − 푏푐 = 1, Ramanujan. 푓 ( ) = (푐푧+푑)2푓(푧), { arithmetical Commenting on the 푐푧 + 푑 32|푐. operations: “unreasonable effec- Note the exponent 2 that appears in this formula. Because tiveness and ubiquity of it, the function 푓(푧) is said to be a modular form of addition, of modular forms,” weight 2 and level 32. The modular forms of (25) attached Martin Eichler once to cubic equations in one variable are of weight 1, but subtraction, wrote, “There are otherwise the parallel of (35) and (36) with (24) and (26) five elementary arith- is striking. The original conjecture of Shimura–Taniyama, multiplication, metical operations: and Weil asserts the same pattern for all elliptic curves: addition, subtraction, division,…and multiplication, divi- Conjecture (Shimura–Taniyama–Weil). Let 퐸 be any ellip- sion,…and modular tic curve. Then modular forms.” ∞ −1 forms.” Equations (26), (38) 휁(퐸; 푠) = 휁(푠 − 1) × ( 푎 푛−푠) , (29), and (30) are just a ∑ 푛 푛=1 few of the many won- 2휋푖푛푧 drous identities which abound, like exotic strains of where 푓퐸(푧) ∶= ∑ 푎푛푒 is a modular form of weight 2. fragrant wild orchids, in what Roger Godement has called The conjecture was actually more precise and predicted the “garden of modular delights.” that the level of 푓퐸—i.e., the integer that appears in the The example above and many others of a similar type transformation property for 푓퐸 as the integers 23 and 32 are described in Jean-Pierre Serre’s delightful monograph in (25) and (37) respectively—is equal to the arithmetic [Se], touching on themes that were also covered in Serre’s conductor of 퐸. This conductor, which is divisible only by lecture at the inaugural Abel Prize ceremony in 2003. primes for which the equation defining 퐸 becomes singular Hecke was able to establish that all cubic polynomials in modulo 푝, is a measure of the arithmetic complexity of one variable are modular; i.e., the coefficients of their zeta 퐸 and can be calculated explicitly from an equation for functions obey patterns just like those of (24) and (25). 퐸 by an algorithm of Tate. An elliptic curve is said to be Wiles’s achievement was to extend this result to a large semistable if its arithmetic conductor is squarefree. This class of cubic diophantine equations in two variables over class of elliptic curves includes those of the form the rational numbers: the elliptic curve equations which (39) 푦2 = 푥(푥 − 푎)(푥 − 푏) can be brought to the form with gcd(푎, 푏) = 1 and 16|푏. The most famous elliptic (31) 푦2 = 푥3 + 푎푥 + 푏 curves in this class are those that ultimately do not exist:

212 Notices of the AMS Volume 64, Number 3 compact subgroup GL2(ℤ푝) for all but finitely many 푝). The shift in emphasis from modular forms to the so-called automorphic representations which they span is decisive. Langlands showed how to attach an 퐿-function to any irreducible automorphic representation of 퐺(픸ℚ) for an arbitrary reductive algebraic group 퐺, of which the matrix groups GL푛 and more general algebraic groups of Lie type are prototypical examples. This greatly enlarges the notion of what it means to be “modular”: a diophantine equation is now said to have this property if its zeta function can be expressed in terms of the Langlands 퐿- functions attached to automorphic representations. One of the fundamental goals in the Langlands program is to establish further cases of the following conjecture: Conjecture. All diophantine equations are modular in the above sense. This conjecture can be viewed as a far-reaching gen- eralisation of quadratic reciprocity and underlies the non-Abelian reciprocity laws that are at the heart of Andrew Wiles’s achievement. Before Wiles’s proof, the following general classes of diophantine equations were known to be modular: Andrew Wiles, Henri Darmon, and Mirela Çiperiani in • Quadratic equations, by Gauss’s law of quadratic June 2016 at Harvard University during a conference reciprocity; in honor of Karl Rubin’s sixtieth birthday. • Cubic equations in one variable, by the work of Hecke and Maass; the “Frey curves” 푦2 = 푥(푥 − 푎푝)(푥 + 푏푝) arising from • Quartic equations in one variable. 푝 푝 푝 putative solutions to Fermat’s equation 푎 + 푏 = 푐 , This last case deserves further comment, since it has whose nonexistence had previously been established in a not been discussed previously and plays a primordial 1 landmark article of Kenneth Ribet under the assumption role in Wiles’s proof. The modularity of quartic equations of their modularity. It is the proof of the Shimura– follows from the seminal work of Langlands and Tunnell. Taniyama–Weil conjecture for semistable elliptic curves While it is beyond the scope of this survey to describe that earned Andrew Wiles the Abel Prize: their methods, it must be emphasised that Langlands and Theorem (Wiles). Let 퐸 be a semistable elliptic curve. Then Tunnell make essential use of the solvability by radicals of the general quartic equation, whose underlying symmetry 퐸 satisfies the Shimura–Taniyama–Weil conjecture. group is contained in the permutation group 푆4 on 4 letters. The semistability assumption in Wiles’s theorem was Solvable extensions are obtained from a succession of later removed by Christophe Breuil, Brian Conrad, Fred Abelian extensions, which fall within the purview of the Diamond, and Richard Taylor around 1999. (See, for class field theory developed in the nineteenth and first instance, the account [Da] that appeared in the Notices at half of the twentieth century. On the other hand, the the time.) modularity of the general equation of degree > 4 in one As a prelude to describing some of the important ideas variable, which cannot be solved by radicals, seemed to in its proof, one must first try to explain why Wiles’s lie well beyond the scope of the techniques that were theorem occupies such a central position in mathematics. available in the “pre-Wiles era.” The reader who perseveres The Langlands program places it in a larger context to the end of this essay will be given a glimpse of how by offering a vast generalisation of what it means for our knowledge of the modularity of the general quintic a diophantine equation to be “associated to a modular equation has progressed dramatically in the wake of form.” The key is to view modular forms attached to cubic Wiles’s breakthrough. equations or to elliptic curves as in (24) or (34) as vectors Prior to Wiles’s proof, modularity was also not known in certain irreducible infinite-dimensional representations for any interesting general class of equations (of degree of the locally compact topological group > 2, say) in more than one variable; in particular it had only been verified for finitely many elliptic curves over ℚ (40) GL (픸 ) = ′ GL (ℚ ) × GL (ℝ) 2 ℚ ∏ 푝 2 푝 2 up to isomorphism over ℚ̄ (including the elliptic curves ′ (where ∏푝 denotes a restricted direct product over all over ℚ with complex multiplication, of which the elliptic the prime numbers consisting of elements (훾푝)푝 for curve of (31) is an instance.) Wiles’s modularity theorem which the 푝th component 훾푝 belongs to the maximal confirmed the Langlands conjectures in the important test case of elliptic curves, which may seem like (and, in 1See the interview with Ribet as the new AMS president in this fact, are) very special diophantine equations, but have issue, page 229. provided a fertile terrain for arithmetic investigations,

March 2017 Notices of the AMS 213 both in theory and in applications (cryptography, coding Like diophantine equations, Galois representations also theory…). give rise to analogous zeta functions. More precisely, the Returning to the main group 퐺ℚ,푆 contains, for each prime 푝 ∉ 푆, a distinguished theme of this report, element called the Frobenius element at 푝, denoted 휎푝. Wiles’s proof is Wiles’s proof is also impor- Strictly speaking, this element is well defined only up tant for having introduced to conjugacy in 퐺ℚ,푆, but this is enough to make the also important a revolutionary new ap- arithmetic sequence proach which has opened 푟 (42) 푁푝푟 (휚) ∶= Trace(휚(휎푝 )) for having the floodgates for many well defined. The zeta function 휁(휚; 푠) packages the further breakthroughs in information from this sequence in exactly the same way introduced a the Langlands program. as in the definition of 휁(풳; 푠). To expand on this revolutionary For example, if 풳 is attached to a polynomial 푃 of point, we need to present degree 푑 in one variable, the action of 퐺 on the another of the dramatis ℚ,푆 new approach roots of 푃 gives rise to a 푑-dimensional permutation personae in Wiles’s proof: representation which has Galois representations. Let ̄ (43) 휚 ∶ 퐺 ⟶ GL (ℚ), opened the 퐺ℚ = Gal(ℚ/ℚ) be the 풳 ℚ,푆 푑 absolute Galois group of and 휁(풳, 푠) = 휁(휚풳, 푠). This connection goes far deeper, floodgates for ℚ, namely, the automor- extending to diophantine equations in 푛 + 1 variables for phism group of the field general 푛 ≥ 0. The glorious insight at the origin of the many further of all algebraic numbers. Weil conjectures is that 휁(풳; 푠) can be expressed in terms It is a profinite group, of the zeta functions of Galois representations arising breakthroughs endowed with a natu- in the étale cohomology of 풳, a cohomology theory with ral topology for which ℓ-adic coefficients which associates to 풳 a collection in the Langlands the subgroups Gal(ℚ/퐿)̄ 푖 {퐻et(풳/ℚ̄ ,ℚℓ)} with 퐿 ranging over the 0≤푖≤2푛 program. finite extensions of ℚ of finite-dimensional ℚℓ-vector spaces endowed with a form a basis of open continuous linear action of 퐺ℚ,푆. (Here 푆 is the set of subgroups. Following the primes 푞 consisting of ℓ and the primes for which the original point of view taken by Galois himself, the group equation of 풳 becomes singular after being reduced 퐺ℚ acts naturally as permutations on the roots of poly- modulo 푞.) These representations generalise the repre- nomials with rational coefficients. Given a finite set 푆 of sentation 휚풳 of (43), insofar as the latter is realised by 0 primes, one may consider only the monic polynomials the action of 퐺ℚ,푆 on 퐻et(풳ℚ̄ ,ℚℓ) after extending the with integer coefficients whose discriminant is divisible coefficients from ℚ to ℚℓ. only by primes ℓ ∈ 푆 (eventually after a change of vari- Theorem (Weil, Grothendieck,…). If 풳 is a diophantine ables). The topological group 퐺ℚ operates on the roots of equation having good reduction outside 푆, there exist Ga- such polynomials through a quotient, denoted 퐺ℚ,푆—the lois representations 휚1 and 휚2 of 퐺ℚ,푆 for which automorphism group of the maximal algebraic extension unramified outside 푆, which can be regarded as the sym- (44) 휁(풳; 푠) = 휁(휚1; 푠)/휁(휚2; 푠). 푖 metry group of all the zero-dimensional varieties over ℚ The representations 휚1 and 휚2 occur in ⊕퐻et(풳/ℚ̄ ,ℚℓ), having “nonsingular reduction outside 푆.” where the direct sum ranges over the odd and even values In addition to the permutation representations of 퐺ℚ of 0 ≤ 푖 ≤ 2푛 for 휚1 and 휚2 respectively. More canonically, that were so essential in Galois’s original formulation there are always irreducible representations 휚1, … , 휚푡 of of his theory, it has become important to study the 퐺ℚ,푠 and integers 푑1, … 푑푡 such that (continuous) linear representations 푡 푑푖 (45) 휁(풳; 푠) = ∏ 휁(휚푖; 푠) , (41) 휚 ∶ 퐺ℚ,푆 ⟶ 퐺퐿푛(퐿) 푖=1 of this Galois group, where 퐿 is a complete field, such as arising from the decompositions of the (semisimplifi- the fields ℝ or ℂ of real or complex numbers, the finite 푖 cation of the) 퐻et(풳ℚ̄ ,ℚℓ) into a sum of irreducible field 픽ℓ푟 equipped with the discrete topology, or a finite representations. The 휁(휚푖, 푠) can be viewed as the ̄ extension 퐿 ⊂ ℚℓ of the field ℚℓ of ℓ-adic numbers. “atomic constituents” of 휁(풳, 푠) and reveal much of Galois representations were an important theme in the the “hidden structure” in the underlying equation. The de- work of Abel and remain central in modern times. Many composition of 휁(풳; 푠) into a product of different 휁(휚푖; 푠) illustrious mathematicians in the twentieth century have is not unlike the decomposition of a wave function into contributed to their study, including three former Abel its simple harmonics. Prize winners: Jean-Pierre Serre, John Tate, and Pierre A Galois representation is said to be modular if its zeta Deligne. Working on Galois representations might seem function can be expressed in terms of generating series to be a prerequisite for an algebraic number theorist to attached to modular forms and automorphic representa- receive the Abel Prize! tions and is said to be geometric if it can be realised in

214 Notices of the AMS Volume 64, Number 3 an étale cohomology group of a diophantine equation as the inverse limit being taken relative to the multiplication- 푛 above. The “main conjecture of the Langlands program” by-3 maps. The groups 퐸[3 ] and 푇3(퐸) are free modules 푛 can now be amended as follows: of rank 2 over (ℤ/3 ℤ) and ℤ3 respectively and are endowed with continuous linear actions of 퐺 , where 푆 Conjecture. All geometric Galois representations of 퐺 ℚ,푆 ℚ,푆 is a set of primes containing 3 and the primes that divide are modular. the conductor of 퐸. One obtains the associated Galois Given a Galois representation representations:

(46) 휚 ∶ 퐺ℚ,푆 ⟶ GL푛(ℤ ) 휚퐸,3̄ ∶ 퐺ℚ,푆 ⟶ Aut(퐸[3]) ≃ GL2(픽3), ℓ (50) with ℓ-adic coefficients, one may consider the resulting 휚퐸,3 ∶ 퐺ℚ,푆 ⟶ GL2(ℤ3). mod ℓ representation The theorem of Langlands and Tunnell about the modular-

(47) 휚̄ ∶ 퐺ℚ,푆 ⟶ GL푛(픽ℓ). ity of the general quartic equation leads to the conclusion that 휚̄ is modular. This rests on the happy circumstance The passage from 휚 to 휚̄ amounts to replacing the 퐸,3 푟 that quantities 푁푝푟 (휚) ∈ ℤℓ as 푝 ranges over all the prime powers with their (51) GL2(픽3)/⟨±1⟩ ≃ 푆4, mod ℓ reduction. and hence that 퐸[3] has essentially the same symmetry Such a passage would Since then, group as the general quartic equation! The isomorphism seem rather contrived in (51) can be realised by considering the action of GL2(픽3) for the sequences “modularity lifting on the set {0, 1, 2, ∞} of points on the projective line over 푁 푟 (풳)—why study 푝 픽3. the solution counts of theorems” have If 퐸 is semistable, Wiles is able to check that both a diophantine equa- 휚퐸,3 and 휚퐸,3̄ satisfy the conditions necessary to apply tion over different proliferated, and the Modularity Lifting Theorem, at least when 휚퐸,3̄ is finite fields, taken irreducible. It then follows that 휚퐸,3 is modular, and modulo ℓ?—if one their study, in ever therefore so is 퐸, since 휁(퐸; 푠) and 휁(휚퐸,3; 푠) are the same. did not know a pri- more general and Note the key role played by the result of Langlands– ori that these counts Tunnell in the above strategy. It is a dramatic illustration arise from ℓ-adic Ga- delicate settings, fo the unity and historical continuity of mathematics that lois representations the solution in radicals of the general quartic equation, with coefficients in ℤℓ. has spawned an one of the great feats of the algebraists of the Italian There is a correspond- industry. Renaissance, is precisely what allowed Langlands, Tunnell, ing notion of what it and Wiles to prove their modularity results more than means for 휚̄ to be mod- five centuries later. ular, namely, that the Having established the modularity of all semistable el- data of 푁 푟 (휚)̄ agrees, very loosely speaking, with the 푝 liptic curves 퐸 for which 휚퐸,3̄ is irreducible, Wiles disposes mod ℓ reduction of similar data arising from an automor- of the others by applying his lifting theorem to the prime phic representation. We can now state Wiles’s celebrated ℓ = 5 instead of ℓ = 3. The Galois representation 휚퐸,5̄ is modularity lifting theorem, which lies at the heart of his always irreducible in this setting, because no elliptic curve strategy: over ℚ can have a rational subgroup of order 15. Nonethe- Wiles’s Modularity Lifting Theorem. Let less, the approach of exploiting ℓ = 5 seems hopeless at first glance, because the Galois representation 퐸[5] is (48) 휚 ∶ 퐺ℚ,푆 ⟶ GL2(ℤℓ) not known to be modular a priori, for much the same be an irreducible geometric Galois representation satisfy- reason that the general quintic equation cannot be solved ing a few technical conditions (involving, for the most part, by radicals. (Indeed, the symmetry group SL2(픽5) is a ̄ double cover of the alternating group 퐴 on 5 letters and the restriction of 휚 to the subgroup 퐺ℚℓ = Gal(ℚℓ/ℚℓ) of 5 퐺ℚ,푆). If 휚̄ is modular and irreducible, then so is 휚. thus is closely related to the symmetry group underlying the general quintic.) To establish the modularity of 퐸[5], This stunning result was completely new at the time: Wiles constructs an auxiliary semistable elliptic curve 퐸′ nothing remotely like it had ever been proved before! Since satisfying then, “modularity lifting theorems” have proliferated, and their study, in ever more general and delicate settings, has (52) 휚퐸̄ ′,5 = 휚퐸,5̄ , 휚퐸̄ ′,3 is irreducible. spawned an industry and led to a plethora of fundamental It then follows from the argument in the previous para- advances in the Langlands program. graph that 퐸′ is modular, hence that 퐸′[5] = 퐸[5] is Let us first explain how Wiles himself parlays his modular as well, putting 퐸 within striking range of the original modularity lifting theorem into a proof of the modularity lifting theorem with ℓ = 5. This lovely epi- Shimura–Taniyama-Weil conjecture for semistable elliptic logue of Wiles’s proof, which came to be known as the curves. Given such an elliptic curve 퐸, consider the groups “3-5 switch,” may have been viewed as an expedient trick (49) at the time. But since then the prime switching argument 푛 푛 푛 퐸[3 ] ∶= {푃 ∈ 퐸(ℚ)̄ ∶ 3 푃 = 0} , 푇3(퐸) ∶= lim 퐸[3 ], ← has become firmly embedded in the subject, and many

March 2017 Notices of the AMS 215 variants of it have been exploited to spectacular effect in • The Sato–Tate conjecture concerning the distribution deriving new modularity results. of the numbers 푁푝(퐸) for an elliptic curve 퐸 as the Wiles’s modularity prime 푝 varies, whose proof was known to follow lifting theorem reveals from the modularity of all the symmetric power The modularity of that “modularity is Galois representations attached to 퐸, was proved in contagious” and can large part by Laurent Clozel, Michael Harris, Nick elliptic curves was often be passed on to Shepherd-Barron, and Richard Taylor around 2006. an ℓ-adic Galois rep- • One can also make sense of what it should mean only the first in a resentation from its for diophantine equations over more general number mod ℓ reduction. It fields to be modular. The modularity of elliptic curves series of is this simple prin- over all real quadratic fields has been proved very ciple that accounts recently by Nuno Freitas, Bao Le Hung, and Samir Siksek spectacular for the tremendous by combining the ever more general and powerful applications. impact that the Modu- modularity lifting theorems currently available with a larity Lifting Theorem, careful diophantine study of the elliptic curves which and the many variants could a priori fall outside the scope of these lifting proved since then, continue to have on the subject. Indeed, theorems. the modularity of elliptic curves was only the first in a • Among the spectacular recent developments building series of spectacular applications of the ideas introduced on Wiles’s ideas is the proof, by Laurent Clozel and by Wiles, and since 1994 the subject has witnessed a Jack Thorne, of the modularity of certain symmetric real golden age, in which open problems that previously powers of the Galois representations attached to seemed completely out of reach have succumbed one by holomorphic modular forms, which is described in one. Thorne’s contribution to this volume. Among these developments, let us mention: These results are just a sampling of the transformative • The two-dimensional Artin conjecture, first formu- impact of modularity lifting theorems. The Langlands pro- lated in 1923, concerns the modularity of all odd, gram remains a lively area, with many alluring mysteries two-dimensional Galois representations yet to be explored. It is hard to predict where the next breakthroughs will come, but surely they will continue to (53) 휚 ∶ 퐺ℚ,푆 ⟶ GL2(ℂ). capitalise on the rich legacy of Andrew Wiles’s marvelous The image of such a 휚 modulo the scalar matrices proof. is isomorphic either to a dihedral group, to 퐴4, to 푆 , or to 퐴 . Thanks to the earlier work of Hecke, 4 5 References Langlands, and Tunnell, only the case of projective [Da] H. Darmon, A proof of the full Shimura-Taniyama-Weil con- image 퐴 remained to be disposed of. Many new cases 5 jecture is announced, Notices of the AMS 46 (1999), no. 11, of the two-dimensional Artin conjecture were proved 1397–1401. MR1723249 in this setting by Kevin Buzzard, Mark Dickinson, Nick [Se] J-P. Serre, Lectures on 푁푋(푝), Chapman & Hall/CRC Re- Shepherd-Barron, and Richard Taylor around 2003 us- search Notes in Mathematics, 11, CRC Press, Boca Raton, ing the modularity of all mod 5 Galois representations FL. MR2920749 arising from elliptic curves as a starting point. • Serre’s conjecture, which was formulated in 1987, Photo Credits asserts the modularity of all odd, two-dimensional Photo of Wiles giving his first lecture is by C. J. Galois representations Mozzochi, courtesy of the Simons Foundation.

(54) 휚 ∶ 퐺ℚ,푆 ⟶ GL2(픽푝푟 ), Photo of the conference in honor of Karl Rubin’s sixtieth birthday is courtesy of Kartik Prasanna. with coefficients in a finite field. This result was proved by Chandrasekhar Khare and Jean-Pierre Wintenberger in 2008 by a glorious extension of the “3-5 switching technique” in which essentially all the primes are used. (See Khare’s report in this volume.) This result also implies the two-dimensional Artin conjecture in the general case. • The two-dimensional Fontaine–Mazur conjecture con- cerning the modularity of odd, two-dimensional 푝-adic Galois representations

(55) 휚 ∶ 퐺ℚ,푆 ⟶ GL2(ℚ̄ 푝) satisfying certain technical conditions with respect to their restrictions to the Galois group of ℚ푝. This theorem was proved in many cases as a consequence of work of Pierre Colmez, Matthew Emerton, and Mark Kisin.

216 Notices of the AMS Volume 64, Number 3 The Mathematical Works of Andrew Wiles Christopher Skinner, with contributions from Karl Rubin, Barry Mazur, Mirela Çiperiani, Chandrashekhar Khare, and Jack Thorne

ir Andrew J. Wiles was awarded the Abel Prize for 2016 for “his stunning proof of Fermat’s Last Theorem by way of the modularity conjecture for semistable elliptic curves, opening a new era in Snumber theory.”1 Andrew Wiles announced his proof of Fermat’s Last Theorem in June of 1993 in a series of three lectures at a conference at the Isaac Newton Institute for Mathematical Sciences at the University of Cambridge. Overnight Wiles and his proof became an international media sensation, making headlines in papers around the world. The story of this proof—the subsequent discovery of a gap and its ultimate and beautiful completion in September of 1994—has entered into popular legend.2 The surprising drama of the proof is told in the 1996 BBC Horizon documentary Fermat’s Last Theorem, directed by Simon Singh, which ably conveys Andrew Wiles with his PhD advisor, John Coates (left); the human side of what is often seen as the distant and his first PhD student Karl Rubin (right); and Ken Ribet (second from right), whose work linking the rarefied world of mathematical research.3 Modularity Conjecture to Fermat’s Last Theorem All this is well known. What is less well known, possibly inspired Wiles to reengage with the problem that had even among number theorists, is that before his proof fascinated him since childhood. This photo was taken of Fermat’s Last Theorem, Wiles had made significant at the Newton Institute in Cambridge, UK, during the contributions to two of the most important problems for conference at which Wiles first announced his proof late-twentieth-century number theory: of the Modularity Conjecture.

Christopher Skinner is professor of mathematics at Princeton Uni- versity. His e-mail address is [email protected]. • Wiles proved, together with John Coates, the first the- 1Citation by the Abel Prize Committee of the Norwegian Acad- oretical evidence for the celebrated Birch–Swinnerton- emy of Science and Letters for the 2016 Abel Prize Laureate: www. Dyer Conjecture; this is now known as the Coates– abelprize.no/c67107/binfil/download.php?tid=67059 Wiles Theorem. • Wiles proved Iwasawa’s Main Conjecture for ℚ, in joint 2No doubt many number theorists share my own experiences of striking up conversations with strangers, who, upon discovering work with Barry Mazur, and for all totally real fields. that I am a mathematician and even a number theorist, ask about Each of these is a landmark result on its own and would “that guy who solved that famous problem—the one who worked be considered the highlight of a distinguished career. in his attic for seven years.” Two of the following contributions describe these works 3I have watched this documentary many times with groups of and their proofs. Karl Rubin, Wiles’s first PhD student, mathematically talented high school students from around the writes about the Coates–Wiles Theorem. Barry Mazur, world. Twenty years later it still inspires questions and conversa- Wiles’s collaborator on his first proof of Iwasawa’s Main tion about what it means to do mathematical research or what a Conjecture for ℚ, writes about Wiles’s work on the Main life spent doing mathematics can be. Conjectures. Anyone seeking to learn about the context, For permission to reprint this article, please contact: significance, and ideas of Wiles’s proof of Fermat’s Last [email protected]. Theorem can do no better than to read Henri Darmon’s DOI: http://dx.doi.org/10.1090/noti1485 Abel Prize lecture in this same issue of the Notices.

March 2017 Notices of the AMS 217 What are you working on now? This is a question that be seen as a vast generalization of the ideas in the proof eminent mathematicians are frequently asked (or so I am of the Coates–Wiles theorem. My own work with Eric reliably informed) and certainly one that Andrew Wiles Urban (2014), which together with Kato’s result proves has repeatedly faced in the years following his proof of much of the Main Conjecture in the Iwaswa theory of Fermat’s Last Theorem. In her contribution below, Mirela elliptic curves, is in large part the natural generalization Çiperiani writes about her collaboration with Andrew to some unitary groups of Wiles’s methods for proving the Wiles, from the mid-2000s, on a very natural Diophantine Iwasawa Main Conjecture for totally real fields. Andrew question: does a genus one curve over ℚ have a rational Wiles’s ideas continue to inform and shape progress on point over a solvable extension of ℚ? As Çiperiani explains, some of the fundamental problems in algebraic number results and techniques arising from the proof of Fermat’s theory. Last Theorem also play a role in this work. What distinguishes a great mathematical proof? There Karl Rubin is, of course, no definitive answer to this question. Cer- tainly proofs of famous or important open conjectures Wiles’s Work on Elliptic Curves with Complex can lay claim to being great. By this measure, Andrew Multiplication Wiles’s proof of Fermat’s Last Theorem is a truly great During my senior year at Princeton, 1975–76, there was proof. But proofs that introduce new ideas or open doors a lot of buzz in the common room about something to progress on problems that were previously viewed as everyone referred to as “Coates–Wiles.” I had a vague idea out of reach also have their claim to greatness. As noted in the Abel Prize citation, Andew Wiles’s proofs also achieve of what an elliptic curve was, but I doubt that I had any greatness by this second measure. clear idea of what John Coates and Andrew Wiles had The new techniques and ideas that led to a suc- done. But I could tell that it was important, and I had a cessful proof of the modularity of semistable elliptic mental image of two very senior mathematicians who had curves have been remarkably robust, also leading to the made a great breakthrough. resolution of a host of problems within the circle of I arrived at Harvard as a graduate student in 1976. the Langlands Program: proofs of Serre’s conjecture, the Wiles arrived a year later, and I discovered that this Artin conjecture for odd two-dimensional representations, “very senior” mathematician was scarcely older than I the Sato–Tate conjecture, meromorphic continuation of was and had been a graduate student at the time of Hasse–Weil 퐿-functions of elliptic curves over totally real this spectacular result. I became his student and had fields, modularity of all elliptic curves over real quadratic the unusual experience of attending my advisor’s thesis fields, and automorphy of small symmetric powers of defense. modular forms, to list just a few. Here’s what the excitement was about. An elliptic curve Much of this progress has been achieved by the efforts 퐸 over the field ℚ of rational numbers is a curve defined by an equation 푦2 = 푥3 + 푎푥 + 푏, with 푎, 푏 ∈ ℚ satisfying of Richard Taylor, who was a PhD student of Wiles and 3 2 later a coauthor of the paper that introduced one of the 4푎 + 27푏 ≠ 0. It is classical that the rational points 4 퐸(퐹) on 퐸 over any field 퐹 containing ℚ form an abelian key ingredients of the proof of modularity; by Wiles’s group, and Mordell proved that 퐸(ℚ) is finitely generated. PhD students; and by Taylor’s collaborators and PhD The rank of 퐸 is the dimension of the ℚ-vector space students. But Wiles’s work also inspired many others to 퐸(ℚ) ⊗ ℚ. push his ideas further. Among these is Chandrashekhar ℤ Khare, who writes below about some of the progress In the late 1950s, after extensive computations, Birch on modularity of two-dimensional Galois representations and Swinnerton-Dyer made the following conjecture. that followed Wiles’s proof and of the use of this in Conjecture (Birch & Swinnerton-Dyer). For every elliptic Khare’s own proof, with Jean-Pierre Wintenberger, of curve 퐸 over ℚ, we have Serre’s conjecture. In a related contribution, Jack Thorne (BSD) rank(퐸(ℚ)) = ord 퐿(퐸, 푠). describes how Wiles’s original techniques evolved and 푠=1 were adapted to proving the automorphy of higher- Here 퐿(퐸, 푠) is the Hasse–Weil 퐿-function of 퐸, defined dimensional Galois representations, leading to his proof, by an Euler product that converges on the complex with Laurent Clozel, of the automorphy of some small half-plane ℜ(푠) > 3/2, and ord푠=1 퐿(퐸, 푠) is its order of symmetric powers of a holomorphic modular form. vanishing at 푠 = 1. This conjecture is still unproved and But we should not lose sight of Wiles’s earlier contribu- is one of the Clay Millennium Problems. tions in the glow of the successes arising from the proof Clearly, to make progress on this conjecture, or even of Fermat’s Last Theorem. The proof of the Coates–Wiles for the statement to make sense, one needs to know that Theorem and Wiles’s proof of Iwasawa’s Main Conjec- 퐿(퐸, 푠) has an analytic continuation at least to 푠 = 1. This ture for totally real fields have inspired similar progress. was already known, thanks to a theorem of Deuring, for For example, Kazuya Kato’s (2004) spectacular success elliptic curves with complex multiplication: we say that 퐸 in constructing an Euler system for elliptic curves and has complex multiplication if the ring of endomorphisms then relating it to the special values of the Hasse–Weil End(퐸) is larger than ℤ, in which case End(퐸) ⊗ℤ ℚ is an 퐿-function of the curve via an explicit reciprocity law can Karl Rubin is Edward and Vivian Thorp Professor of Mathematics 4Now generally known as the Taylor–Wiles method. at UC Irvine. His e-mail address is [email protected].

218 Notices of the AMS Volume 64, Number 3 imaginary quadratic field. (The case End(퐸) = ℤ is much Conjecture and on the modularity of elliptic curves (see more common.) For example, the elliptic curve 푦2 = 푥3+푎푥 the contributions by Barry Mazur and Henri Darmon, has complex multiplication because (푥, 푦) ↦ (−푥, 푖푦) is respectively, in this issue), have continued to play an an automorphism of 퐸 of order 4. Deuring proved that if 퐸 important role in progress on the Birch and Swinnerton- has complex multiplication, then 퐿(퐸, 푠) can be identified Dyer conjecture. with the 퐿-function attached to a Hecke character and • Kolyvagin (1990) recognized that elliptic units form hence has an analytic continuation to the entire complex what he calls an Euler system. (In fact, as one of very plane. Further, in this case Damerell proved that there is few known examples, elliptic units helped him to an explicit Ω ∈ ℝ× such that 퐿(퐸, 1)/Ω ∈ ℤ. formulate the concept of an Euler system.) Combining By the mid-1970s little was known about (BSD) beyond the methods of Coates and Wiles with Kolyvagin’s computational examples. That was how things stood when Euler system machinery led to my 1991 proof of Coates and Wiles made their breakthrough: Iwasawa’s Main Conjecture for imaginary quadratic fields. Theorem 1 (Coates and Wiles, 1977 [2]). Suppose 퐸 has • Using a quite different Euler system of Heegner complex multiplication. If 퐸(ℚ) is infinite, then 퐿(퐸, 1) = 0. points, the combined results in the 1980s of Koly- In other words, if rank(퐸(ℚ)) > 0, then ord푠=1 퐿(퐸, 푠) > vagin, Gross–Zagier, Bump–Friedberg–Hoffstein, and 0. Murty–Murty showed that (BSD) holds if 퐸 is modular The key to the proof of Theorem 1 is the use of and ord푠=1 퐿(퐸, 푠) ≤ 1. Robert’s elliptic units to provide the crucial link between • The work of Wiles on modularity [6], completed by the algebraic and analytic sides of (BSD). Elliptic units are Taylor–Wiles [4] and Breuil–Conrad–Diamond–Taylor global units in abelian extensions of imaginary quadratic [1], showed in the 1990s that every elliptic curve over fields, defined by analytic functions, so they live in both ℚ is modular. Hence we have the following result for the algebraic and analytic worlds. all elliptic curves over ℚ, with or without complex Suppose 퐸 has complex multiplication, and let 퐾 multiplication. be the imaginary quadratic field End(퐸) ⊗ ℚ. Suppose ℤ Theorem 2. If ord 퐿(퐸, 푠) ≤ 1, then rank(퐸(ℚ)) = 푝 ≥ 5 is a prime where 퐸 has good reduction, and 푠=1 ord 퐿(퐸, 푠). 푝 factors as 푝 = 휋 ̄휋∈ End(퐸) ⊂ 퐾. For 푛 ≥ 1 let 푠=1 퐸[휋푛] ⊂ 퐸(퐾)̄ denote the kernel of the endomorphism Along with some Iwasawa-theoretic results due to Kato 푛 푛 휋 , let 퐾푛 ∶= 퐾(퐸[휋 ]), and let (2004) and Skinner and Urban (2014), this is currently the best result in the direction of the Birch and Swinnerton- Γ ∶= Gal(퐾 /퐾) ≅ (ℤ/푝푛ℤ)×. 푛 푛 Dyer conjecture for elliptic curves over ℚ. Since 퐾푛 is an abelian extension of 퐾, there is a subgroup × 풞푛 ⊂ 퐾푛 of elliptic units. We also let 푈푛 denote the units References in the ring of integers of the completion of 퐾푛 at the [1] C. Breuil, B. Conrad, F. Diamond, and R. Taylor, On the unique prime above 휋. modularity of elliptic curves over ℚ: wild 3-adic exercises, Coates and Wiles constructed a “logarithmic derivative” J. Amer. Math. Soc. 14 (2001), 843–939. MR1839918 푛 homomorphism 휓푛 ∶ 푈푛 → 퐸[휋 ] and computed that [2] J. Coates and A. Wiles, On the conjecture of Birch and Swinnerton-Dyer, Invent. Math. 39 (1977), 223–251. 퐿(퐸, 1) 푛 휓푛(풞푛) = 퐸[휋 ]. MR0463176 Ω [3] K. Rubin, The “main conjectures” of Iwasawa theory for Under a mild additional assumption on 푝, they showed imaginary quadratic fields, Invent. Math. 103 (1991), 25–68. that the group of Γ푛-equivariant homomorphisms MR1079839 푛 푛 [4] C. Skinner and E. Urban, The Iwasawa main conjectures for HomΓ푛 (푈푛, 퐸[휋 ]) is cyclic of order 푝 , generated by 휓푛, and therefore GL2, Invent. Math. 195 (2014), 1–277. MR3148103 [5] R. Taylor and A. Wiles, Ring-theoretic properties of cer- 푛 푛 퐿(퐸,1) (1) HomΓ푛 (푈푛/풞푛, 퐸[휋 ]) ≅ ℤ/(푝 , Ω )ℤ. tain Hecke algebras, Ann. of Math. 141 (1995), 553–572. Now suppose 퐸(ℚ) is infinite. Fix a point 푃 ∈ 퐸(ℚ) of MR1333036 [6] A. Wiles, Modular elliptic curves and Fermat’s last theorem, infinite order, and for every positive integer 푛 choose a 푛 Ann. of Math. 141 (1995), 443–551. MR1333035 point 푄푛 ∈ 퐸(퐾)̄ such that 휋 (푄푛) = 푃. The “Kummer map” that sends 휎 ∈ Gal(퐾/퐾̄ 푛) to 휎(푄푛) − 푄푛 defines a homomorphism Barry Mazur 휅 ∈ Hom (Gal(퐾/퐾̄ ), 퐸[휋푛]). 푛 Γ푛 푛 Andrew Wiles’s Work on the Main Conjecture Using class field theory, 휅푛 induces a homomorphism of Iwasawa Theory 푛 What a joy it was to work with Andrew on the Main 휅푛̃ ∈ HomΓ푛 (푈푛/풞푛, 퐸[휋 ]), Conjecture of Iwasawa theory over ℚ, and how great that and Coates and Wiles showed that there is an integer 푘, 푛−푘 Andrew went on to establish it, more generally, for totally independent of 푛, such that 휅푛̃ has order 푝 for all 푛 ≥ 푘. real fields. Comparing this with (1) as 푛 grows proves Theorem 1. The ideas in the proof of Theorem 1, along with Barry Mazur is Gerhard Gade University Professor at Harvard methods Wiles developed for his work on Iwasawa’s Main University. His e-mail address is [email protected].

March 2017 Notices of the AMS 219 푎 The basic idea behind the Main Conjecture might be 휁 ↦ 휁 for any 휁 ∈ 휇푛. Any automorphism of 휇푛 extends thought of as having, as a starting place, the classical uniquely to an automorphism of the field ℚ[휇푛], giving analytic formulas of number theory, such as Dirichlet’s us canonical isomorphisms: Class Number Theorem for quadratic imaginary fields: ∗ ≃ ≃ (ℤ/푛ℤ) ⟶ 퐴푢푡(휇푛) ⟶ Gal(ℚ[휇푛]/ℚ). 퐿(휒, 1) ℎ(−푑) = , Fixing a prime 푝 (for simplicity, suppose 푝 > 2) and letting 2휋 푤√푑 푛 run through powers of 푝, form the 푝-cyclotomic tower where 휒 is the quadratic Dirichlet character cutting out the of (abelian Galois) extensions quadratic imaginary field ℚ(√−푑) of (integer) conductor ℚ ⊂ ℚ[휇 ] ⊂ ℚ[휇 2 ] ⊂ ℚ[휇 3 ] ⊂ … −푑 < 0; ℎ(−푑) is the order of the ideal class group of that 푝 푝 푝 ∞ field; 푤 is the number of roots of unity in it; and 퐿(휒, 푠) and put ℚ[휇푝∞ ] ∶= ⋃휈=1 ℚ[휇푝휈 ]. We have that 휈 ∗ ∗ is the Dirichlet 퐿-function. One of the striking aspects Gal(ℚ[휇푝∞ ]/ℚ) = lim (ℤ/푝 ℤ) = ℤ푝 , of this formula is that the left-hand side is analytic; the 휈→∞ ∗ right-hand side is arithmetic. the latter, ℤ푝 , being the profinite topological group of ∗ The “Main Conjecture” is not a misnomer: for any 푝-adic units, which decomposes as a product, ℤ푝 = ∗ ∗ prime number 푝 the conjecture asserts a fundamental F푝 ×{1+푝ℤ푝}, where F푝 ≃ Gal(ℚ[휇푝]/ℚ) is a cyclic group ∗ relationship that establishes a close tie between of order 푝−1, and the subgroup Γ ∶= {1+푝Z푝} ⊂ ℤ푝 is an • the 푝-adic 퐿-functions of a number field 푘—these being infinite cyclic pro-푝-group. A neat topological generator ∗ 푝-adic analytic objects closely related to the classical to choose for Γ is the 푝-adic unit 훾 ∶= (1 + 푝) ∈ Γ ⊂ ℤ푝 . (complex) 퐿-functions of 푘, The field ℚ[휇푝∞ ] is generated by two linearly disjoint • the deep arithmetic of that number field—namely, the subfields: ℚ[휇푝] and a field, call it ℚ∞, Galois over ℚ with 푝-primary parts of ideal class groups of certain abelian Galois group ∗ extensions of 푘. Gal(ℚ∞/ℚ) = Γ ∶= 1 + 푝Z푝 ⊂ Z푝 . Slightly more specifically, the Main Conjecture identifies The topological group Γ is our “group of operators.” the zeroes of 푝-adic 퐿-functions of a number field 푘 with the eigenvalues of an operator on a 푝-adic vector The Vector Space Containing Basic Arithmetic space constructed from the 푝-primary parts of ideal Data Related to Number Fields class groups of the abelian extensions of 푘 alluded to above or constructed from some closely related arithmetic Let 푘 ⊂ 퐾 be number fields, contained in 풞, linearly disjoint from ℚ , with 퐾 totally real, and 퐾/푘 a cyclic objects.1 ∞ Galois extension. Put 퐾 = 퐾 ⋅ ℚ ⊂ 풞. So Gal(퐾 /퐾) = This puts the conjecture somewhat in the spirit of a ∞ ∞ ∞ Gal(ℚ /ℚ) = Γ. suggestion attributed to Hilbert and Pólya that a complex ∞ Let 퐿 denote the maximal unramified pro-푝 abelian 퐿-function of a number field might arise naturally as ∞ extension of 퐾 . Let 푋 ∶= Gal(퐿 /퐾 ), which, since it is a the characteristic series attached to a certain unbounded ∞ ∞ ∞ projective limit of 푝-abelian groups, we can view naturally operator on a (naturally defined) Hilbert space. (Their as a ℤ -module. Also, Γ = Gal(퐾 /퐾) acts naturally (and suggestion, though, goes further by noting that if these 푝 ∞ ℤ -linearly) on 푋. The action of an element in Γ on operators were self-adjoint, this would relate well to 푝 푋 is defined by lifting it to an element in Gal(퐿 /퐾) the Riemann Hypothesis.) It also is in the spirit of the ∞ and then noting that conjugation by that lifted element classical theory of 퐿-functions attached to varieties over doesn’t depend on the lifting and induces a well-defined finite fields, for such an 퐿-function can also be thought of automorphism of 푋. as the characteristic polynomial of the Frobenius operator ̄ One knows that 푉 ∶= 푋 ⊗ℤ푝 ℚ푝 is a finite-dimensional on an appropriate étale cohomology group. ∗ To outline an example of the Main Conjecture, we’ll vector space. If 휒 ∶ Gal(퐾/푘) ↪ 풞 is an odd (faithful) discuss the relevant group of operators, vector space, 푝- character cutting out the field extension 퐾/푘, we will be adic 퐿-functions, and the method for the construction of considering the action of Γ on the 휒-part of 푉, i.e., on 푉휒 ∶= {푣 ∈ 푉 | 푔(푣) = 휒(푔) ⋅ 푣}. We want to get as full an the relevant arithmetic objects. understanding of the 푉휒 as possible and specifically the eigenvalues of 1 − 훾 on 푉휒 where 훾 ∈ Γ is a topological The 푝-Cyclotomic Tower and the Fundamental generator. Operator in the Main Conjecture Let 푛 ≥ 1. Consider the finite field extension ℚ[휇푛]/ℚ 퐿-Functions obtained by adjoining the group of 푛th roots of unity, By interpolating special values of the classical complex 2휋푖푎/푛 ∗ 휇푛 ∶= {푒 | 푎 = 0, 1, … , 푛 − 1} ⊂ 풞 , 퐿-functions, Kubota and Leopoldt defined the 푝-adic 퐿- to the field ℚ of rational numbers. The group (ℤ/푛ℤ)∗ is functions over the field 푘 = ℚ, and subsequently Deligne and Ribet defined them over totally real fields 푘. Briefly, canonically isomorphic to the group 퐴푢푡(휇푛) of automor- in the context above, let 휁 (휎, 푠) denote the partial phisms of the cyclic group 휇푛, this isomorphism being 푘 defined by sending 푎 ∈ (ℤ/푛ℤ)∗ to the automorphism zeta-function of 푘 associated to elements 휎 ∈ Gal(퐾/푘). The special values 휁푘(휎, 1 − 푛) are rational numbers, as 1For example, from abelian extensions of 푝-power degree unrami- proved by Klingen and Siegel. Let 휓 be a one-dimensional ∗ −1 푛 fied over those alluded-to extensions of 푘. character over 푘 with values in ℚ푝 and 휒푛 ∶= 휓 휔

220 Notices of the AMS Volume 64, Number 3 where 휔 is the Teichmüller character. For any integer References 푛 ≥ 1 one puts [1] B. Mazur and A. Wiles, Class fields of abelian extensions of ℚ, Invent. Math. 76 (1984), 179–330. MR742853 퐿푝(1 − 푛, 휓) [2] B. Mazur, How can we construct abelian extensions of number fields?, Bulletin of the A.M.S. 48 (2011), 155–209. = 휒 (휎)휁 (휎, 1 − 푛) ⋅ (1 − 휒 (푃)푁(푃)1−푛), ∑ 푛 푘 ∏ 푛 MR2774089 휎∈Gal(퐾/푘) 푃 |푝 [3] A. Wiles, The Iwasawa conjecture for totally real fields, Ann. of Math. 131 (1990), 493–540. MR1053488 where “∏푃 |푝” is taken over all primes 푃 of 푘 lying above 푝, and 푁(푃) is the norm of the ideal 푃. These special values, {1 − 푛 ↦ 퐿푝(1 − 푛, 휓) ∈ ℚ̄ 푝}, Mirela Çiperiani interpolate to produce a 푝-adic analytic function 퐿푝(푠, 휓) on ℤ푝 if 휓 is not trivial and on ℤ푝 − {1} with a simple Solvable Points on Genus One Curves pole at 푠 = 1 when 휓 is trivial. Moreover, there is a In the spring of 2002, as a graduate student at Princeton, unique power series 퐺휓(푇) ∈ ℤ̄푝[[푇]] such that for the I approached Andrew Wiles to ask about the possibility ∗ topological generator 훾 ∈ Γ ⊂ ℤ푝 (viewed as an element of working under his supervision. He suggested that ∗ of ℤ푝 ) we have I think about a problem that was thrillingly natural and beautiful: Does every genus one curve, defined over 푠 퐿푝(푠, 휓) = 퐺휓(훾 − 1). the rational numbers ℚ, have a point over some solvable extension of ℚ? The Main Conjecture, then, for 휒 odd and 푝 > 2 Solvable extensions of ℚ—Galois extensions with solv- identifies the characteristic polynomial of 훾 acting on 푉휒 able Galois group—are, concretely, fields contained in as described above, with the Weierstrass polynomial of −1 −1 root extensions of ℚ, i.e., in fields obtained by starting the power series 퐺휓(훾(1 + 푇) − 1) where 휓 = 휒 휔. with ℚ and considering successive extensions of the That is, as Andrew proves [3], the eigenvalues of 훾 acting form F( 푛 훼)/F for some 훼 ∈ F and 푛 ∈ ℕ. This relation- 휒 √ on 푉 are identified with the zeroes of 퐿푝(푠, 휓). ship between solvability of the Galois group and iterated adjunction of roots is the Galois-theoretic criterion for Method solvability by radicals of a polynomial equation in one The essential issue in proving the Main Conjecture is to variable. Thus, as shown by Abel and Galois, equations construct (by means of Galois representations attached to in one variable with coefficients in ℚ and degree at modular forms) as many abelian unramified extensions least 5 need not be solvable by radicals. This connection as would be predicted from the analytic side of the to classical Galois theory is one of the appeals of the conjectured formula. A version of the classical analytic problem. formula then allows one to conclude the conjecture. When An obvious stumbling block is that, while this is evi- the field 푘 is ℚ, we did this [1] by a two-step approach, dently a diophantine equations problem, it is much less tangible than that of solving a quintic: we don’t have a starting from a marvelous idea of Ken Ribet relating generic way of writing the equations that describe genus divisibility of the 푝-adic 퐿-function by 푝 to a similar one curves defined over ℚ. A priori, a genus one curve divisibility of the order of a specific ideal class group defined over ℚ is cut out by several homogeneous poly- by 푝. (For a leisurely discussion of Ribet’s method and nomial equations with rational coefficients. The famous connection with the earlier work of Herbrand, see [2].) Weierstrass cubic equations 푦2푧 = 푥3 + 푎푥푧2 + 푏푧3 (here Briefly, the 푝-adic 퐿-function 퐿푝(푠, 휒) (times an elementary 푎, 푏 ∈ ℚ such that 푥3 + 푎푥 + 푏 has no repeated roots) de- nonzero factor) occurs as the constant term of a 푝-adic scribe elliptic curves, i.e., genus one curves which do have Eisenstein series (of 푝-adic weight determined by the a point over their field of definition, namely, (0 ∶ 1 ∶ 0). value of 푠). Whenever the 푝-adic 퐿-function vanishes, However, each genus one curve is a torsor for an elliptic this Eisenstein series has constant term zero and can be curve, its Jacobian; i.e., the Jacobian acts on the genus one shown, therefore, to be congruent modulo arbitrarily high curve, and over ℚ that action becomes freely transitive. powers of 푝 to cuspidal eigenforms. Moreover, since these Conversely, every torsor for an elliptic curve E is a genus cuspidal modular eigenforms are congruent modulo a one curve with Jacobian E. In light of this correspon- high power of 푝 to Eisenstein series, their associated 푝-adic dence, genus one curves over ℚ with fixed Jacobian E are Galois representations are extremely well behaved modulo parametrized by a (Galois) cohomology group, the Weil– 1 those powers of 푝 and can be seen to cut out larger and Châtelet group H (Gal(ℚ/ℚ),̄ E(ℚ))̄ . So now we dispense with defining equations of genus one curves and work larger unramified 푝-power abelian extensions of 퐾∞. The procedure employed by Wiles [3] for the totally real case is with elements of the Weil–Châtelet group. a good deal more delicate than in the case of 푘 = ℚ, in that In 2006, in joint work with Wiles, we showed that one now uses the Galois representations of eigenforms under certain restrictions, Wiles’s original question has a positive answer, as stated momentarily. The removal of on Hilbert–Blumenthal moduli spaces to construct the these restrictions is the subject of ongoing joint work. desired unramified abelian extensions. Andrew works systematically over the relevant weight space constructing Mirela Çiperiani is associate professor of mathematics at the the appropriate cuspidal Λ-adic Hilbert modular forms, University of Texas at Austin. Her e-mail address is mirela@ using Hida’s theory, and [3] proves much more. math.utexas.edu.

March 2017 Notices of the AMS 221 Theorem ([ÇW]). Let C be a genus one curve defined over sufficiently many cohomology classes ramified at ℚ, with Jacobian E, such that primes in 풬 depends on the finiteness of all the

(1) C has a point defined over the ℓ-adics ℚℓ for all 푝-primary components of the Tate–Shafarevich rational primes ℓ; and group of E/ℚ. This is only known for elliptic (2) one of the following conditions holds: curves E/ℚ of analytic rank less than or equal to (a) the analytic rank of E/ℚ is less than or equal 1 (by the modularity theorem, and the combined to 1 or and celebrated work of Kolyvagin, Gross–Zagier, (b) E is semistable over ℚ. Bump–Friedberg–Hoffstein, and Murty–Murty). Then C has a point over a solvable extension of ℚ. When conditions (1) and (2b) hold, we can still find nontrivial Heegner points, using work of Cornut-Vatsal, The analytic rank of an elliptic curve E is the order by viewing C as a genus one curve over a nontrivial of vanishing of its 퐿-series 퐿(E, 푠) at 푠 = 1. As for extension of ℚ. We now attempt to apply the unramified semistability, after lifting an elliptic curve from ℚ to ℤ and under ramified principle. However, while this field exten- then reducing modulo a prime, it may remain nonsingular sion enables us to construct ramified Kolyvagin classes, or it may acquire a node or cusp. Semistability disallows because of issue (ii) we are not able to see that we have cusps. sufficiently many of them. I will now describe two fundamental ideas of Wiles It is in circumventing the potential existence of an which provide the frame of the proof. The first is referred infinite 푝-primary component of the Tate–Shafarevich to as the “unramified under ramified principle.” Fix an group that we need the second idea. It is referred to as elliptic curve E defined over ℚ. Genus one curves C defined the “patching method.” A similar method was used in over ℚ with E as their Jacobian, and with points over the proof of the modularity theorem. Wiles’s idea is the ℚℓ for every rational prime ℓ, form a subgroup of the following. We construct as many ramified classes as we Weil–Châtelet group. (This subgroup contains the Tate– can for each in an infinite sequence of field extensions Shafarevich group, which consists of genus one curves {퐹푛}. Observe that in order to do this we must choose that have both ℓ-adic and real points.) We in fact work with unrelated sets of primes 풬푛 for the fields 퐹푛, all with the their preimages in the Selmer group. These Selmer classes same cardinality. We consider groups 푀푛 of cohomology become trivial after an extension of ℚ unramified at all classes over 퐹푛 ramified at primes in 풬푛 (actually, 푀푛 but a finite set of primes ℓ. According to the unramified is viewed as a module over a ring related to 퐹푛). There under ramified principle, if we find a finite set of primes is no natural containment between these modules, but 풬 obeying certain conditions and if we have sufficiently each of them contains all the classes over ℚ that we want many cohomology classes which are ramified at primes to capture. However, our Kolyvagin classes generate a ′ ′ in 풬, then under the cohomology group structure they submodule 푀푛 ⊆ 푀푛, and for no 푛 can we see that 푀푛 will generate all the unramified classes. contains the desired classes. By considering their mod- We will apply this principle in the case where the genus ule structure (and ignoring their content), we construct one curve C satisfies conditions (1) and (2a) of our theorem injective maps 푀푛 → 푀푛+1. This gives rise to a module by constructing ramified classes which correspond to lim 푀 over an Iwasawa algebra. Miraculously, a structure ⎯⎯⎯⎯⎯→ 푛 genus one curves with points over some solvable extension theorem from Iwasawa of ℚ. These ramified classes will then generate a group theory shows us that containing all genus one curves that have the same I developed each of our genus Jacobian as C and satisfy conditions (1) and (2a). Hence one curves C satisfy- C has a solvable point. Thus the task is to choose the enormous ing conditions (1) and set 풬 and to construct sufficiently many ramified classes ′ (2b) lies in some 푀푛 for which have solvable points. This is achieved by using admiration of some 푛. Thus 퐶 has a the cohomology classes constructed by Kolyvagin. The solvable point. construction of these classes makes use of Heegner Wiles for his Working with Wiles points. These points are defined on modular curves and has been a wonderful pushed forward to the elliptic curve via its modular generosity and experience. At the be- parametrization—whose existence is known by the work modesty, and awe ginning, when I barely of Wiles extended by Breuil, Conrad, Diamond, and Taylor. knew what an ellip- The unramified under ramified principle is sufficient for his ability to tic curve was, I felt to prove the theorem in the case when the analytic rank lucky but humbled and of E/ℚ is less than or equal to 1. It is not sufficient to see to the heart of daunted to be en- prove it in general, for the following two reasons: trusted with such a (i) If the analytic rank of E/ℚ is greater than 1, then the matter. fantastic problem. I the relevant Heegner points are trivial. Hence, in had to rapidly learn this case we cannot construct enough nontrivial the background material needed to begin thinking about Kolyvagin classes. the problem and to understand Wiles’s proposal to use (ii) The set of primes 풬 depends on the order of the unramified under ramified principle. Later, at times the curve C viewed as an element of the Weil– when the problem seemed impossible, I was buoyed by Châtelet group of E/ℚ, and the existence of the Wiles’s confidence that we could solve it. I developed

222 Notices of the AMS Volume 64, Number 3 enormous admiration of Wiles for his generosity and ℓ. Wiles’s theorem “lifts modularity” in that it asserts that modesty, and awe for his ability to see to the heart of the if 휌̄ is modular, then the lift 휌 of 휌̄ is also modular. matter—feelings that stay with me today. Wiles’s theorem allows for representations 휌 where ℤ푝 is replaced by more general 푝-adic rings. In particular, References it allows for homomorphisms 휌 ∶ 퐺ℚ → GL2(ℚ푝); a [ÇW] M. Çiperiani and A. Wiles, Solvable points on genus one continuity argument associates to such a 휌 a residual curves, Duke Math. J. 142 (2008), 381–464. MR2412044 representation 휌̄ ∶ 퐺ℚ → GL2(픽푝). Much earlier results of Shimura and Deligne attach to each eigenform 푓 and Chandrashekhar Khare embedding 휄 ∶ ℚ → ℚ푝 a Galois representation

휌푓,휄 ∶ 퐺ℚ → GL2(ℚ푝). Modularity of GL2 Galois Representations The spirit of Wiles’s modularity lifting theorem is: if and the Work of Andrew Wiles 휌̄ ≅ 휌푓,휄̄ for some eigenform 푓, then 휌 ≅ 휌푔,휄′ for some The work of Andrew Wiles on the modularity of elliptic eigenform 푔 (not necessarily the same as 푓). More about curves provided some of the key ideas and techniques this theorem and its context can be found in Darmon’s which led to the proof of Serre’s modularity conjecture. lecture in this issue and in the article by Thorne. Moreover, and equally importantly, it psychologically Wiles’s strategy for proving his marvelous modularity made it possible to imagine that there could be a strat- lifting theorem can very roughly be paraphrased as follows. egy to prove results that before Wiles’s work seemed The kernel of the mod 푝 reduction map GL2(ℤ푝) → GL2(픽푝) completely inaccessible. is prosolvable (the inverse limit of solvable groups), which To retrace the path from Wiles’s work [5] on modularity enables the use of duality theorems in Galois cohomology of elliptic curves to the proof of Serre’s conjecture [2], we and congruences between modular forms to bootstrap describe briefly Wiles’s modularity lifting theorem and the modularity property from 휌̄ to 휌. his strategy for proving it. One spectacular application of Wiles’s modularity lift- Let 퐺ℚ = Gal(ℚ/ℚ) be the Galois group of an algebraic ing theorem was the proof of the modularity of semistable closure ℚ of ℚ. Wiles’s modularity lifting theorem is elliptic curves defined over ℚ (and hence Fermat’s Last about 2-dimensional 푝-adic Galois representations for a Theorem!). Elliptic curves are often encountered as the prime 푝 > 2. These are certain homomorphisms projective curves defined by an (affine) Weierstrass equa- 2 3 휌 ∶ 퐺ℚ → GL2(ℤ푝), tion: 푦 = 푥 + 푎푥 + 푏 for constants 푎 and 푏. The curve is defined over ℚ if 푎, 푏 ∈ ℚ. The points on an elliptic curve including those that arise from the action of 퐺ℚ on the torsion points of an elliptic curve defined over ℚ. Wiles 퐸 form an abelian group, with the addition law defined by proved [5], [4] that under certain hypotheses on 휌 and on rational functions in 푥 and 푦; the identity element of the its residual representation group is the unique point at infinity (the point (0 ∶ 1 ∶ 0) in projective coordinates). The torsion points 퐸[푁] on 퐸 휌̄ ∶ 퐺ℚ → GL2(픽푝) of (positive integer) order 푁 form a group isomorphic to (the reduction of 휌 modulo 푝), the representation 휌 is ℤ/푁ℤ × ℤ/푁ℤ. If the elliptic curve is defined over ℚ, then modular. the Galois group 퐺ℚ acts on 퐸[푁] by its action on the The hypotheses on the residual representation 휌̄ in coordinates of the points. The 푝-adic Tate module 푇푝퐸 of Wiles’s theorem require that 휌̄ be odd (the image of com- 퐸 is then the inverse limit of the groups 퐸[푝푛] of 푝-power 푛 2 plex conjugation has eigenvalues +1 and −1), irreducible, torsion points: 푇푝퐸 = lim 퐸[푝 ] ≅ ℤ푝. The action of 퐺ℚ ←⎯⎯⎯⎯⎯푛 and modular. For 휌,̄ being modular means that there is a on 푇푝퐸 determines a 푝-adic Galois representation cuspidal modular eigenform (an analytic function on the 휌 ∶ 퐺 → GL (ℤ ). complex upper-half-plane with many symmetries) 퐸,푝 ℚ 2 푝 ∞ The residual representation 휌퐸,푝̄ of 휌퐸,푝 is just the rep- 2휋푛휏 푓(휏) = ∑ 푎푛(푓)푒 resentation of 퐺ℚ on the group of 푝-torsion points 푛=1 2 퐸[푝] ≅ 픽푝. The modularity of an elliptic curve can be such that for all but finitely many primes ℓ, the Fourier interpreted as the 푝-adic Galois representation 휌퐸,푝 being coefficient 푎ℓ(푓) can be matched with the trace of 휌̄ modular for some prime 푝 (equivalently, all primes 푝). evaluated on a Frobenius element in 퐺ℚ for the prime ℓ. The application of the modularity lifting theorem to More precisely, the Fourier coefficients 푎푛(푓) are all alge- modularity of elliptic curves over ℚ comes by taking 휌 = braic integers (so belong to ℚ) and for some embedding 휌퐸,푝. For 푝 = 2 or 3 the image of the representation 휌퐸,푝 휄 ∶ ℚ ↪ ℚ푝 of ℚ into an algebraic closure ℚ푝 of ℚ푝, for is prosolvable (the inverse limit of finite solvable groups). almost all primes ℓ the image of 휄(푎ℓ(푓)) in the residue This is one of the two places in his argument where Wiles field 픽푝 of ℚ푝 equals the trace of 휌̄ on a Frobenius element uses lucky accidents which happen for small primes. He for ℓ. Similarly, 휌 is modular if there is an eigenform 푓 uses results of Langlands and Tunnell which imply that and an embedding 휄 such that for almost all primes ℓ, an odd irreducible representation 휌̄ ∶ 퐺ℚ → GL2(픽3) is modular. This uses the solvability of GL (픽 ) for 푝 = 3 휄(푎ℓ(푓)) equals the trace of 휌 on a Frobenius element for 2 푝 (indeed, PGL2(픽3) is isomorphic to 푆4) and the fact that Chandrashekhar Khare is professor of mathematics at UCLA. His the map GL2(ℤ푝) to GL2(픽푝) splits for 푝 = 3; both these e-mail address is [email protected]. statements are false for 푝 > 3!

March 2017 Notices of the AMS 223 At this point, the modularity lifting theorem allows these descent results imply that modularity of 휌̄ follows

Wiles to conclude modularity of semistable 퐸 unless by showing that 휌|̄ 퐺퐹 arises from a Hilbert modular form 휌퐸,3̄ is reducible. To deal with the case when 휌퐸,3 is for a solvable, totally real extension 퐹/ℚ. reducible, Wiles plays a “3-5” trick. He constructs another To study general representations 휌̄ as in Serre’s con- semistable elliptic curve 퐸′ over ℚ, whose mod 3 repre- jecture, Taylor considered moduli spaces over ℚ whose sentation surjects onto GL2(픽3) and such that the mod points over a number field 퐾 correspond to abelian va- 5 representations arising from 퐸 and 퐸′ are isomorphic rieties over 퐾 (with real multiplication) which give rise and irreducible. Here again the use of the small prime to 휌|̄ 퐺퐾 , and at an auxiliary place ℓ ≠ 푝 give rise to a 5 is vital, as the moduli space he considers of elliptic mod ℓ dihedral representation. The latter are known to curves with level 5 structure isomorphic to 퐸[5] turns be modular by an old result of Hecke. Taylor used a out to be of genus zero, and in fact the projective line theorem of Moret–Bailly to produce points of the moduli over ℚ, and thus has many rational points. Then applying spaces he considered over totally real fields 퐹 (but which the modularity lifting theorem he deduces that 휌퐸′,3 is could not be guaranteed to be solvable over ℚ). Then modular. This implies that 휌퐸′,5 is also modular, hence modularity lifting theorems for representations of 퐺퐹

휌퐸,5̄ = 휌퐸̄ ′,5 is modular, and then by another application yield that 휌|̄ 퐺퐹 arises from a Hilbert modular form over of the theorem, that 휌퐸,5, and therefore 퐸, is modular. 퐹. This may be regarded as a potential version of Serre’s Wiles’s work was generalized by Breuil, Conrad, Dia- conjecture. Together with automorphic descent for cyclic mond, and Taylor (2001) to prove the modularity of all prime degree extensions of totally real number fields, this elliptic curves over ℚ. Recently, Freitas, Hung, and Siksek led to the meromorphic continuation of the Hasse–Weil (2015) proved the modularity of elliptic curves defined 퐿-series attached to elliptic curves over all totally real over all real quadratic fields, using a “3-5-7” trick. fields. Before the work of Wiles there was no path from a 푝-adic To proceed along these lines to prove Serre’s conjecture Galois representation to a modular form. His completely in the general case, it would be necessary either to new method of modularity lifting showed that if only a find a general procedure to show existence of totally small quotient of a 푝-adic representation arose from a real solvable points on geometrically irreducible smooth modular form, then the entire Galois representation did. projective varieties of general type over ℚ or to prove This has turned out to be a very powerful method to nonsolvable automorphic descent for Hilbert modular prove modularity of Galois representations! forms.

Conjectures of Serre and Fontaine–Mazur Proof of Serre’s Conjecture My proof with Wintenberger of Serre’s conjecture [2] took Serre conjectured [3] that for any 휌̄ ∶ 퐺ℚ → GL2(픽푝) that is continuous, odd, and irreducible, there is an eigenform a different path and used as a starting point results of Tate and Serre that proved Serre’s conjecture for 푓 such that 휌̄ ≃ 휌푓,휄̄ ; i.e., 휌̄ is modular. J.-M. Fontaine and B. Mazur conjectured [1] that if 휌 ∶ representations 휌̄ of residue characteristic 푝 ≤ 3 with limited ramification, with no a priori assumptions on the 퐺ℚ → GL2(ℚ푝) is continuous, odd, irreducible, unramified outside a finite set of primes, and potentially semistable image of 휌.̄ Tate and Serre proved that any representation with Hodge–Tate weights (푎, 푏), say 푎 ≤ 푏, then the 휌̄ ∶ 퐺ℚ → GL2(픽푝) unramified outside 푝 and with 푝 ≤ 3 is cyclotomic twist 휌(−푎) is modular. reducible. This is another instance of the magic of small Wiles’s modularity lifting results were in the direction primes! of showing that the conjecture of Serre implies that of Our proof measured the complexity of the represen- Fontaine–Mazur. These modularity lifting results were tation 휌̄ in terms of its ramification at 푝 (Serre’s weight) improved in various crucial ways by Diamond, Fujiwara, and the ramification away from 푝 (its Artin conductor 푁). and Kisin, including generalizations with the base field A double induction on (푝, 푁) was used to reduce Serre’s ℚ replaced by a totally real field 퐹. In an important conjecture to the results of Tate and Serre. A principle of the proof is to use potential modularity to produce development, Skinner and Wiles lifted the condition that compatible systems of representations that lift 휌̄ such as 휌̄ is irreducible in the case when the lift 휌 is ordinary at would exist if it were known that 휌̄ were modular. 푝. All these developments were crucial in our later work Our proof of Serre’s conjecture owes its existence to on Serre’s conjecture. the modularity lifting theorems initiated by Wiles, a tool to attack modularity which was as powerful as it was Potential Version of Serre’s Conjecture unexpected when it was introduced, and also to Wiles’s The automorphic descent results of Saito and Shintani, and prime switching trick in his proof of the modularity of Langlands are an important ingredient in the applications elliptic curves. of modularity lifting theorems. These show that given a cyclic extension of totally real number fields 퐾/퐹 of References prime degree and a cuspidal automorphic representation [1] Jean-Marc Fontaine and Barry Mazur, Geometric Ga- of 휋 of GL2(픸퐾) which is a discrete series at the infinite lois representations, Elliptic Curves, Modular Forms, & places and invariant under 휎 ∈ Gal(퐾/퐹), then 휋 is the Fermat’s Last Theorem (Hong Kong, 1993), Ser. Number base change of a cuspidal automorphic representation of Theory, I, Internat. Press, Cambridge, MA, 1995, pp. 41–78. GL2(픸퐹). When combined with modularity lifting results, MR1363495

224 Notices of the AMS Volume 64, Number 3 [2] Chandrashekhar Khare and Jean-Pierre Wintenberger, Passing to spectra, one can think of Spec 핋휚,푆 as a closed Serre’s modularity conjecture (I), Invent. Math. 178 (2009), subspace of Spec 푅휚,푆: it is the locus of the modular no. 3, 485–504. MR2551763 Galois representations inside the space of all Galois [3] Jean-Pierre Serre, Sur les représentations modulaires de de- representations. In particular, the existence of this map gré 2 de Gal(Q/Q), Duke Math. J. 54 (1987), no. 1, 179–230. expresses the existence of Galois representations attached MR885783 to modular forms, itself a highly nontrivial fact. Diagrams [4] Richard Taylor and Andrew Wiles, Ring-theoretic proper- ties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), such as (4) have achieved an iconic status in algebraic no. 3, 553–572. MR1333036 number theory. [5] Andrew Wiles, Modular elliptic curves and Fermat’s last The truth of the modularity lifting theorem is implied theorem, Ann. of Math. (2) 141 (1995), no. 3, 443–551. by the much more refined statement that the map (4) is an MR1333035 isomorphism. This goes some way towards explaining the importance of the universal deformation ring 푅휚,푆, which Jack Thorne was first introduced by Mazur. A large part of Wiles’s fundamental 1995 work is taken up with introducing the Modularity of 푛-Dimensional Galois Repre- tools necessary to effectively study the map (4), putting sentations him in a position to prove ‘푅 = 핋’ in many cases. One tool that has turned out to be surprisingly versatile More than twenty years have passed since Andrew Wiles is the Taylor–Wiles method, introduced in the companion proved the first modularity lifting theorems and deduced paper [3]. Roughly speaking, this is an effective machine to Fermat’s Last Theorem as a consequence. His theorems fo- study (4) in the so-called minimal case where 푆 is as small cussed on proving the modularity of certain 2-dimensional as possible. To pass from this case to the general case, Galois representations. These were the homomorphisms Wiles introduced a numerical isomorphism criterion to (1) 휚 ∶ 퐺ℚ,푆 → GL2(ℤℓ) compare the situation for varying sets 푆. The verification arising from elliptic curves 퐸, 퐺ℚ,푆 being the Galois group of this criterion then involves delicate calculations in of the maximal extension of ℚ unramified outside some Galois cohomology and with modular forms. finite set of primes 푆. Despite this focus, the ideas of the The first modularity lifting theorems for Galois rep- proof have proved powerful and flexible enough that they resentations of dimension 푛 > 2 were proved by Clozel, are still a driving force of our understanding of much Harris, and Taylor in a paper published in 2008, which was more general Galois representations today. heavily influenced by an earlier unpublished manuscript The first key hypothesis imposed on 휚 is the modularity of Harris and Taylor. This was made possible thanks to of the residual representation the construction of 푛-dimensional Galois representations attached to modular forms on unitary groups, itself ini- (2) 휚 ∶ 퐺 → GL (픽 ) ℚ,푆 2 ℓ tiated by Clozel and Kottwitz and then studied in great that is the reduction of 휚 modulo ℓ. In other words, one detail by Harris and Taylor in their proof of the local assumes at the outset the existence of a cuspidal modular Langlands conjectures for GL푛, allowing one to write the eigenform 푛-dimensional analogue of the map 푅휚,푆 → 핋휚,푆. 푛 The Taylor–Wiles method was generalized by Clozel, (3) 푓 = ∑ 푎푛(푓)푞 푛≥1 Harris, and Taylor in order to prove a modularity lifting which is matched with 휚, in the sense that for almost theorem in the minimal case. However, such a restriction on ramification makes these theorems very difficult to all primes 푝, the Fourier coefficient 푎푝(푓) is congruent apply in interesting situations. The general case was modulo ℓ to the integer 푎푝(퐸) = 푝 + 1 − #퐸(픽푝). This allows the introduction of a first key player, the Hecke treated using a generalization of the numerical criterion of Wiles, but only conditional on a conjecture (referred algebra 핋휚,푆, which acts faithfully on a space of cuspidal modular forms, all of whose Fourier coefficients agree to colloquially as Ihara’s lemma) that remains unproven modulo ℓ with those of 푓 and which have level supported today. at the primes of 푆. Kisin had earlier developed a generalization of the The second key player is the universal deformation ring Taylor–Wiles method in his study of modularity for GL2, in a work published in 2009. He enlarged the diagram (4) of 휚, which we call 푅휚,푆. It is a complete local ring which is characterized by a universal property. For example, the to a diagram set of homomorphisms 푅휚,푆 → ℤℓ is in bijection with the ̂ ′ (5) ⨂푝∈푆푅휚,푝 → 푅휚,푆 → 핋휚,푆, set of equivalence classes of lifts 휚 ∶ 퐺ℚ,푆 → GL2(ℤℓ) of 휚 that one hopes to prove are modular. where each ring 푅휚,푝 is an object parameterizing deforma- These key players are related by a surjective ring tions of the restriction of 휚 to a decomposition group at homomorphism 푝 (in other words, a local Galois group 퐷푝 = Gal(ℚ푝/ℚ푝)). In this point of view, the geometry of the rings 푅 begins (4) 푅휚,푆 → 핋휚,푆. 휚,푝 to play a key role. The Taylor–Wiles–Kisin method finally Jack Thorne is a Clay Research Fellow and Reader in Number allows one to link the modularity of Galois representa- Theory at the University of Cambridge. His e-mail address is tions 휚1, 휚2 ∶ 퐺ℚ,푆 → GL푛(ℤℓ) which have the property [email protected]. that the representations 휚1|퐷푝 , 휚2|퐷푝 determine points on

March 2017 Notices of the AMS 225 the same irreducible component of Spec 푅휚,푝 for each [2] Richard Taylor, Automorphy for some 푙-adic lifts of au- prime 푝 ∈ 푆. tomorphic mod 푙 Galois representations. II, Publ. Math. Inst. Building on this, Taylor [2] made a detailed study of Hautes Études Sci. No. 108 (2008), 183–239. MR2470688 the irreducible components of certain ‘local’ deformation (2010j:11085) [3] Richard Taylor and Andrew Wiles, Ring-theoretic proper- rings and introduced a very surprising trick that allowed ties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), him to circumvent completely the conjectural Ihara’s no. 3, 553–572. MR1333036 (96d:11072) lemma and prove the first modularity lifting theorems for GL푛 without restriction on the permitted ramification of Photo Credits 휚 relative to 휚. Photo taken at the Newton Institute is courtesy of Ken These theorems have had spectacular applica- Ribet. tions. Combined with the earlier work of Harris, Photo of Christopher Skinner is courtesy of William Crow/ Shepherd-Barron, and Taylor, they implied that the even- Princeton University. dimensional symmetric power Galois representations Photo of Barry Mazur is courtesy of Jim Harrison. 푛−1 (6) Sym 휚 ∶ 퐺ℚ,푆 → GL푛(ℤℓ) Photo of Mirela Çiperiani is by C. J. Mozzochi, courtesy of the Simons Foundation. are potentially modular: there exists a number field 퐹/ℚ 푛−1 Photo of Chandrashekhar Khare is by David Weisbart. such that the restriction Sym 휚|퐺퐹 to the absolute Galois group of 퐹 is modular, in the sense of being associated to modular (or automorphic) forms on GL푛,퐹. These ideas in turn led to the proof of the Sato–Tate conjecture for elliptic curves over ℚ: Theorem 1. Let 퐸 be an elliptic curve over ℚ without complex multiplication. Then the quantities 푎푝(퐸)/2√푝 ∈ [−1, 1] are equidistributed as 푝 → ∞ with respect to the Sato–Tate measure 2 (7) √1 − 푡2 푑푡. 휋 There are many fruitful directions that remain to be explored. For example, can one show that the symmetric powers (6) are modular and not just potentially modular? In joint work with Clozel [1], I showed that the answer to this question is affirmative for 푛 ≤ 9. A major part of our proof is a generalization of an important modularity lifting theorem of Skinner and Wiles which applies to Galois representations 휚 for which the residual representation 휚 is reducible. This modularity lifting theorem also played a major part in the proof of Serre’s conjecture. The proof of the Skinner–Wiles theo- The work of Wiles rem employs many ideas which appear in has had a Andrew Wiles’s most famous works, such transforming as 푝-adic families of modular forms and effect. congruences between Eisenstein series and cuspidal modular forms, as well as many other ideas whose importance would become apparent only later: we mention in particular an emphasis on the geometry of Galois deformation rings. The work of Wiles has had a transforming effect on this corner of number theory, and his influence continues to be felt throughout the subject.

References [1] Laurent Clozel and Jack A. Thorne, Level raising and sym- metric power functoriality, II, Ann. of Math. (2) 181 (2015), no. 1, 303–359. MR3272927

226 Notices of the AMS Volume 64, Number 3 ABOUT THE AUTHORS

Christopher Skinner studied at Karl Rubin was Andrew Wiles’s Princeton with Andrew Wiles in first PhD student. the mid-1990s.

Christopher Skinner Karl Rubin

Barry Mazur delights in the Mirela Çiperiani completed memories of the times Andrew her PhD in 2006 under the and he had at Harvard, as supervision of Andrew Wiles. colleagues.

Barry Mazur Mirela Çiperiani

Using techniques and ideas Jack Thorne studied at Harvard that grew out of Wiles’s proof with Richard Taylor, himself a of Fermat’s Last Theorem, student of Andrew Wiles. Chandrashekhar Khare (with Jean-Pierre Wintenberger) proved Serre’s Conjecture.

Chandrashekhar Jack Thorne Khare

March 2017 Notices of the AMS 227

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Interview with New AMS President Kenneth A. Ribet

Education and the AMS Allyn Jackson Notices: First I’d like to ask you about education. You have been devoted to teaching all your career. You won two teaching awards at UC Berkeley, one early on in your time as a faculty member there, and another one just recently, in 2013. How do you see the role of the AMS in education? Ribet: The AMS has a big voice and can look at trends in education and make policy statements about practices we should be adopting in our classrooms. For example, many peo- ple are interested in strategies for active There is a real learning.1 Students to- day are uneasy going hunger among the into a classroom and general public for having a passive expe- Ken Ribet (back row, third from left) often meets for rience. It is a challenge fairly sophisticated breakfast with groups of students, such as this one to engage students and that met with him in November of last year (from left get them involved in mathematics. to right): Xuefei Lei, Rahul Malayappan, Aileen Ho, the course and the ma- Karam Samplay, Gauri Powale, Emily Hsiao, and terial. If the AMS has a role in shaping how people see such Genaro Hernandez Salgado. challenges, that would be a very positive development. One of the committee meetings I have enjoyed the most is that of the AMS Committee on Education, which It’s thoroughly illuminating. I am by training a com- meets in Washington DC, typically every October. The pletely traditional teacher. Nonetheless, the exposure that committee brings in speakers who talk about educational I’ve had to these different points of view has influenced practices. the way that I interact with my students. Currently I am teaching a class where the enrollment is about 375. It’s my job to engage with those students Allyn Jackson is senior writer and deputy editor of the Notices. Her e-mail address is [email protected]. as well as I can. I try to make sure the students come 1See “What Does Active Learning Mean For Mathematicians?” by to the lectures—I call them class meetings instead of Benjamin Braun, Priscilla Bremser, Art M. Duval, Elise Lockwood, lectures—and to make them as interactive as possible. I and Diana White, Notices, February 2017. run around the room and solicit questions, I make an For permission to reprint this article, please contact: effort to call students by name when they raise their [email protected]. hands. This seems to limit absences and eliminate texting DOI: http://dx.doi.org/10.1090/noti1488 and web surfing in class.

March 2017 Notices of the AMS 229 COMMUNICATION

Notices: You said students don’t want a passive experi- I think we mathematicians have gotten much better ence in the classroom. Is that very different from when you at presenting ourselves to the outside world. Mathemati- first started at Berkeley? cians, who fifty years ago were solitary figures pacing the Ribet: When I first started at Berkeley, students would floor, are interacting with people outside their immediate go into this old room with lots of wood and chalkboards circle. It’s perhaps part of the general trend where math- and sit there. The professor would lecture, and the ematics has become more collaborative. People are more students would take notes. The experience was completely used to explaining themselves to others who are not nec- scripted. No one questioned the procedure. Now, students essarily “living inside their brains,” as my wife says. More get input through a constant barrage of media—through and more mathematicians have taken up positions in their tablets and phones and laptops. It’s perceived as business and industry and government, where they have cool to ditch class and try to make up for it at the very to explain their work to colleagues who have no mathe- last minute by binge-watching recorded lectures. matical training. And they have to listen to descriptions In my courses, I look for ways to interact with the of problems that are fundamentally mathematical, from students informally, to get a sense of what would appeal people who don’t view those problems in mathematical to them, what reaches them. I try to make mathematics terms. something that they can relate to. There is a related trend. More and more Public Awareness of Mathematics The world of math mathematics is ap- Notices: Andrew Wiles’s proof of Fermat’s Last Theorem is much less plied. Thirty years ago, [featured in this issue of the Notices], to which your work if you talked about contributed crucially, attracted huge attention from the orderly than it applied mathematics, general public. Since that time, the public’s interest in it meant differential mathematics has grown quite a bit. Why? And what role used to be. equations and fluid can the AMS play here? dynamics. Nowadays, Ribet: I was in Cambridge, England, in June 1993, when applied math can be Wiles announced a proof of Fermat’s Last Theorem. When anything—number theory, or combinatorics, or algebraic I returned to Berkeley, people at MSRI [Mathematical geometry applied to robotics. There is a wonderful report, Sciences Research Institute]—Bill Thurston, Lenore Blum, Mathematical Sciences in 2025 [National Academy Press, and others—suggested doing a public event around FLT. 2013], that talks about all the ways in which math has There was a huge gathering in San Francisco where a influenced daily life. Quaternions in electric toothbrushes group of mathematicians—I was one of them, as were and video games—this is just great! Obviously we have Karl Rubin, Bob Osserman, Lenore Blum, and others—got something to offer to the outside world. The AMS is right up in front of 2,000 people and talked about Wiles’s proof.2 at the center of the picture. We had a post-event discussion at MSRI the next week, Notices: What do you think the AMS can do to capitalize and Thurston said: “This is great. We have to do more on this upswell of public interest in math? things like this. What do we do now?” I remember being Ribet: The AMS is part of the reason there is this public skeptical. I thought the proof of FLT was such a singular interest. We have the wonderful AMS Public Awareness development that there would be no way to maintain Office, which is tweeting all the time and has a big presence the public’s interest in research mathematics. Now, over on Facebook. We also have the AMS office in Washington twenty years later, it’s been incredibly gratifying to see DC, which helps communicate about mathematics with 4 the growth in interest in FLT and many other wonderful people on Capitol Hill. It’s one of the fundamental mathematical results. activities of the AMS now, to look beyond the immediate Recently I watched a Numberphile3 video on a combi- community of academic mathematicians. natorial problem called the Josephus problem. The video featured a young mathematician named Daniel Erman. He New Generations, New Technologies explained something that is fairly sophisticated, using Notices: You mentioned in your candidacy statement for induction and binary expansion. At the time I watched AMS president that technology is bringing rapid changes it, the number of views of that video was up to about and unprecedented opportunities to mathematics. Can you a quarter of a million, and it had been out only a few say more about what you mean by that, what kind of tech- days. That’s one of many indications of a real hunger nology you are referring to? among some segments of the general public for fairly Ribet: It’s clear that one of the challenges for the AMS sophisticated mathematics. now is to continue to maintain its relevance in a con- text where young mathematicians are using technology 2See “Fermat Fest Draws a Crowd,” by Allyn Jackson, Notices, to share and gather information with less of the formal October 1993. structure that older mathematicians like myself are used 3Sponsored by MSRI, Numberphile (www.numberphile.com) is a website featuring videos made by Brady Haran, in which math- 4Karen Saxe, who started as director of the AMS Washington Of- ematicians talk about mathematical ideas and concepts. The fice in January of this year, was interviewed by Notices associate videos are personal, humorous, quirky, insightful, and a lot of fun. editor Harriet Pollatsek in the December 2016 issue.

230 Notices of the AMS Volume 64, Number 3 COMMUNICATION to. People ask questions and get answers on MathOver- flow. Many people feel that the definitive version of a manuscript is the one deposited on arXiv. Traditional journals are sometimes seen as unimportant. People of- ten bypass MathSciNet® when trying to get information. They collaborate through blogs. The world of math is much less orderly than it used to be. When I was a young mathematician, I would run to the library every month to look at the latest printed volume of Math Reviews. I would wait for the Notices and the Bulletin and the Abstracts to arrive in my mailbox. I would go to the library to look at the latest issues of Inventiones, the Duke Mathemati- cal Journal, and Annals of Math. Now, many results are disseminated without that structure. Notices: What can the AMS do to make itself relevant to young people who are developing as mathematicians in Ken Ribet. this very different environment? Ribet: An initiative ongoing for the past three or four often scramble to prepare themselves for such careers years, through strategic planning, aims to reinvent the by increasing their programming skills or learning basic AMS in some sense—to make sure it remains relevant statistics. and to meet changing needs of younger mathematicians. People who make this transition report back and say, The AMS has carried out interviews and surveys to find “It’s fantastic, I should have done this from the beginning out what mathematicians think of and need from the of my graduate career, this is a much more rewarding AMS. It’s hard for me to summarize the findings in a few experience than I imagined when I started on that road.” short sentences, but it is clear that in the next few years, At every JMM [Joint Mathematics Meetings] there is a panel the AMS will evolve in lots of ways in response to this about industry jobs, and the panelists are just glowing, strategic planning. The books will look different. There saying how wonderful their work is. But somehow, the will be new journals. The AMS will respond to what we word is not percolating to the level of first-year graduate have heard from our membership. students, who are still very much of the mindset that they want to be exactly like their professors. I think the AMS Preparing for Careers in Business, Industry, can have an important role in changing the mindset. Government Notices: What other challenges face the AMS? Memories of Serge Lang Ribet: One challenge is that the majority of the young Notices: You were a great friend of Serge Lang.6 He re- people who are in graduate programs or are new PhDs will signed from the AMS about ten years before his death. pursue careers outside of academia. It’s important for the What do you think he would have said about your becom- AMS to make people aware of this reality and to prepare ing AMS president? them for transitions they’ll need to make, to ensure that Ribet: Serge had an expression he would use to describe, 5 graduate programs are providing good preparation. say, a speech by a prominent university administrator My wife, Lisa Goldberg, is a mathematician who works or some other august figure. He would refer to these in industry as a financial economist and is on the faculty speeches as “big time bullshit.” This was one of his of UC Berkeley in economics and statistics. Graduate favorite descriptors, and he used it a lot. Maybe twenty- students come into those departments, and from Day five years ago, I was asked to run for the AMS Council. One, the majority of them are thinking about jobs in Serge was in Berkeley every summer, and he was here industry, biotech, consulting, government. Very few of when I wrote my candidate’s statement. I showed it to them consider that their goal is to be like their professors. Serge and asked him what he thought. He shook his head In mathematics, which is a neighboring discipline, it’s and looked very displeased. Using Serge’s expression, I completely different. If you talk to a first- or second-year said, “Big time bullshit?” And he said, “Not big time”! graduate student in mathematics, you find that person’s I lost that election. More recently, I was asked again goal is to, say, transform the study of Banach algebras to run for the Council. I had just been to an informal through the use of derived algebraic geometry. With few gathering of the Council after one of its long meetings, exceptions, they think in terms of academic careers. What and I had an incredibly positive impression of the whole happens is that a majority of those people, soon after enterprise. I loved the people and what they were doing. their PhDs, realize that they would be a lot happier I agreed enthusiastically to make a second run for the in a business or industry or government setting. They 6Serge Lang (1927–2005) was a French-born mathematician who 5See “Math PhD Careers: New Opportunities Emerging Amidst Cri- spent most of his career at Yale University. Read more about sis,” by Yuliy Baryshnikov, Lee DeVille, and Richard Laugesen in him in the obituary that appeared in the May 2006 issue of the this issue of the Notices. Notices.

March 2017 Notices of the AMS 231 COMMUNICATION

Council. I was elected to the Council and was later chosen profession for many years. Somehow, they snuck my name for the Executive Committee. Then I was asked to run for in. I got to wear a strange cloak and participate in all sorts president—and I was elected. of amazing rituals. I really felt like an impostor! After the I think that Serge, despite the fact that he had negative induction ceremony, we were invited to a seven-course views of various things, had a great deal of respect for banquet, and for every course they would bring out about people’s passion. He liked me very much, and he respected a dozen St. Emilion wines. me. If he were around today and saw that I was passionate Notices: A dozen for each course? about working for the AMS, he would respect that. Ribet: A dozen for each course. There were many Just to summarize what happened with Serge’s res- glasses on the table, and I came close to sampling all of ignation from the AMS: In the 1970s, Serge started his the wines. A few years later I went back to one of their “file-making,” in which he developed the theme that there banquets with my wife, were systematic abuses of power by people at the top of who was also com- the scientific establishment who would quash opposing When I was a pletely amazed. I’ve views if those views conflicted with their own ambitions. never had anything He thought that HIV/AIDS was one of those subjects where graduate student, I comparable in my the people in power were quashing opposing viewpoints. had a moment of experience. What prompted Serge’s resignation from the AMS was one I no longer have particular article in the Notices about the mathematics doubt and thought the palate of the pro- of HIV.7 This was most unfortunate, because it was an fessional, and I no abrupt break with an organization that he had supported about becoming a longer follow every- through his whole career, over one particular incident. So thing that’s available, it was certainly not the case that Serge was opposed to disc jockey. but I still have the AMS in general. pretty strong prefer- Notices: He was an interesting character. ences when looking at wine lists in restaurants. I Ribet: He really was. I still have stacks of his files in typically go for wines that are exceptional values in a my office. When my office was renovated, I had a great given price range. deal of fun filling up barrels with paper to be recycled or Notices: Did you think about becoming a sommelier? shredded. But I kept all of Lang’s files. Ribet: I never thought about becoming a sommelier. But when I was a graduate student, I had a moment of Budding Sommelier? doubt: “Why am I doing this? I’ll never do any research. Notices: In 1988 you were named Vigneron d’Honneur de Even if I write a thesis, it won’t be very good.” I thought St. Emilion. What does this mean? about becoming a disc jockey, because when I was an Ribet: I have been interested in wine ever since I was undergraduate I spent all my time on the campus radio. a graduate student and then a Sloan Fellow in Paris. So I turned to my friends who had gone into the radio From the beginning, I had very strong opinions about the industry and asked if they had a job for me. They said, quality of wines. I could discern all kinds of tastes in “You’re really good in math, why don’t you stick with wines and was completely impassioned. In Paris, through that? The radio industry is pretty hard.” So I decided to a chance encounter, I met someone who was a sommelier, try again. and I ended up becoming an affiliate member of the At the end of my graduate career, I was very surprised Association des Sommeliers de Paris. Once I was on their that people were actually offering me jobs. I thought, mailing list, I got invited to a ton of industry tastings in “Who cares? I’m doing this thing with abelian varieties, Paris. There would be five or six per week. Of course I it’s not likely to attract any attention.” didn’t go to every tasting, but I went to quite a few of Notices: So everybody goes through those doubts. them. They would typically be held at 4 pm or 5 pm in the Ribet: Absolutely. Doing mathematics is just a perma- afternoon, between the two meal services at restaurants. nent situation of being wracked with self-doubt. Where is It was a fantastic education. I became acquainted with my next theorem? Am I still any good? Is this proof really every single alchoholic product on the French market at right? Will anyone care about it? the time—this was the late 1980s. Notices: Those are comforting words for the young peo- During this period, some of the mathematicians in ple entering the profession. They know they are not alone. Bordeaux felt that it was incredibly unusual to have a Ribet: You bet! professional mathematician who also had a professional Notices: And will you be in charge of the wine at the education in wine. So they brought my name to the AMS banquet? attention of the people in St. Emilion who have an annual Ribet: I’ll be glad to help if they need me. induction ceremony for honorary winemakers. Most of these winemakers are wine critics from the newspapers, Photo Credits or industry figures—people who have served in the wine Photo of Ken Ribet with students is courtesy of Jerry Fowler. 7“Using Mathematics to Understand HIV Immune Dynamics,” No- Photo of Ken Ribet is courtesy of Gauri Powale. tices, February 1996; see also Lang’s response on receiving the 1999 Steele Prize for Exposition, Notices, April 1999.

232 Notices of the AMS Volume 64, Number 3 American Mathematical Society

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Math in Brazil: Sowing the Seeds

Marcelo Viana

Brazil is soon to host the first International Congress of Mathematicians ever to be held in the Southern Hemi- sphere, ICM 2018. About a year before that, Rio de Janeiro will also be the stage for the first International Mathe- matical Olympiad to happen in the country, IMO 2017. Taking advantage of this circumstance, the mathematical community is promoting the Biennium of Mathematics 2017–2018, a broad and ambitious initiative sponsored by the National Congress to disseminate and popularize mathematics in the whole society—children and their families, students and their teachers. All this pays tribute to the remarkable development Figure 2. The first Brazilian Mathematical Colloquium experienced by Brazilian mathematics over the last de- was held in 1957. cades. And none of it could have been foreseen only a generation ago, back in 1986 when I joined IMPA–Instituto should anyone have told us that in less than three decades de Matemática Pura e Aplicada for grad school. Back then, a young Brazilian raised and educated in the country would win the Fields Medal, we would all have smiled in utter disbelief. Except, perhaps, for then seven-year-old Artur Avila himself. Because the fact is that mathematics, and science in general, in Brazil is a very young idea. Brazilians love to blame it on the kings of Portugal. The conventional story goes that a jealous colonial power, eager to prevent the development and spread of “subversive ideas,” forbade the printing and circulation of journals and newspapers, thus hampering the development of a knowledgeable society. A more nuanced truth is that many such policies remained in place after independence under the two emperors and, to a lesser extent, even after the republic had replaced the monarchy. As a result, most scientific institutions Figure 1. Rio de Janeiro will host the International were created rather late. Besides, not surprisingly, the Mathematical Olympiad in 2017 and the International earlier ones were concerned with such issues as tropical Congress of Mathematicians in 2018. diseases, sanitation, and public health. Mathematics was not a priority at that stage. The first person to go on record as a mathematical Marcelo Viana is director of IMPA. His e-mail address is viana@ researcher was Joaquim Gomes de Souza, born in 1829 in impa.br. the northeastern state of Maranhão. After getting a degree For permission to reprint this article, please contact: from the Military School in Rio de Janeiro, the only insti- [email protected]. tution in the country offering higher education in math- DOI: http://dx.doi.org/10.1090/noti1481 ematics, Souza traveled to Paris, where he presented his

234 Notices of the AMS Volume 64, Number 3 COMMUNICATION mathematical papers to the Académie des Sciences. Deeply disappointed with the lack of response (the committee, chaired by Cauchy, never even met), Souza returned to his country, quit mathematics for politics and poetry, and died at the age of thirty-five, leaving no academic heirs. Another century would pass before regular activities in mathematics could take off in the 1950s. That is when Brazil joined the IMU (International Mathematical Union), the Brazilian Mathematical Colloquium was first held (see Figure 2), and a number of important institutions were founded, including IMPA. By then a network of public universities was in place around the country, most notably the University of São Paulo and the University of Brazil in Rio de Janeiro. Leopoldo Nachbin and Mauricio Peixoto graduated in engineering from the latter and went on to co-found IMPA and to be the first Brazilian mathema- ticians giving invited addresses at the ICM in 1962 and 1974, respectively. Figure 3. Students of the indigenous school Tuparã, at The contribution of Brazilian mathematicians to the Nova Ubiratã, state of Mato Grosso, participating in the ICMs has become more and more regular, with a current 2015 Mathematical Olympiad. total of three plenary lectures and fourteen invited ad- dresses. A record was broken in Seoul 2014 with one ple- nary (Fernando Codá Marques) and three invited speakers (in dynamics, geometry, and probability).

Figure 4. Students participating in the 2014 Mathematical Olympiad: in the front row, triplets and Table 1. The number of mathematics papers by gold medalists Fabia, Fabiele, and Fabiola Monteiro. Brazilian authors has risen from 255 in 1985 to 2,076 All this reflects the rapid growth of mathematical re- in 2015. search that is so clearly displayed in Tables 1 and 2: Brazil now accounts for about 2.1 percent of the world’s total production. As a comparison, both the GDP and the popu- lation of Brazil are around 2.9 percent of the world’s total. There are now nearly sixty graduate programs in math- ematics and statistics. They train an increasing number of Brazilian students and a substantial number of foreigners, especially from Latin America and, increasingly, Asia, Europe, and North America. A major concern is that, de- spite steady growth, the total number of doctoral degrees granted in mathematics and statistics every year (about 180 currently) still falls short of the needs presented by Brazil’s expanding university system, not to mention in- dustry. The difficulty in attracting good graduate students is also a reflection of deeper problems in basic education. Table 2. The percentage of mathematics papers by Indeed, official data about the performance of students Brazilian authors has risen from a half percent in 1985 in elementary, middle, and high schools just released by to over 2 percent today. the Ministry of Education show that, while there has been

March 2017 Notices of the AMS 235 COMMUNICATION

for a dialogue that has been missing for decades and is crucial for dealing with the problems of education, espe- cially at the high school level. Brazil has two major mathematical Olympiads. The OBM (Brazilian Mathematical Olympiad) was created in 1979 at the initiative of the SBM, promoting regional and national mathematical competitions involving about 500,000 students from middle school to college level as well as Brazil’s very successful participation in Interna- tional Mathematical Olympiads. The OBMEP (Brazilian Mathematical Olympiad for Public Schools) was started by IMPA and the federal government in 2005 and now reaches about eighteen million secondary school students every year. Over little more than a decade, it has become a major event in our academic calendar Figure 5. At the 2007 Mathematical Olympiad award and one with a particularly high profile. As a sign of its ceremony, Brazil president Lula pays tribute to Ricardo prestige, the award ceremony for the gold medalists is Oliveira, a survivor of spinal amyotrophy, who was usually chaired by the president of Brazil. Efforts are cur- carried to school in a wheelbarrow and went on to win rently under way to integrate the two competitions more a gold medal. Ricardo will soon graduate in industrial effectively, starting in 2017. mechatronics. Brazilian mathematics was born open to international cooperation by necessity as much as by design. IMPA some improvement in early years, progress at the high keeps a vigorous visitor program totaling over eight hun- school level has stalled. Results in the PISA (Programme dred visitor-months per year. A highlight is the cycle of for International Student Assessment), an international meetings run by the SBM and the SBMAC jointly with their performance test held every third year for fifteen-year-old counterparts in some of our major partner countries. The students, are equally unflattering. Despite some modest first Brazil-Spain meeting was held in Fortaleza in Decem- progress, Brazil still performs well below the OECD (Or- ber 2015, and the Brazil-Italy meeting hosted by IMPA in ganization for Economic Cooperation and Development) August 2016 was also remarkably successful. Preparations average. In fact, in 2012 Brazil ranked 58th among the 65 for the Brazil-France meeting in 2019 are well advanced. countries taking the test. These facts prompted the federal The idea of a joint meeting with the United States in 2020 government to propose a reform of high school education, has been floated, and conversations were initiated with which is currently being discussed in Congress. our Portuguese colleagues, having in mind that 2022 will The SBM (Brazilian Mathematical Society) was founded mark the 200th anniversary of Brazil’s independence from in 1969. Other mathematical societies have been es- Portugal. tablished since: the Brazilian Society for Applied and Preparations for the ICM 2018 and the IMO 2017, as Computational Mathematics (SBMAC), the Brazilian Sta- well as for the IMU General Assembly 2018, in São Paulo, tistics Association, the Brazilian Society for Mathematical are well under way. Education, and the Brazilian Society for the History of Rio de Janeiro has gotten used to hosting major in- Mathematics. ternational events: the 2013 Catholic Journey of Youth, The SBM has about two thousand associates and is a nonprofit publishing house for mathematical books and journals. It also runs several initiatives of broad interest, such as PROFMAT, the nationwide master’s Brazilian program for school- teachers. PROFMAT mathematics operates through a na- tionwide network of was born over seventy universi- ties and institutes in open to all twenty-seven Bra- zilian states and has international already granted Mas- cooperation. ter’s degrees to more than 2,600 high school teachers. It is also help- ing bring universities Figure 6. Around 300 mathematicians attended the and schools together Brazil-Italy joint meeting at IMPA in August 2016.

236 Notices of the AMS Volume 64, Number 3 COMMUNICATION AmericAn mAthemAticAl Society the 2014 FIFA World Cup, and the 2016 Olympic and Paralympic Games, just to mention the largest ones. Major infrastructure was overhauled in several parts of town, most especially in the western neighborhood of Barra da Tijuca, pictured in Figure 1, where ICM 2018 will take place. Barra da Tijuca was a major site for the Olympic Games and now boasts over 12,000 hotel rooms and a brand-new transportation system. FEATURED TITLES FROM Following a rough year in 2015 and despite an economic situation that remains adverse, the general mood in Brazil has been improving substantially as the country’s polit- ical and economic troubles seem to be subsiding. Both the achievements of Brazilian mathematicians and the shortcomings of mathematical education in the country regularly put mathematics in the headlines, which makes ALGEBRAIC the timing for the upcoming IMO and ICM especially GEOMETRY II fortunate. All this makes me confident that we will have David Mumford, Brown University, Providence, RI, and a great Biennium of Mathematics 2017–2018 and that it Tadao Oda, Tohoku University, will sow the seeds of a new era for mathematics in Brazil. Japan I’m really looking forward to the official launching of the Several generations of students of Biennium at the first Brazilian Math Festival, April 27–30, algebraic geometry have learned the subject from David Mumford’s 2017. Check out the website festivaldamatematica. fabled “Red Book,” which con- org.br and come participate! tains notes of his lectures at Harvard University. Their genesis Photo Credits and evolution are described by Mumford in the preface. Figure 1 is courtesy of the ICM2018 Organizing Commit- 2015; 516 pages; Hardcover; ISBN: 978-93-80250-80-9; List US$76; AMS tee. members US$60.80; Order code HIN/70 Figures 2, and 6, as well as Marcelo Viana's photo are courtesy of IMPA. OPERATORS ON Figures 3, 4, and 5 are courtesy of IMPA/OBMEP. HILBERT SPACE Tables 1 and 2 are courtesy of MathSciNet. V. S. Sunder, Institute of Mathematical Sciences, Chennai, India This book’s principal goals are: (i) to present the spectral theorem as a statement on the existence of a unique continuous and mea- surable functional calculus, (ii) to ABOUT THE AUTHOR present a proof without digress- ing into a course on the Gelfand Marcelo Viana, director of IMPA, re- theory of commutative Banach ceived the inaugural ICTP Ramanu- algebras, (iii) to introduce the jan Prize in 2005 and served as pres- reader to the basic facts concerning the various von Neumann- Schatten ideals, the compact operators, the trace-class operators ident of the Brazilian Mathematical and all bounded operators, and finally, (iv) to serve as a primer Society from 2013 to 2015. Being a on the theory of bounded linear operators on separable Hilbert father of two (six and nine) is rein- space. troducing him to mathematics from 2015; 110 pages; Softcover; ISBN: 978-93-80250-74-8; List US$40; AMS members US$32; Order code HIN/69 Marcelo Viana a whole new angle.

VIEW MORE TITLES AT BOOKSTORE.AMS.ORG/HIN

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March 2017 Notices of the AMS 237 AMERICAN MATHEMATICAL SOCIETY FELLOWS of the ical Society American Mathemat

The nomination period is February 1 through March 31.

Learn how to make or support a nomination in the Requirements and Nominations Guide at: www.ams.org/profession/ams-fellows

Questions: Contact AMS staff at 800-321-4267, ext. 4113 or by email at [email protected] THE GRADUATE STUDENT SECTION

Ryan Haskett Interview

Haskett: My parents for sure. I had good high school teachers who were very supportive and many college pro- fessors who took the teaching aspect of their jobs very seriously. My PhD advisor was instrumental in keeping me in grad school. Diaz-Lopez: How would you describe your work to a graduate student? Haskett: I’m currently taking a career break and travel- ing, but let me talk about my last job. My main goal was to build a system that would minimize the effects of random currency price spikes on our global portfolio of invest- ments, at a reasonable cost. Building and improving this semi-automated system kept me fairly busy but also left time to explore some research questions for the firm. For instance, what are metrics to judge whether a hedge fund was good, or just lucky? Or, how do you understand the interactions between really long-term investments, like private equity, and more-common financial investments, like stocks and bonds? I mainly use linear algebra and scientific computing and Ryan Haskett is a mathematician and finance little else to directly justify all the differential equations professional and currently about halfway through a and asymptotics I studied in graduate school. Still, my year of extreme vagrancy. He and his partner have just time as a grad student taught me how to work with messy finished an RV tour of the Rocky Mountains and are off data sets and also apply results from academic papers, with backpacks to Europe and North Africa. even from other fields. These skills have been extremely useful in finance. Diaz-Lopez: You finished your PhD, and shortly after- ward you started working in the financial sector. What Diaz-Lopez: When/how did you know you wanted to be a message would you give to doctoral students and profes- mathematician? sional mathematicians thinking about having a career in Haskett: That is a hard question for me. I’m not sure I this sector? remember a time when I wasn’t interested in math or com- Haskett: The first thing to note is just how easy the puter science. Games were the early draw for understand- transition was. I had taken only one economics class ing computers, as in those days often a good amount of during college and grad school combined, but companies expertise was required just to get games working on a PC. were more than willing to help to get more employees In math, maybe the draw was orderliness. I like how with strong mathematical backgrounds. The Chartered systems work together. Also, at my college (at the time) Financial Analyst (CFA) program was useful as well, and the professors of mathematics were much more engaging most companies will cover the fees. The CFA designation than the professors of computer science. is billed as an entry to a community of financial profes- Diaz-Lopez: Who encouraged or inspired you (math- sionals. However, the financial community seems to think ematically or otherwise)? of it as a series of three hard tests that are meant to create a class of people who passed, versus a class who did not. That sounds fairly awful, but what the CFA does exception- DOI: http://dx.doi.org/10.1090/noti1492 ally well is assemble the whole background one needs to

MArch 2017 Notices of the AMs 239 THE GRADUATE STUDENT SECTION really get into finance and a series of tests to make sure you learn it thoroughly. Looking Back on a Year of Interviews To answer your question: obviously, the salaries are fairly exceptional, but the big draw for me was the time- The Graduate Student Section was launched with the frame of projects. In grad school, projects took years, and January 2017 issue of the Notices. Since then, each in- I would often get bored working on the same thing for that stallment of the section has included an interview with long. In finance, projects usually take weeks to months, a mathematician. Below is a list of the Graduate Student which is enough time to really dive into an area, but not Section interviews that have appeared to date. too long to make it tedious. Also, the feedback on your work tends to be fast and numerically obvious. Comments and suggestions for future interviews are Diaz-Lopez: All mathematicians feel discouraged oc- welcome. Please post comments on the Notices website casionally. How do you deal with discouragement? at www.ams.org/notices. Please send suggestions Haskett: This seems unrelated, but regular exercise and for interviews, or for other topics you would like to see sufficient sleep seem to make the real difference between covered in the Graduate Student Section, to those who succeed and those who don’t, both as grad [email protected]. students and professionals. It works wonders for people's January 2016: Ian Agol mental states. A lot of research has come out recently February 2016: Fernando Codá Marques showing the variety of benefits from both. March 2016: Elisenda Grigsby Diaz-Lopez: If you were not a mathematician, what April 2016: Arlie Petters would you be? May 2016: Melanie Wood Haskett: Computer programmer for sure, probably in June/July 2016: Jordan Ellenberg the Valley. August 2016: Helen Moore Diaz-Lopez: If you could recommend one lecture (book, September 2016: Po-Shen Loh paper, article, etc.) to graduate students, what would it be? October 2016: Timothy Gowers Haskett: For fun, I love the “Crash Course World His- November 2016: Colin Adams tory” lecture series on YouTube. December 2016: Gigliola Staffilani January 2017: Arthur Benjamin Photo Credit February 2017: Tom Grandine Photo of Ryan Haskett is courtesy of Ryan Haskett.

Alexander Diaz-Lopez, having earned his PhD at the University of Notre Dame, is now visiting assistant professor at Swarthmore College. Diaz-Lopez was the first graduate student member of the Notices Editorial Board.

240 Notices of the AMS Volume 64, Number 3 THEGRADUATESTUDENTSECTION

WHATIS… an Elliptic Curve? Harris B. Daniels and Álvaro Lozano-Robledo Communicated by Steven J. Miller and Cesar E. Silva

Elliptic curves are ubiquitous in number theory, algebraic geometry, complex analysis, cryptography, physics, and beyond. They lie at the forefront of arithmetic geometry, as shown in the feature on Andrew Wiles and his proof of Fermat’s Last Theorem that appears in this issue of the Notices. The goal of arithmetic geometry, in general, is to determine the set of 퐾-rational points on an algebraic variety 퐶 (e.g., a curve given by polynomial equations) defined over 퐾, where 퐾 is a field, and the 퐾-rational points, denoted by 퐶(퐾), are those points on 퐶 with coordinates in 퐾. For instance, Fermat’s Last Theorem states that the algebraic variety 푋푛 + 푌푛 = 푍푛 has only trivial solutions (one with 푋, 푌, or 푍 = 0) over Figure 1. A curve of genus 1 over the complex ℚ when 푛 ≥ 3. Here we will concentrate on the case numbers is a Riemann surface with one hole. of a 1-dimensional algebraic variety, that is, a curve 퐶, and a number field 퐾 (such as the rationals ℚ or the For example, the curve Gaussian rationals ℚ(푖)). Curves are classified by their 퐶 ∶ 3푋3 + 4푌3 = 5 geometric genus as complex Riemann surfaces. When the genus of 퐶 is 0, as for lines and conics, the classical has no ℚ-rational points, but the local methods we use methods of Euclid, Diophantus, Brahmagupta, Legendre, in the genus 0 case to Gauss, Hasse, and Minkowski, among others, completely rule out global points determine the 퐾-rational points on 퐶. For example, fail here.1 A goal of The study of the theory of elliptic 퐶 ∶ 37푋 + 39푌 = 1 and 퐶 ∶ 푋2 − 13푌2 = 1 elliptic curves 1 2 curves is to find all have infinitely many rational points that can be completely the 퐾-rational points grew in the 1980s. determined via elementary methods. However, when the on curves of genus genus of 퐶 is 1, we are in general not even able to decide one. 2 whether 퐶 has 퐾-rational points, much less determine all An elliptic curve 퐸 is a smooth projective curve of the points that belong to 퐶(퐾). genus 1 defined over a field 퐾, with at least one 퐾- rational point (i.e., there is at least one point 푃 on 퐸 with coordinates in 퐾). If the field 퐾 is of characteristic 0 (e.g., Harris B. Daniels is assistant professor at Amherst College. His e-mail address is [email protected]. 1퐶 ∶ 3푋3 + 4푌3 = 5 is an example of Selmer where the local-to- Álvaro Lozano-Robledo is associate professor at the University global principle fails. This means that there are points on 퐶 over of Connecticut. His e-mail address is alvaro.lozano-robledo@ every completion of ℚ—i.e., over ℝ and the 푝-adics ℚ for every uconn.edu. 푝 prime 푝—but not over ℚ itself. For permission to reprint this article, please contact: 2Curves are considered in projective space ℙ2(퐾), where, in addi- [email protected]. tion to the affine points, there may be some points of the curve at DOI: http://dx.doi.org/10.1090/noti1490 infinity.

March 2017 Notices of the AMS 241 THEGRADUATESTUDENTSECTION number fields) or characteristic 푝 > 3, then every elliptic The open questions about the rank of an elliptic curve can be given by a nice choice of coordinates, called curve are central to what makes the 퐾-rational points on a short Weierstrass model, of the form elliptic curves so hard to determine. The difficulty arises 퐸 ∶ 푦2 = 푥3 + 퐴푥 + 퐵, from the failure of the local-to-global principle (or Hasse principle) on curves of genus greater than or equal to 1 3 2 with 퐴 and 퐵 in 퐾 (and 4퐴 + 27퐵 ≠ 0 for smoothness). (see footnote 1). For an elliptic curve 퐸/퐾, one defines In this model there is only one 퐾-rational point at infinity, the Tate–Shafarevich group X = X(퐸/퐾) to measure the denoted by 풪. One aspect that makes the theory of failure of the Hasse principle on 퐸. In a sense, X plays the elliptic curves so rich is that the set 퐸(퐾) can be equipped role of the ideal class group of a number field. However, with an Abelian group structure, geometric in nature (see we do not know that X(퐸/퐾) is always a finite group.3 If Figure 2), where 풪 is the zero element (in other words, we knew that X is always finite, then a method Fermat elliptic curves are 1-dimensional Abelian varieties). inaugurated, called descent, would presumably yield an algorithm to determine all the 퐾-rational points on 퐸. In the 1960s, Birch and Swinnerton-Dyer conjectured an analytic approach to computing the rank of an elliptic 푃 + 푄 curve. Later, their conjecture was refined in terms of the Hasse–Weil 퐿-function of an elliptic curve 퐸 (over ℚ for 푃 simplicity), which is defined by an Euler product: −푠 −1 퐿(퐸, 푠) = ∏ 퐿푝(퐸, 푝 ) , 푝 prime 2 푄 where 퐿푝(퐸, 푇) = 1 − 푎푝푇 + 푝푇 for all but finitely many primes, 푎푝 = 푝 + 1 − #퐸(픽푝), and #퐸(픽푝) is the number of 푅 points on 퐸 considered as a curve over 픽푝. Thus defined, 퐿(퐸, 푠) converges as long as Re(푠) > 3/2. In fact, Hasse conjectured more: any 퐿-function of an elliptic curve over Figure 2. The addition law on an elliptic curve. ℚ has an analytic continuation to the whole complex plane. This has now been proved as a consequence of the The Abelian group 퐸(퐾) was conjectured to be finitely modularity theorem that we discuss below. The Birch and generated by Poincaré in the early 1900s and proved to Swinnerton-Dyer conjecture (BSD) claims that the order be so by Mordell for 퐾 = ℚ in 1922. The result was of vanishing of generalized to Abelian varieties over number fields by 퐿(퐸, 푠) at 푠 = 1 Weil in 1928 (a result widely known as the Mordell–Weil An elliptic curve can is equal to 푅퐸/ℚ, Theorem). The classification of finitely generated Abelian the rank of 퐸(ℚ). groups tells us that 퐸(퐾) is the direct sum of two groups: be equipped with an In fact, the conjec- its torsion subgroup and a free Abelian group of rank Abelian group ture also predicts 푅 ≥ 0, i.e., the residue at 푠 = 푅 퐸(퐾) ≅ 퐸(퐾)tors ⊕ ℤ . structure. 1 in terms of in- We then call 푅 = 푅퐸/퐾 the rank of the elliptic curve 퐸/퐾. variants of 퐸/ℚ. For instance, for For instance, the 2 3 3 퐸 ∶ 푦2 + 푦 = 푥3 + 푥2 − 10푥 + 10, curve 퐸 ∶ 푦 +푦 = 푥 −7푥+6 is of rank 3, with 퐸(ℚ) ≅ ℤ , and the graph of 퐿(퐸, 푥) for 0 ≤ 푥 ≤ 3 is displayed in the group 퐸(ℚ) is generated by 푃 = (2, −2) and 푄 = Figure 3. The BSD conjecture is known to hold only in (−4, 1). Here 푃 is a point of order 5 and 푄 is of infinite certain cases of elliptic curves of rank 0 and 1, by work order, and so 퐸(ℚ) ≅ ℤ/5ℤ ⊕ ℤ. of Coates and Wiles, Gross and Zagier, Kolyvagin, Rubin, Which finitely generated Abelian groups can arise as Skinner and Urban, among others. However, Bhargava, the group structure of an elliptic curve over a fixed field Skinner, and Zhang have shown that BSD is true for at 퐾? The possible torsion subgroups 퐸(퐾)tors that can occur least 66 percent of all elliptic curves over the rationals. have been determined only when 퐾 = ℚ, or when 퐾 is The study of elliptic curves grew in popularity in the a quadratic or cubic number field (e.g., 퐾 = ℚ(푖), or 3 1980s when Hellegouarch, Frey, and Serre outlined a road 퐾 = ℚ( √2)). For 퐾 = ℚ, the list of torsion subgroups map to prove Fermat’s Last Theorem by proposing that was conjectured by Levi in 1908, later reconjectured by a certain elliptic curve cannot exist. Roughly speaking, Ogg in 1970, and finally proved in 1976 by Mazur: if 푝 ≥ 11 and 푎푝 + 푏푝 = 푐푝 is a nontrivial solution ℤ/푁ℤ for 1≤푁≤10, or 푁=12, of Fermat’s equation 푋푝 + 푌푝 = 푍푝, then the so-called 퐸(ℚ) ≅ { 2 푝 푝 tors ℤ/2ℤ⊕ℤ/2푀ℤ for 1≤1푀≤4. Frey–Hellegouarch curve 푦 = 푥(푥 − 푎 )(푥 + 푏 ) would have two properties thought to be contradictory. First, In contrast, the list of possible ranks 푅퐸/퐾 is completely the curve would be semistable, which is a mild technical unknown, even over ℚ. We do not even know if this list is finite or infinite for any fixed number field. The largest 3The finiteness of X is known only in certain cases with rank ≤ 1, rank known over ℚ is 28, for a curve found by Elkies. by work of Kolyvagin and Rubin.

242 Notices of the AMS Volume 64, Number 3 THEGRADUATESTUDENTSECTION condition about the type of curves 퐸/픽푝 that we get Freitas, Le Hung, and Siksek extended the modularity the- by reducing the coefficients of E modulo 푝. Second, orem to real quadratic fields. Modularity fits into a much the curve would be modular, a property we explain larger context, together with the Langlands program and in the next paragraph. The statement that the Frey– the Fontaine–Mazur conjecture, which was described in Hellegouarch curve is semistable but not modular was Mark Kisin’s “What Is a Galois Representation?” (Notices, first formalized by Serre and then proved by Ribet. June/July 2007). With Ribet’s result, to prove Fermat’s Last Theorem, one The canonical starting point for a graduate student needed to prove “only” that all semistable elliptic curves interested in learning more about elliptic curves is Sil- over ℚ are modular. This statement emerged in the verman’s The Arithmetic of Elliptic Curves [1]. A more 1950s and is sometimes called the modularity conjecture, elementary approach is Silverman and Tate’s Rational or Taniyama–Shimura–Weil conjecture.4 The modularity Points on Elliptic Curves. conjecture relates two seemingly very distinct objects: elliptic curves and modular forms. References [1] J. H. Silverman, The Arithmetic of Elliptic Curves, 2nd Edition, Springer-Verlag, New York, 2009. MR2514094 [2] Andrew Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. 141 (1995), no. 3, 443–551. MR1333035

Figure and Photo Credits Figure 1/public domain, https://commons.wikimedia. org/wiki/File:Torus_illustration.png Figure 3 was created by Àlvaro Lozano-Robledo with SageMath. Photo of Harris B. Daniels and Àlvaro Lozano-Robledo is by Keith Conrad, courtesy of Àlvaro Lozano-Robledo.

Figure 3. An 퐿-function 퐿(퐸, 푥) with a zero of order 3 at 푥 = 1. ABOUT THE AUTHORS A modular form is a complex-analytic function 푓 Harris B. Daniels (right) re- on the upper-half complex plane that satisfies cer- ceived his PhD in mathematics tain symmetries. In particular, 푓(푠) admits a Fourier in 2013 from the University of series expansion 푓(푠) = 푛 Connecticut and is currently ∑푛≥0 푎푛푞 , where 푞 = assistant professor at Amherst 2휋푖푠 Modularity fits 푒 , and we can attach College in Massachusetts. to the modular form 푓 into a much an 퐿-function 퐿(푓, 푠) = Álvaro Lozano-Robledo 푠 ∑푛≥0 푎푛/푛 . The mod- larger context, Álvaro Lozano-Robledo (left) and Harris B. Daniels ularity conjecture says is a father of two real children, that every elliptic curve together with the one academic child, and is expecting a second (academic) 퐸 is associated to a descendent in spring 2017. He is the author of Elliptic modular form 푓 such Langlands Curves, Modular Forms, and Their L-functions (AMS, that 퐿(퐸, 푠) = 퐿(푓, 푠); program and the 2011) and has published over twenty-five research arti- i.e., their 퐿-functions cles related to the theory of elliptic curves. coincide. In particular, this implies that 퐿(퐸, 푠) Fontaine–Mazur has an analytic contin- conjecture. uation to ℂ, because 퐿(푓, 푠) is known to have one. In 1993 Wiles [2] announced a proof of the modularity conjecture in the semistable case, but a flaw was found in the proof, which was fixed in 1995 by Taylor and Wiles. In 2001 the full conjecture was proved for all elliptic curves over ℚ by Brueil, Conrad, Diamond, and Taylor. In 2015

4See Lang’s article in the Notices, November 1995, for a detailed historical account of the modularity conjecture.

March 2017 Notices of the AMS 243 AMERICAN MATHEMATICAL SOCIETY Math in Moscow Scholarship Program

Study mathematics the Russian way in English The American Mathematical Society invites undergraduate mathematics and computer science majors in the U.S. to apply for a special scholarship to attend a semester in the Math in Moscow program, run by the Independent University of Moscow.

Features of the Math in Moscow program: • 15-week semester-long study at an elite institution • Study with internationally recognized research mathematicians • Courses are taught in English

Application deadlines for scholarships: September 15 for spring semesters and April 15 for fall semesters.

For more information about the Math in Moscow program, visit: mccme.ru/mathinmoscow

For more information about the scholarship program, visit ams.org/programs/travel-grants/mimoscow OPINION The Common Vision Project: Four Reactions

Communicated by Harriet Pollatsek

Note: The opinions expressed here are not necessarily those of Notices. Responses on the Notices webpage are invited.

The Common Vision project seeks to modernize under- graduate programs in the mathematical sciences, espe- cially courses taken in the first two years. It is a joint effort of the American Mathematical Association of Two-Year Colleges, the American Mathematical Society, the Ameri- can Statistical Association, the Mathematical Association of America, and the Society for Industrial and Applied Mathematics. Their report1 reflects a synthesis of common themes in seven curricular guides published by these five associations along with more recent research and input. The report calls on the community to: (1) update curricula, (2) articulate clear pathways between curricula driven by changes at the K–12 level and the first courses students take in college, (3) scale up the use of evidence-based pedagogical methods, A group of students from the University of Arizona (4) find ways to remove barriers facing students at with William Yslas Vélez at Montezuma Castle National critical transition points (e.g., placement, transfer), and Monument. They were en route to the 2011 SUNMARC (5) establish stronger connections with other disci- conference at Northern Arizona University. plines. It urges institutions to provide faculty with training, re- William Yslas Vélez sources, and rewards for their efforts to meet these goals. The Notices asked four mathematicians to write short A document that brings together the thoughts of different pieces summarizing their reactions to the Common Vision mathematical societies into one united vision is certainly report. These pieces appear below. useful. This is especially important when these societies express a common vision. The data presented in the

1www.maa.org/common-vision William Yslas Vélez is University Distinguished Professor, asso- For permission to reprint this article, please contact: ciate head for undergraduate affairs, and Math Center director [email protected]. at the University of Arizona. His e-mail address is velez@math. DOI: http://dx.doi.org/10.1090/noti1482 arizona.edu.

March 2017 Notices of the AMS 245 Opinion

introduction are compelling and serve notice that much technology in the teaching of mathematics, how do we needs to be done to be more inclusive. expect these same new faculty to adopt new teaching As someone who has devoted considerable energy techniques? Those departments that have a higher per- to increasing diversity centage of graduates pursuing teaching should be taking in mathematics and in- the lead in providing the training that is necessary to creasing the number of move forward. There appears to mathematics majors, I am well aware of the Amy Cohen be a disconnect content described in between what the the first thirty pages of Although “workforce preparation and faculty develop- this report. There was ment” is one of the four areas identified as requiring societies think is only one piece of in- “significant further action,” it is not until page 26 that formation that I found the Common Vision report mentions changes in instruc- important and surprising: “A primary tional staffing over the last two decades that challenge point emphasized by the achievement of the project’s goals and more generally how rank-and-file all the guides is that the challenge our profession. status quo is unaccept- Tenured research-active faculty members now teach a faculty operate. able.” smaller proportion of students in the calculus sequence, 2 My opinion is that linear algebra, and differential equations. Their place faculty do not seem to is increasingly taken by full-time non-tenure-track (FT have bought into such a strong statement. There appears NTT) doctoral faculty or by graduate students. Hiring in to be a disconnect between what the societies think is the category of FT NTT doctoral faculty by PhD-granting important and how rank-and-file faculty operate. I give departments is growing. Hiring of “tenure-eligible” faculty talks to many mathematics departments and regularly am has stagnated since 2008. Even upper-level math major part of academic program reviews of departments, yet I courses in universities are frequently taught by FT NTT don’t encounter such strong feelings expressed during mathematicians. those interactions. Departmental culture and policy about course offer- This difference is reflected in the section on “Scaling.” ings, hiring, and promotion are still set by tenured faculty. The authors pose three questions concerning the issue FT NTT faculty with teaching loads of four courses per of sustainability/transportability of successful programs. term cannot reasonably be expected to design or imple- I think that to these three questions must be added the ment new courses, coordinate courses, do advising, and/ first question: “How do we get departments interested in or mentor new colleagues without support from senior these questions?” How do we convince departments that faculty. they should reallocate their limited resources to address The growing enrollment in mathematics from soph- these questions when they have so many other pressing omore year onward is largely driven by the lure of a matters to address? growing STEM workforce. Students whose primary majors The last section on “Moving Forward” is of central are outside math realize that a second major or a strong importance. Indicating what the problems are is the first minor in math will enhance their preparation for graduate step; the second step should be how to mobilize depart- education and/or employment after college. Mathemati- ments to be concerned about finding solutions to these cians whose undergraduate studies were in elite schools problems. Then and only then will we be able to begin to (inside or outside the United States) are often surprised develop solutions. that future high school teachers of mathematics enroll in In looking toward the future, we recognize that depart- upper-level math courses (e.g., analysis, algebra, geometry, ments have different missions and that the issues they and/or probability) alongside future engineers, microbiol- choose to address will be different, depending upon their ogists, and economists. goals. Bachelor’s-granting institutions are not the same Mathematicians recognize the power of generalization as doctoral-granting institutions. However, they have one and abstraction for a broad variety of applications. Un- thing in common. Doctoral mathematical science depart- dergraduates need to learn how to generalize from the ments train all these faculty, and it is this training that concrete to the abstract and then how to specialize back should be central to moving forward. to concrete problems. Pure mathematicians may have In an opinion piece in the 2015 June/July Notices, I difficulty in teaching this process successfully without spoke of gathering data on our graduate programs, and I want to promote that idea again here because of its impact on creating the next generations of faculty. Departments Amy Cohen is professor of mathematics (emerita) at Rutgers Uni- versity. Her e-mail address is [email protected]. should record where their graduates are going. If these graduates are going into the teaching profession, then 2R. Blair, E. Kirkman, J. Maxwell, Statistical Abstract of Under- their training should reflect this path. On-the-job train- graduate Programs in the Mathematical Sciences in the US, Con- ing in the many facets indicated in this Common Vision ference Board of the Mathematical Sciences, AMS, 2013. See also is simply not practical. If all a student sees in their grad- Cohen’s article on “Disruptions of the academic math employment uate training is the standard lecture format and lack of market” in the October 2016 Notices.

246 Notices of the AMS Volume 64, Number 3 Opinion a culture of professional development including collab- In the “Course Structure” section of the report the au- oration across disciplines. Without concurrent “faculty thors identify “a general call to move away from the use development” I fear that the new courses the Common of a traditional lecture as the sole instructional delivery Vision report advocates, if they find instructors at all, may method in undergraduate mathematics courses.” The fail for lack of effective instruction. evidence for the efficacy of active-learning approaches What forms might effective faculty development take? is strong and growing; it should inform our professional Professional societies in the mathematical sciences spon- development efforts as well. I’m encouraged that the term sor relevant conference sessions, short courses, and re- “workshop” appears often in Chapter 3, but I’ve partici- lated projects. A very limited fraction of faculty (whether pated in “workshops” that were essentially lectures and on tenure track or not) find institutional funding to par- minimally effective as a result. How might we design a ticipate. Professional societies should offer such programs workshop grounded in modern understanding of how also at their local and regional meetings. Individual insti- people learn? If we insist that the goal of a short course be tutions and their statewide systems could do the same. not mere transmission of information but rather genuine Online professional learning communities potentially cognitive engagement with the problem at hand, what offer another mode of outreach. Collaboration among changes in format and execution can we expect to see? university administration, departmental leadership and For that matter, how might we invite administrators to faculty, and funding sources is essential so that faculty be active participants in the improvements we seek? At development occurs and improves teaching and learning the end of the first year in the Vermont Mathematics Ini- in undergraduate mathematics. tiative, a professional development program for teachers, each teacher is required to bring a supervising adminis- Priscilla Bremser trator to a leadership training retreat. It’s intriguing to imagine a Modeling Across the Curriculum workshop on The Common Vision report puts forth an ambitious economic mobility with deans working alongside faculty agenda for change in mathematics instruction. It sets ex- and students. Instead of poster sessions for policymakers, pectations for admin- how about bring-your-own-data service-learning projects, istrators, departments, where statistics students and faculty work with legislators and instructors, some to interpret information relevant to current issues? How might of whom will need A corollary then to “the status quo is unacceptable” we design to be convinced that is this: traditional ways of promoting improvements in their participation is undergraduate mathematics education are not enough. a workshop vital. Fortunately, the Our potential collaborators in and out of the academy will report is a collabora- not hop on board if all we offer is a passive experience, grounded tive effort, endorsed whether it’s a printed report, a PowerPoint presentation, or by all of the relevant a MOOC. Instead, let’s invite them into carefully structured in modern major professional or- and intellectually challenging activities. ganizations. That in it- understanding of self is no guarantee of Douglas Mupasiri success, however. The how people learn? community, meaning The Department of Mathematics at the University of those of us who are Northern Iowa, of which I am head, has just undergone eager to move into the its required septennial academic program review, and that implementation phase, will have to be intentional and process has brought the issues discussed in the Common creative in approaching our work. In order to modernize Vision project into sharp focus. Here I discuss four of the undergraduate programs in the mathematical sciences, biggest challenges I see, organized by the themes men- we must bring modern approaches to curricular design, tioned in all seven guides written by the five professional professional development, and outreach activities. associations that participated in the Common Vision proj- The verdict that “the status quo is unacceptable” won’t ect. I also consider student diversity, mentioned in some get much argument from the various constituencies. The but not all the guides. data presented in the report confirm what we all know: Curricula: It is hard to see how departments will be able too many college students fail mathematics courses, to modernize their curricula in the ways contemplated in and too many others lose interest in mathematics. To its the report without an infusion of new resources. Many credit, the report does not place blame for the situation departments will have to make some hard choices about on any group. Chapter 3, “Moving Forward,” presents con- which changes to make and which not, based on their local crete “ways in which faculty members, deans, and other contexts. An essential part of any effort with a chance of administrators can create an environment that supports succeeding will be developing appropriate curriculum improvements in undergraduate mathematical sciences models for different types of institutions, publicizing and education” (page 31). disseminating them, and then scaling them up.

Priscilla Bremser is professor of mathematics at Middlebury Col- Douglas Mupasiri is professor of mathematics at the University of lege. Her e-mail address is [email protected]. Northern Iowa. His e-mail address is [email protected].

March 2017 Notices of the AMS 247 Opinion

Course structure: Most Also: mathematics faculty still use AMS Blog On Teaching and Learning Mathematics, Septem- The onus is the traditional lecture, but ber 10, 20; October 1, 10, 2015. there is increasing recogni- Photo Credits on the entire tion of the need for change to mathematical more active forms of instruc- Photo of William Yslas Vélez and students is courtesy of tion (Freeman et al., 2014). William Yslas Vélez. sciences The recently published CBMS Photo of William Yslas Vélez is courtesy of William Yslas statement on active learning Vélez. community to is but the latest example. The Photo of Douglas Mupasiri is courtesy of Douglas Mupasiri. projects Progress through Cal- Photo of Amy Cohen is courtesy of Cecilia Arias. heed the call. culus, Transforming-Post Sec- Photo of Priscilla Bremser is courtesy of Priscilla Bremser. ondary Education in Mathemat- ics, and others are promising efforts to address this challenge. Workforce preparation: Should mathematical sciences departments train their students for specific jobs or ABOUT THE AUTHORS concentrate their attention on providing them with the William Yslas Vélez has a simple competencies and critical thinking skills that will give goal in life: to convince every stu- them the flexibility to successfully move from one job to dent at his university to be a math another? The fact is: It is possible to do both. Departments major or math minor. William can complement their existing course offerings with de- Yslas Vélez vices such as guided research and strategically arranged internships to design programs that aim to “narrow the gap between today’s mathematics as it is practiced in the Amy Cohen’s major mathemati- academy, industry and government and how it is expe- cal interests are PDEs with soliton rienced in higher education’s instructional programs.”3 solutions and making education in Student diversity: The authors of the Common Vision mathematics more satisfying for report write, “The fact that our community has been un- faculty as well as for students. able to attract and retain a diverse student population in the mathematical sciences is a dreadful shortcoming that Amy Cohen must be remedied.” Beyond the moral case for recruiting students from groups that have been traditionally un- Priscilla Bremser enjoys bicycling, derrepresented in the mathematical sciences, including cross-country skiing, and hiking in women, there is growing consensus that there is also an Vermont. economic and national security imperative. But addressing this issue won’t be easy, as anyone who has tried it will attest. There are, however, some potentially hopeful signs on the horizon. Increasing press coverage of the growing income gap between the rich and the poor the world over is bringing more attention to this issue. Professional or- ganizations are also giving greater focus to the matter. Priscilla Bremser The AMS recently named Helen G. Grundman director of its newly created Department of Education and Diversity. Douglas Mupasiri has been on the SIAM has a Diversity Advisory Committee. faculty of the University of North- The Common Vision project has brilliantly laid a pred- ern Iowa for twenty-three years and icate for a compelling call to action. The onus is on the head of the department of mathe- entire mathematical sciences community to heed the call. matics for the past six years. He was “Are we equal to the challenge?” I think yes. born in Zimbabwe and came to the United States to go to college and Related Notices articles: eventually became a naturalized A Common Vision for Undergraduate Mathematics, by Tara Holm citizen. and Karen Saxe, June/July, 2016. Douglas Mupasiri Are Mathematics Faculty Ready for Common Vision? by Marcus Jorgensen, November 2016. What Does Active Learning Mean for Mathematicians? by Benjamin Braun, Priscilla Bremser, Art M. Duval, Elise Lockwood, and Diana White, February 2017.

3 From the Executive Summary of the Common Vision Report.

248 Notices of the AMS Volume 64, Number 3 AMERICAN MATHEMATICAL SOCIETY

MATHEMATICAL MOMENTS

Dis-playing the Game of Thrones

Who is the real hero of the television series Game of Thrones and book series A Song of Ice and Fire? To help answer this, mathematicians used network science (applied graph theory) to analyze and create a pictorial representation of the third book’s characters and the connections among them. They then looked at different measures of importance—in graph theory terms, centrality—and Tyrion Lannister emerged as the leader in all but one of the measures. The answer may not be resolved for the entire series, but one thing is clear: Thinking about Game of Thrones is a lot safer than playing it!

The researchers constructed this network by first gathering data based on the proximity of characters’ names in the text of the book. Probability, combi- natorics, and numerical Andrew Beveridge, Macalester College approximation were used to detect group- ings among characters, and the algorithm successfully identi- fied seven logical and coherent communities without any prompts or hints. The resulting network, despite being Image: based on fiction, shares Andrew Beveridge, many characteristics with courtesy of real-world networks, such as a the Mathematical Association of America. small number of people playing an outsized role in the network. Discovering this information is nice, but network and graph theory have many other applications of greater significance, such as understanding the flow of information and modeling the spread of disease.

For More Information: “Network of Thrones,” Andrew Beveridge and Jie Shan, Math Horizons, April 2016.

Listen Up! The Mathematical Moments program promotes appreciation and understanding of the role mathematics plays in science, nature, technology, and human culture.

MM/123 www.ams.org/mathmoments

Trimming Taxiing Time

It’s hard to choose which part of air travel is the most fun— body scans, removing your shoes, fighting for the armrest, middle seats—but waiting on the runway is a worthy candidate. Controllers often let jets leave the gate when ready, regardless of existing runway lines, which can lead to long waits. Mathematical models that rely on probability and dynamic programming estimate travel time to the runway and wait time on the runway, allowing controllers to see the effect that the different options open to them have on flights’ departure times. In tests at different airports the models have demonstrated the ability to shorten runway wait times, which reduces congestion and saves tons of fuel.

The models are very accurate: always predicting the number of aircraft in runway queues to within two. And despite their complexity (they involve many variables, such as weather conditions and runway configuration), the models also are very quick: controllers can get real-time updates on anticipated queues every 15 minutes.

Hamsa Balakrishnan, The models aren’t yet in use everywhere, but they may be soon because with an air MIT system that is expected to be stretched to capacity in about five years, analysts say that managing departures is a good way to improve airport and airline efficiency.

For More Information: “A Queuing Model of the Airport Departure Process,” Transportation Science, Ioannis Simaiakis and Hamsa Balakrishnan, Vol. 50, No. 1 (2015).

Listen Up! The Mathematical Moments program promotes appreciation and understanding of the role mathematics plays in science, nature, technology, and human culture.

MM/124 www.ams.org/mathmoments

Making Art Work

Great art looks good no matter where you’re standing, but for art with perspective there’s a special point where your view of the scene will match that of the artist. He or she uses geometry to maintain perspective, while the rest of us can use algebra and geometry to find exactly where to stand so that the flat image on the two- dimensional surface expands into the three-dimensional world of the artist. For this printout, unfortunately, your eye needs to be an uncomfortable three inches away from the red dot. This helps explain why art looks better in museums than in books and why large-screen movie theaters are so popular.

Thinking about perspec- tive led to projective geometry, in which “parallel” lines meet “at infinity”. That may sound strange, but ideas Annalisa Crannell, Franklin & Marshall College from projective geom- etry are useful in art, computer vision, and in determining where the camera was for a given photo. Perhaps even stranger but remarkable: Theorems in projective planes remain true when two of the most funda- mental ideas, point and line, are interchanged.

For More Information: Viewpoints: Mathematical Perspective and Fractal Geometry in Art, Marc Frantz and Annalisa Crannell, 2011.

Image St. Jerome in His Study, by Albrecht Dürer.

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Explaining Rainbows

Even without a pot of gold at the end, rainbows are still pretty fascinating. The details of how they form are fascinating, too. Light is refracted upon entering and exiting raindrops, and reflected within raindrops, sometimes more than once. The angle of refraction and thus the position of the rainbow—not a fixed place but at an angle of elevation of about 42 degrees from the line connecting your eye to your head’s shadow—can be figured out using trigonometry. Because different colors of light have different wavelengths, they are refracted at different angles, which produces a rainbow (or two).

If you look closely, you’ll see lighter colors inside the inner violet band (inset), which appear because of the interference of light waves, with some waves reinforcing each other. The explanation for these bands isn’t obvious and they weren’t accounted for in early theories about rainbows. Proving that these lighter bands should appear required the wave theory of light and their precise description involved an integral,

John Adam, Old Dominion (the Airy integral) that was numerically evaluated using infinite series. Curiosity University about rainbows has led to many other discoveries in mathematics and physics, including “rainbows” formed by scattered atoms and nuclei.

For More Information: The Rainbow Bridge: Rainbows in Art, Myth, and Science, Raymond L. Lee, Jr. and Alistair B. Frasier, 2001. Eric Rolph at English Wikipedia Eric Rolph at English Image: Listen Up! The Mathematical Moments program promotes appreciation and understanding of the role mathematics plays in science, nature, technology, and human culture.

ways MM/126 www.ams.org/mathmoments

Maintaining a Balance

Tipping points are important—and well-named—features of many complex systems, including financial and ecological systems. At a tipping point, a small change in conditions in the system can result in drastic changes overall (just as a small change in the weight of one end of a see-saw can reverse positions). Most such systems are studied using mathematical models based on collections of differen- tial equations. The variables in the equations are related, leading to feedback in the system and potentially substantial changes, such as economic collapse. Research is now being done to recognize tipping points in hopes that something can be done before it is too late.

Some catastrophic changes have occurred when Earth’s natural systems have been disrupted. One happened over 200 million

Daniel Rothman, MIT years ago when more than 90% of the planet’s species became extinct. Mathematics helped in the formulation of a new theory for the cause of the die-off: a methane-producing microbe that thrived on nickel produced by active volcanoes in Siberia. Faster-than-exponential growth in carbon levels at the time pointed math is to a biological trigger, and compu- tational genomics showed that that strain of microbe came into existence at about the same time as the extinctions. This is a case in which a tipping point was in fact about the size of a point.

For More Information: “Climate, Past, Present, and Future ,” Dana Mackenzie, What’s Happening in the Mathematical Sciences, Vol. 10, 2015.

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Farming Better

Farming, not an easy job in the first place, now requires more analytics and technology to meet the increasing demand for food. In one case, in California, mathematicians, hydrologists, and farmers met to design a plan that would minimize water used for crops but still make a profit for the farmers and meet consumer making a demand. The mathematical model that was created incorporated data such as plant growth properties and water requirements of different crops to identify which ones to plant, the best time to plant them, and which areas to leave unplanted. The farmers were happy to use their own resources and those of the community wisely, while the mathematicians were happy to work with experts in the field.

The application of math and high-tech approaches to farming is called precision agriculture. It involves collecting much more data than before, such as the weight of each hen in a chicken coop, and using models to find the best course of action to remedy any deficiencies in the production process. One aspect of farming that has become more efficient as a result is the use of fertilizer. Using GPS-equipped machines that sample the soil, farmers know exactly where more fertilizer is needed, thus overcoming the natural tendency to over-fertilize. As a result, more food is grown and less fertilizer is wasted, which means fewer nitrates in watershed run-off.

For More Information: “A Role for Modeling, Simulation, and Optimization in an Agricultural Water Crisis,” Eleanor Jenkins and Kathleen Fowler, SIAM News, December 2014.

Listen Up! The Mathematical Moments program promotes appreciation and understanding of the role mathematics plays in science, nature, technology, and human culture. 7 MM/128 www.ams.org/mathmoments difference See over 100 Mathematical Moments, hear people talk about how they use math on the job in the modern world, and read translations in 13 languages at www.ams.org/mathmoments MM2016 A m e r i c a n M a t h e m a t i c a l S o c i e t y Women Doing Mathematics Some work independently, others work in collaboration as members of interdisciplinary teams. Many of them also teach at the college or university level, while others are employed in industrial or government laboratories. Whether they do mathematics for the sheer intellectual challenge, or for the critical insights it brings to solving important theoretical and real-world problems, women mathematicians love what they do. Highlighted here are just a few of the women doing mathematics today. Please visit www.ams.org/samplings/ posters to order this in its *original poster format, the upcoming 2017 Women Doing Mathematics, or any other AMS poster that promotes awareness of mathematics, its beauty, and its many applications.

*edited for Notices 64:03 spread Rebecca Goldin, PhD

Her research is in symplectic geometry, group actions and related combinatorics. She is Director of Research for the Statistical Assessment Service, a nonprofit media education and watchdog group affiliated with George Mason University concerned with the media’s use of statistics and mathematics.

“Mathematics for me touches on all the core joys of the human mind. It has rules, patterns, and structure, yet leaves so much room for creativity and invention. It impacts society deeply—from the roots of our education to the leading edge of science and technology—yet distinguishes itself by its sheer purity and abstraction. And there is almost nothing like the “aha!”moments that come with learning, teaching, understanding, or discovering something new in mathematics.” Trachette Jackson, PhD

She uses mathematical models, computer simulations and model-driven experiments to advance the current understanding of tumor growth and angiogenesis and to quantify the relative impact of new, cell- specific treatment strategies on the pathobiology of cancer. She received the Blackwell-Tapia Prize in 2010.

“Many of the challenges of contemporary biology and medicine lie at the intersection of the mathematical and biological sciences. Working at this interface and continually striving to further integrate the fields of mathematics and biology is both exciting and rewarding. There’s nothing I’d rather be doing with my career!” Sommer Gentry, PhD

Her research in optimization in kidney transplantation has been profiled in Science and TIME magazine and on television. She serves as an advisor to both the US and Canada in their efforts to create national paired donation registries, and her research group helped lobby Congress to clarify the legal status of kidney paired donation.

“I chose operations research because I wanted to make a difference to people’s lives. Operations research is like a toolkit that’s used everywhere: transportation, network security, credit cards, international relations, robotics, transplantation—I’ve worked in all these areas! If you master mathematical modeling you can be a contributor to almost any area of human endeavor.” Abigail Thompson, PhD

Her current research is in low-dimensional topology and knot theory. As a consequence of her work, the concept of ‘thin position’ has emerged as a major tool for attacking some of the fundamental problems in the study of 3-manifolds.

“Some human constructions are unreasonably appealing, resonating with the part of our brain that recognizes beauty: suspension bridges, kites, sailboats, cellos, and elegant mathematical arguments.” A m e r i c a n M a t h e m a t i c a l S o c i e t y

Ivelisse Rubio, PhD

Her research interests are applications of computational algebra, finite fields, latin squares, and coding Women Doing Mathematics theory, which has applications in the internet, deep-space telecommunications, satellite broadcasting, and data storage. She has also organized and directed many undergraduate research programs and projects. photo by Angel Garcia photo by “Mathematics is a world where there is no certainty; Nothing is true until you prove it. I love the challenge of facing a problem that has not been solved, that it is not easy to solve and maybe no one will solve! Working in mathematics is also my way of being different.”

Andrea L. Bertozzi, PhD

She develops mathematical methods and frameworks necessary to solve a diverse host of modern problems such as analyzing crime patterns, control of robotic vehicles, and fundamental physics of complex fluids. Her research brings together ideas from differential equations, inverse problems, and statistical physics.

“I really enjoy working with students on applied mathematics research. It’s very rewarding to train photo by Mary Watkins/UCLA photo by Mary students to make an impact in diverse areas of science and engineering using the mathematics that they develop.”

Bryna Kra, PhD

Her research lies in dynamical systems and ergodic theory, with a focus on problems related to combinatorics and number theory. She was awarded a Centennial Fellowship of the American Mathematical Society in 2006 and the Conant Prize in 2010. She is a member of the AMS Board of Trustees.

“Every addition to our collective mathematical knowledge is a small triumph, from a child discovering a pattern, to a student solving an exercise, to a researcher taking a step in the proof of a new theorem. But nothing compares to the pure exhilaration that comes with proving an old conjecture or drawing a connection between seemingly unrelated concepts. Mathematics is the language for communicating such insights, connecting centuries of past research with future advances.” Melanie Wood, PhD

While a high school student she became the first female American to make the US International Math Olympiad Team. While at Duke University she won a Gates Cambridge Scholarship, Fulbright Fellowship, and a National Science Foundation Graduate Fellowship, became the first American woman to be named a Putnam Fellow, and also pursued her interest in theater. Wood is assistant professor at the University of Wisconsin–Madison and an American Institute of Mathematics Five-Year Fellow.

“Insight. Originality. Inspiration. New perspectives. Opening your mind. Finding a different way. Playing around. That is mathematics. There is a myth that mathematics is about memorization, technicalities, formulas, and equations—there is only one correct answer. This picture utterly fails to describe the creative process that is professional mathematics.” Maria Chudnovsky, PhD

She studies the properties of graphs and was part of a team of researchers that proved a hypothesis in graph theory that had stumped mathematicians for 40 years. In addition to its mathematical beauty, graph theory can be a useful tool in operations research, computer science, and engineering. Chudnovsky is a MacArthur Fellow.

“One of the best things about mathematics is that it teaches you to think clearly, no matter what you are thinking about.”

For more information on women mathematicians and their work see www.ams.org/women-mathematicians COMMUNICATION

100 Years Ago: Joan Clarke

One hundred years ago, in June 1917, Joan Clarke was born in London. Her outstanding mathematical talent led her to become one of the very few women cryptanalysts working at Bletchley Park during World War II. Bletchley was the nerve center of a mammoth operation, carried out in utmost secrecy, to decode German and other enemy military communications. Decades later, as the secrecy surrounding Bletchley Park has lifted, the impact of the work of codebreakers like Clarke has become clear. Clarke’s story echoes that of the women in Hidden Figures: The American Dream and the Untold Story of the Black Women Mathematicians Who Helped Win the Space Race, a book by Margot Lee Shetterly that appeared last fall and was also made into a movie. The book recounts the stories of several mathematically talented women who worked as data analysts for NASA from the 1940s to the 1960s. Like the “hidden figures” in Shetterly’s book, Women working on Enigma ciphers in Hut 6 at Clarke has also become better known through a movie, Bletchley Park. An Enigma machine is on the table in when her character was portrayed by Keira Knightley in the foreground. By 1942, 7,000 people worked at the 2014 film The Imitation Game. Bletchley, two-thirds of them women. Clarke entered Cam- bridge University in Her equality with 1936 and the follow- which appeared in the 1993 book Codebreakers, Clarke ing year obtained a modestly noted that her quick rise from the ranks of “the the men was never first in Part I of the girls,” as the female staff was universally called, “was in question, even university’s legendary obviously because of my degree, and before I had had any Mathematics Tripos. It chance of proving myself.” in those had not been very Clarke earned less than her male counterparts but long since women were worked with them as an equal. Her talent, ingenuity, unenlightened even permitted to take and perseverence earned her great respect among her the Mathematical Tri- coworkers. After the war she went to work at Bletchley’s days. pos; the first was successor organization, the Government Communications Charlotte Angas Scott Headquarters (GCHQ). In his essay “Hut 8 from the Inside,” in 1881. In 1939, Clarke received a first in Part II, under which appeared in The Bletchley Park Codebreakers (2011), the supervision of W. Gordon Welchman, and that same Rolf Noskwith writes of Clarke: “It was a tribute to her year she earned her BA degree. It was actually called a ability that her equality with the men was never in “title of degree”; Cambridge began awarding full-fledged question, even in those unenlightened days.” degrees to women only after the end of World War II. What follows is a condensed and slightly edited version Clarke started at Bletchley in 1940. By 1942, the number of an obituary for Clarke that appeared in IEEE Annals of people working there reached over 7,000, and two- of the History of Computing, January-March 20011. The thirds of them were women. They worked in a multitude obituary was written by Ralph Erskine, I. J. (Jack) Good, of roles, as intercept operators, transcribers, typists, and Eric A. Weiss. Erskine is a leading historian of encoders, linguists, punched card machine operators, cryptography specializing in the history of World War II administrators, secretaries—and as codebreakers. A week cryptography and was co-editor (with Michael Smith) of the after joining the staff of Hut 8 at Bletchley, Clarke was The Bletchley Park Codebreakers. In 2002, Erskine gave the moved to a room where she worked directly with Alan Gauss Lecture of the Deutsche Mathematiker Vereinigung. Turing and two other cryptanalysts, Tony Kendrick and Good, who died in 2009, worked as a cryptanalyst at Peter Twinn. In an essay about her work at Bletchley, 1The original obituary is copyrighted by the IEEE Computer Soci- For permission to reprint this article, please contact: ety, which has kindly given permission for the Notices to publish [email protected]. the version appearing here. The Notices also thanks Ralph Erskine DOI: http://dx.doi.org/10.1090/noti1489 for his permission.

252 Notices of the AMS Volume 64, Number 3 COMMUNICATION

Bletchley Park in Hut 8, together with Clarke, and later while Hut 6 was working on those of the German Air Force became a professor of statistics at Virginia Tech. Weiss, and Army. who died last year at the age of 99, was a writer, editor, and electrical engineer who served as the biographies Decoding Enigma-coded Messages editor for IEEE Annals of the History of Computing. The German military believed that the signal contents were secret since they had been encrypted on the Enigma — Allyn Jackson machines before being transmitted by radio. The electro- mechanical Enigma machine—which had been available in commercial form since 1919—changed each letter in Obituary: Joan Elisabeth Lowther Clarke Murray a complex manner depending on a daily changing key. Joan Elisabeth Lowther Clarke Murray was born June 24, The German military knew that the commercial Enigma 1917 in London, England, and died September 4, 1996 in could be broken, but believed that the military machine, Headington, Oxfordshire, England. She was a cryptanalyst which had a plugboard (Stecker board), was unbreakable while at the Government Code and Cypher School (GCCS) in practice. The plugboard interchanged letters in pairs, at Bletchley Park, Buckinghamshire, England, from 1940 which added 150 trillion possible settings to a key (for to 1945 and worked at the Government Communications the ten letter pairs that were generally used). Headquarters, England (1945 to 1952 and 1962 to 1977). Even Alastair Den- niston (head of GCCS Before Bletchley Park at Bletchley at the Clarke was the youngest daughter of William Kemp time) shared the Ger- Lowther Clarke (a London clergyman) and Dorothy Elisa- man belief that the beth Clarke. She matriculated to Cambridge University in military Enigma was 1936. She chose Newnham College, half a mile from the invincible, telling his university’s heart, and not Girton, then Cambridge’s only codebreakers that “all other women’s college. German codes were Early in 1940, Gordon Welchman, who joined the GCCS, unbreakable.” But the recruited Clarke to join the codebreakers. Without telling new codebreakers at her what the job was, he said that it did not really Bletchley Park, Clarke need mathematics but that mathematicians tended to among them, were de- be good at the job. Cryptanalysis requires imagination termined to prove him and accurate thinking. Some mathematics is often used, wrong by reading Ger- especially probability and statistics, but at that time the man Enigma signals. mathematics needed was usually not very advanced. However, German con- She was to start work in June 1940 after she had fidence in the machine completed Part III of her studies and was to be paid was not wholly mis- 2 pounds (then about US$8) a week. Men with her placed. Some Enigma qualifications were getting paid much more. cipher versions, such as Aegir (code-named Joan Clarke in 1936. Summer 1940 Pike by Bletchley Park On June 17, 1940, the day before France capitulated, and and used by merchant raiders) and TGD (used by the after receiving a “pass” Gestapo), were never broken during the war. in Part III at Cam- Matched plaintext and Enigma ciphertext for April 23- bridge, Clarke arrived Clarke fully held 26, 1940, captured from the German patrol boat Schiff at Bletchley Park. Alan 26 off the Norwegian coast, had become available in early Turing, six years her her own in the May and were being used to test a new key-finding aid senior, whom she had male-dominated called the Bombe. Clarke started checking the output of previously met as a the first Bombe—optimistically named Victory—on the friend of her older world of data from Schiff 26. The Bombe had entered service on brother Martin, re- March 18, 1940. The plugboard connections for one day cruited her to work in codebreaking. were then apparently found to be already available on a his Hut 8 instead of piece of paper from Schiff 26, enabling the solution of with Welchman in Hut that day to be completed manually. 6. The grounds of the big estate of Bletchley Plark were gradually being covered with hastily constructed, barn- More Decoding and Banburismus like huts—one-story, non-uniform, and wooden—usually Early in July 1940, Turing figured out how to make the about 18 by 48 feet in size, internally partitioned into Bombes test 26 hypotheses at once, using a process known rooms of various shapes. A hut number designated a as “simultaneous scanning,” rather than one at a time. He building and also indicated the work being done in it. Hut explained his brainstorm to Clarke and discovered that 8 was attacking the Enigma signals of the German Navy, he first had to tell her how electrical relays worked. Later

March 2017 Notices of the AMS 253 COMMUNICATION that summer, when Turing wrote his account of Enigma Stecker of the letter E, as determined in the first stage, theory for the use of new recruits to Huts 6 and 8, locally assumed that all the plaintext consisted of the letter E, called the “Prof’s book,” Clarke was the guinea pig who and proceeded with had to read it to test whether it was understandable to this through a triangu- lesser mortals. Her work on the lar dottery. Her name Hut 8 made little progress against the Naval Enigma was not attached to the until the capture, in February and May 1941, of the keys Naval Enigma invention, so the entire and special bigram (digraph) substitution tables used to helped to shorten Stecker-recovery pro- encipher Naval Enigma message keys, which were the cedure was named starting positions for Enigma’s rotors in specific signals. the war and saved Yoxallismus, although Hut 8 cryptanalysts were therefore sometimes deployed Yoxall now disclaims against other versions of Enigma and other cryptographic many lives on both all memory of in- machines. Following Lieutenant-Colonel John Tiltman’s venting the second recovery of the plaintext of signals enciphered on the sides of the conflict. stage. Railway Enigma in July 1940, Clarke helped to find At the time, Clarke the wiring of its specially wired rotors and worked on was told that she had merely rediscovered “pure Dillyis- breaking traffic using it. Based on the commercial Enigma, mus,” a reference to the highly talented Dillwyn (Dilly) the Railway Enigma-cipher—named Rocket by Bletchley Knox, a founding member of GCCS and a former mem- Park—lacked a plugboard, making the traffic relatively ber of Room 40, the Royal Navy’s highly successful easy to solve. codebreaking unit during World War I. In August 1941, following another series of codebook captures, Hut 8 started to use the codebreaking technique Engagement on and off called Banburismus that Turing had developed in mid- In the spring of 1941, Alan Turing and Joan Clarke got 1940. It had been impossible to employ it earlier, since engaged. He gave her a ring, and they arranged for formal it depended on Bletchley Park having the bigram tables. introductions to both families. However, they kept the Banburismus was a Bayesian sequential procedure for engagement secret, and Clarke did not wear her ring producing and handling a probability network consisting in the Hut. After taking their holiday together at the of thousands of weights of evidence. It used paper sheets end of August, they broke off the engagement by mutual that were punched with representations of the messages agreement. Turing believed the marriage would be a failure under attack and then moved across each other to find owing to his homosexuality. Clarke and Turing continued both single-letter and multi-letter repeats of coincident to be friends and corresponded during Turing’s trip to holes to obtain a probabilistic piece of information about the United States in the winter of 1942-1943. Clarke said the two messages. that they had a special friendship, and her warm feelings Banburismus was invaluable in breaking the main Naval for Turing lasted throughout her life. Enigma cipher, Heimisch—later called Hydra and code- named Dolphin by Bletchley Park. Without Banburismus, End of the War breaking the Naval Enigma would have been greatly slowed down, since Bombes were in short supply until the The labor-intensive Banburismus attack was discontinued US Navy’s four-rotor Bombes came into service in August in mid-September 1943 when three-rotor Bombes became 1943. more freely available following the entry into service of Clarke was good at Banburismus and was so enthu- the US Navy’s four-rotor Bombes. By the end of 1943, siastic and enthralled that she would sometimes not the US Navy codebreaking unit (Op-20-G) had assumed hand over her work to the next shift but would stay responsibility for breaking the four-rotor cipher, Triton to see if a few more tests would give a result. One of (Shark to Bletchley Park), that the Atlantic U-boats used. Clarke’s important contributions to the work of Hut 8 This led to many of the Hut 8 staff being transferred accelerated the solution of Naval Enigma Offizier signals, to other work, but Clarke remained in Hut 8 as a highly which were often extremely difficult to break. Some were capable member of a small team that broke Naval Enigma never solved, since they were reenciphered with a second ciphers until the end of the war. set of plugboard settings. Leslie Yoxall devised the first stage of an ingenious statistical method for recovering Postwar Period Stecker settings from messages of about two hundred After the war, Clarke transferred to the successor of both letters or more. It involved a graphical procedure and Bletchley Park and GCCS, the Government Communica- was called a dottery, because the cryptanalyst penciled tions Headquarters (GCHQ) at Eastcote, where she met a dots into the cells of a square matrix to represent letter colleague named Lieutenant-Colonel J. (Jock) K.R. Murray, correspondences. whom she married in 1952. She was appointed a Member A day or two after Yoxall’s invention, Clarke invented a of the British Empire in 1947 for her codebreaking work method for greatly speeding up the “routine method,” the during the war. Because of Murray’s poor health, the Mur- first stage of the dottery procedure. Clarke’s contribution rays moved to Scotland, where they developed a shared was the second stage of the procedure. It took the interest in Scottish history.

254 Notices of the AMS Volume 64, Number 3 COMMUNICATION Celebrating After her retirement, Clarke helped Sir Harry Hinsley on what became Appendix 30 to volume 3, part 2 of the 1988 British Intelligence in the Second World War—a substantially revised assessment of the Polish, French, Women's History Month and British contributions to breaking Enigma.

Summary Looking Back a Clarke was a lady in the tradition of her day. She was congenial but shy, kind, gentle, truthful, nonaggressive, Quarter-Century agreeable to all, and always subordinate to the men in her life, except in Hut 8, where she was treated as Twenty-six years ago, the September 1991 issue an equal. Although she recognized and later mentioned of the Notices carried a special issue devoted to some of the impediments and unfairness she met as the theme “Women in Mathematics.” Below are a woman at Cambridge and Bletchley Park, she never directly confronted them. some of the pieces from that issue: She was a highly intelligent person of exceptional gifts. Clarke was fully able to hold her own in the “In Her Own Words: Six Mathematicians Reflect male-dominated world of outstanding British wartime on Their Lives and Careers’’ codebreaking—she was one of only three female cryptan- (with contributions by Joan S. Birman, Deborah alysts who worked on the different Enigma machines. She was an enthusiastic and encouraging colleague who was Haimo, Susan Landau, Bhama Srinivasan, Vera much admired by all who worked with her at Bletchley Pless, and Jean E. Taylor) Park and GCHQ. Although the remaining secrecy associated with crypt- “The Past, Present, & Future of Academic analysis still makes it impossible to be more specific Women in Math Sciences’’ by L. Billard about her accomplishments, it is clear that her work on the Naval Enigma helped to shorten the war and saved many lives on both sides of the conflict. “Top Producers of Women Mathematics Doctorates’’ by Allyn Jackson —Ralph Erskine, I.J. (Jack) Good, Eric A. Weiss “Mathematics and Women: The Undergraduate School and Pipeline’’ Photo Credits Photo of women working on codebreaking in Bletchley by D. J. Lewis Park are ©Crown. Reproduced by kind permission, Director, GCHQ. “Merging and Emerging Lives: Photo of Joan Clarke reproduced, with permission of the Women in Mathematics’’ Clarke family, from www.bletchleyparkresearch.co. by Claudia Henrion uk/research-notes/women-codebreakers/. “The Escher Staircase’’ by Jenny Harrison

“Mathematics and Women: Perspectives and Progress’’ by Alice T. Schafer

“A Brief History of the Association for Women in Mathematics: The Presidents’ Perspectives’’ by Lenore Blum [Available on the AWM web site at www.awm-math.org/articles/ notices/199107/blum/]

March 2017 Notices of the AMS 255 COMMUNICATION

Commutative Algebra Provides a Big Surprise for Craig Huneke’s Birthday

Irena Swanson Communicated by Tom Garrity

A commutative algebra conference in July 2016 on the occasion of Craig Huneke’s sixty-fifth birthday, included a major mathematical surprise. Craig Huneke has been at the forefront of research in commutative algebra, introducing and advancing several influential notions, such as d-sequences, licci ideals, symbolic powers, ho- mological methods, During the computational meth- ods, tight closure, banquet, Hochster uniform bounds, and read his poem prime characteristic methods. He has men- honoring Huneke. tored twenty-four PhD Group photo of the conference participants, Craig students and many Huneke and Melvin Hochster in the center. Claudia postdocs; he has Polini said that commutative algebra in general is coorganized conferences, such as the Kansas-Missouri- friendly to women (possibly due to Emmy Noether Nebraska KUMUNU commutative algebra conference; he being one of us) and in particular how Huneke has has served on the Executive Committee of the AMS and been a tremendous role model as a teacher, on the Board of Trustees of the MSRI. collaborator, mentor, and organizer. Almost all the talks were directly related to Huneke’s work, and most speakers started their talks describing how Huneke affected their work as a mathematician and as a person: they talked about his mathematical productivity, Clowes, professor of Slavic languages and literatures, and extensive collaborations, productive and precious lunches their children Sam and Ned, has been a shining example with napkin notes, excellent and influential talks, his for possibilities in home and professional life for women advising, mentoring, his friendly competitiveness, and so in academia. During the banquet, Hochster read his poem on. Claudia Polini addressed how commutative algebra honoring Huneke. The most colorful roast was given in general is friendly to women (possibly due to Emmy by Hailong Dao, who mused how he had turned down Noether being one of us) and in particular how Huneke has a postdoc offer from the University of Kansas, where been a tremendous role model as a teacher, collaborator, Huneke was based at the time, and when Hailong later mentor, and organizer. His own family life, with wife Edith joined the University of Kansas in a tenure-track position, Irena Swanson is professor of mathematics at Reed College. Her Huneke “won” the postdoc standoff by leaving after e-mail address is [email protected]. Hailong’s three years there. Neal Epstein, Karl Schwede, For permission to reprint this article, please contact: and I wrote a song that the whole room sang for Huneke [email protected]. to the tune of Woodie Guthrie’s “This Land Is Your Land,” DOI: http://dx.doi.org/10.1090/noti1480 one of Huneke’s favorite songs.

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Huneke’s postdocs, from left to right: Karen Smith, Alberto Corso, Graham Leuschke, (Craig Huneke), Luis Núñez-Betancourt, Ian Aberbach, Adela Vraciu, Melvin Hochster read his touching poem and Jeff Mermin. Standing in the background: PhD presented his plaque to Craig Huneke. students Janet Striuli and Branden Stone.

There were twenty-nine talks over six days, but the we do not yet have a decent bound on the regularity of really big deal was two consecutive talks on day five by all reduced equidimensional ideals. The Eisenbud-Goto Jason McCullough and Irena Peeva on Rees-like algebras. Conjecture was aiming to fill in this gap. Special cases As an organizer I was aware that the two wanted to give have been proved. consecutive talks, but I was not aware of their research In their talks at the conference, McCullough and Peeva collaboration, nor did their abstracts hint at anything reported on their new techniques for handling projective field-changing. Whereas their talks were spectacular in dimension and Betti numbers: Rees-like algebras and content and in the changes they are bringing to the field, step-by-step homogenization. their abstracts were clever decoys, Peeva’s just mentioning The classical Rees algebra of an ideal is a workhorse “some open questions.” for analyzing powers of that ideal in one fell swoop, and It turns out that Rees-like algebras are very interest- the projective scheme of a Rees algebra is the blowing up ing and powerful and that the discussion of the open of the spectrum of the ring along the subscheme defined questions was the big deal! by the ideal. Explicitly, the Rees algebra of an ideal 퐼 McCullough started with an announcement that Peeva in a commutative ring 푆 is a subring of the polynomial and he were coordinating their talks and tried hard to ring 푆[푡] in one variable 푡, and it is generated over 푆 by get them ready for the conference as a birthday surprise elements 푎푡 as 푎 varies over 퐼. The classical extended of a sort, which meant that the audience would have to Rees algebra of 퐼 is 푆[퐼푡, 푡−1], which is generated over come back for Peeva’s talk to get the rest of the story. 푆[퐼푡] by 푡−1. Huneke has done great work with Rees Early in his talk McCullough brought up the Castelnuovo- algebras, including analyzing them for ideals generated Mumford regularity (the smallest integer such that related by d-sequences, determining their Cohen-Macaulay and cohomology modules vanish in shifted higher degrees) integral closure properties, and exploring their analogues and some open questions in this area, such as the Eisenbud- for modules with Eisenbud and Ulrich (2002). Goto conjecture and the Bayer-Stillman conjecture. The McCullough and Peeva defined and developed a new audience was now abuzz: is the surprise about resolving Rees-like algebra 푆[퐼푡, 푡2] of 퐼, generated over 푆[퐼푡] by 푡2. one of these two? Which one? Or is it something else? Is This notion is a generalization of a local example due the answer positive or negative? to Hochster where he produced a family of prime ideals The Eisenbud-Goto conjecture ([EG], 1984) estimates with fixed embedding dimension and fixed Hilbert-Samuel the Castelnuovo-Mumford regularity reg (푃) of a homo- multiplicity but arbitrarily many minimal generators. geneous prime ideal 푃 in a polynomial ring 푆, such as For the context of the conjectures, let 푆 be a polynomial ℂ[푥1, … , 푥푛], in terms of the degree (multiplicity) of 푆/푃 ring over a field; let 퐼 be an ideal in 푆 generated by and the codimension of 푃: homogeneous elements 푓1, … , 푓푚; let 푇 be the polynomial ring generated over 푆 by 푚 + 1 variables 푦1, … , 푦푚, 푧; let 2 reg (푃) ≤ deg(푆/푃) − codim (푃) + 1. 휑 ∶ 푇 → 푆[퐼푡, 푡 ] be the 푆-algebra map taking 푦푖 ↦ 푓푖푡 and 2 Bayer and Mumford (1993) had discussed that the 푧 ↦ 푡 ; and finally, let 푄 be the kernel of 휑. McCullough missing link between the sharp doubly exponential bound and Peeva proved the following: on regularity of arbitrary homogeneous ideals and the 1. 푄 is a prime ideal (minimally) generated by nice bounds on regularity in the smooth case is that the elements {푦푖푦푗 − 푧푓푖푓푗 ∶ 1 ≤ 푖, 푗 ≤ 푚} and

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number of generators as 푄, and its dimension, depth, pro- jective dimension, Castelnuovo-Mumford regularity, and degree are all well understood and well behaved. For example, the homogenized prime’s quotient has degree 푚 equal to 2 ∏푖=1(deg(푓푖) + 1), and its projective dimension equals 푚 − 1 plus the projective dimension of 푆/퐼. This is a beautiful and powerful construction, all by itself! We got this far in the second talk, delivered by Peeva. We were close to the inevitable big announcement, but Peeva paused right there, leaving us in suspense a little longer, to address the honoree of the conference with high praise for his research, service, and enormous impact on the field. The context made her words all the more powerful. Huneke’s PhD students at the conference, from left to And then she went for what their birthday surprise was right: Yongwei Yao, 2002; Manoj Kummini, 2008; Neil all about. Namely, starting with an arbitrary homogeneous Epstein, 2005; Giulio Caviglia, 2004; Branden Stone, ideal in a polynomial ring 푆, via step-by-step homogeniza- 2012; Craig Huneke; Irena Swanson, 1992; Janet tion of the defining ideal of its Rees-like algebra, one Striuli, 2005; Ilya Smirnov, 2015; Elóisa Grifo, current gets a prime ideal 푃 in a polynomial ring 푆 with precise student; Alessandro De Stefani, 2016. control on the dimension of 푆, on the number of gener- ators of 푃 and their degrees, on the dimension, depth, multiplicity, regularity, and projective dimension of 푆/푃. 푚 푚 {∑푖=1 푐푖푗푦푖 ∶ ∑푖=1 푐푖푗푓푖 = 0}. Thus we have an ex- In particular, if they start with the famous Mayr-Meyer plicit set of generators of 푄, the latter kind arising ideals (1982) or variant ideals due to Koh (1998) and from the first syzygies of 퐼. (In contrast, the Bayer and Stillman (1988), the prime ideals constructed generators of the presenting ideal of the classical via this machinery yield counterexamples to the Eisenbud- Rees algebras are not well understood at all.) Goto conjecture and the Bayer-Stillman conjecture. And 2. If we assign deg(푦푖) = deg(푓푖) and deg(푧) = 2, worse, McCullough and Peeva showed that the regularity then 푄 is a homogeneous ideal in 푇. Furthermore, if of reduced irreducible schemes is not bounded above by deg(푓푖) ≥ 2 for all 푖, the graded Betti numbers of 푇/푄 any polynomial in the degree. Peeva finished her talk with over 푇 equal the graded Betti numbers of 푇/(푄+(푧)) a discussion of compelling future redirections. over 푇/(푧) since 푧 is a non-zerodivisor. These Betti Peeva’s and McCullough’s lectures were masterfully numbers are well understood because the minimal delivered: they carried amazing content, and the suspense free resolution of 푇/(푄 + (푧)) over 푇/(푧) is a was just right. The two got huge applause. mapping cone with ingredients being the minimal 푆- Huneke, instead of the usual questions for the speakers, resolution of 퐼 and the well-known Koszul resolution turned to the audience and said: “When I was seven years and the Eagon-Northcott resolution. old, I wanted a bicycle. I really, really wanted a bicycle. The only “problem” with Rees-like algebras is that the And I got it for my birthday, and that was the best birthday constructed polynomial ring 푇 is not standard graded and present to me ever—until this present right now.” so not immediately relevant for resolving the Eisenbud- These results of McCullough and Peeva are reshaping Goto and the Bayer-Stillman conjectures. The classical the field of commutative algebra: their counterexamples approach is to change the degrees of the variables to to the Eisenbud–Goto conjecture not only show that the 1 and then homogenize the ideal. However, Betti num- conjectured bound does not hold but also that there bers change under homogenization, and so all of the is no bound on the regularity of irreducible varieties information about the Rees-like algebra would be lost in this way. McCullough and Peeva invented a way to that is a polynomial function of the degree. Furthermore, remove the nonstandard obstacle, and they called their McCullough and Peeva produced a beautiful and general process step-by-step homogenization. They homogenize machinery that is worthy of deeper study, and in fact they the ideal repeatedly, one variable at a time. The striking are working with collaborators on further papers. discovery is that this process preserves the Betti numbers The summer of 2016 was amazing for commutative if the ideal is prime, as it is in their case. In contrast to algebra: McCullough and Peeva resolved a thirty-two- classical homogenization, where one needs to homoge- year-old problem, then Yves André proved the direct nize a Gröbner basis in order to obtain generators of the summand conjecture which was posed in its general form homogenized ideal, for the step-by-step homogenization by Mel Hochster forty-seven years ago, Bhargav Bhatt it suffices to homogenize the generators of 푄. Without significantly shortened the proof, and Tigran Ananyan going into too much detail, the homogenized 푄 is a prime and Mel Hochster gave a positive answer to Stillman’s ideal in a standard graded polynomial ring, it has the same question.

258 Notices of the AMS Volume 64, Number 3 COMMUNICATION Celebrating Acknowledgments. I thank Melvin Hochster for the lion’s share of the organization of the conference, in- cluding convincing Huneke that a conference was in order. I also thank Craig Huneke, Jason McCullough, Women's History Month Irena Peeva, Marilina Rossi, and Bernd Ulrich for feed- back on early versions of this article, and Bernd Ulrich for sharing the slides of his talk. Links for Women References [EG] D. Eisenbud and S. Goto, Linear free resolutions and mini- in Mathematics mal multiplicity, J. Algebra 88 (1984), 89–133. MR 0741934

Photo Credits African Women in Mathematics Association: Group photo of the conference participants is courtesy of africanwomeninmath.org/ Margie Morris. Photo of Melvin Hochster reading his poem is courtesy of Margie Morris. Association for Women in Mathematics: Photo of Huneke’s postdocs is courtesy of Irena Swanson. https://sites.google.com/site/awmmath/home Photo of Huneke’s PhDs students at the conference is courtesy of Margie Morris. Photo of Irena Swanson is courtesy of Irena Swanson. Association for Women in Science: www.awis.org

European Women in Mathematics: www.europeanwomeninmaths.org/

IMU Committee for Women in Mathematics: www.mathunion.org/cwm

Women in Math Project: darkwing.uoregon.edu/~wmnmath/

2017 Women in Mathematics, Science, and Technology Conference: www.millersville.edu/wmsc/

2017 Workshop “WIN4 Women in Numbers 4”: www.birs.ca/events/2017/5-day- workshops/17w5083

2017 Workshop “Women in Control: New Trends in Infinite Dimensions”: www.birs.ca/events/2017/ 5-day-workshops/17w5123

March 2017 Notices of the AMS 259 COMMUNICATION

Math PhD Careers: New Opportunities Emerging Amidst Crisis

Yuliy Baryshnikov, Lee DeVille, and Richard Laugesen

smart and industrious people), but few take this route well prepared. Often these graduates fall into their jobs by chance, after some struggle, and frequently with a feeling that an indelible stigma is attached to their nonacademic career choice. This is a sign of our failure as mentors. Regardless of one’s opinion on the proper goals of PhD study, the fact that a significant fraction of graduates One of the most troubling trends in the mathematics from institutions of all types will spend the bulk of their profession is the shortage of regular positions in academia professional lives outside mathematical academia forces for recent PhDs, as described in Amy Cohen’s op-ed in upon us the moral the October Notices [Coh16]. responsibility of pro- Our view is that the demand for PhD mathematicians is Academia is not a viding opportunities there, but we just fail to recognize it. True, many PhD grad- that will prepare grad- uates will leave mathematical academia, but that does not closed ecosystem. uate students for mean they must leave mathematics. That mathematics is intellectually challeng- critically relevant to the needs and challenges of twenty- ing and professionally rewarding careers that continue to first-century society is a cliché, but still true. Scientific involve mathematics. and engineering fields have become more quantitative The path from academia to a new career is rarely and computational, and the opportunities for mathemati- smooth. Hurdles and challenges—from relatively trivial cians to contribute wherever sophisticated models are ones such as learning to code to such existential ones as used—be it in big data, neuroscience, climate science, abandoning the quest for pure knowledge—are abundant telecommunications—are expanding rapidly. These op- along the way. We as advisors rarely have relevant experi- portunities are often not explicitly acknowledged by us ence ourselves. More often than not, we faculty members in our roles as mentors and hence can be hard for our are unable to guide our students at this critical juncture mentees to identify and pursue. in their careers. Thus mathematics graduates often pursue such op- We cannot expect to overcome this hurdle by having all portunities on their own through a slow and sometimes mathematical faculty suddenly develop industry connec- painful osmosis-like process. Most of us know personally or by sight the students from our departments who have tions (although it doesn’t hurt to try). But we can choose “gone into industry,” in many cases after growing weary of to speak out in our own departments, in the common the rat race in pursuit of an elusive tenure-track position. rooms, and in faculty meetings, and advising meetings, Many become quite successful in their new careers (being to say that academia is not a closed ecosystem and that our graduates should explore the full range of options, Yuliy Baryshnikov is professor of mathematics and electrical and including the intellectually rewarding, satisfying, and sta- computer engineering at the University of Illinois, Urbana– ble jobs outside universities that rely on mathematical Champaign. His e-mail address is [email protected]. expertise and have tangible impacts on our economy Lee DeVille is professor of mathematics at the University of Illinois, and society. Students can seek careers in national labs, Urbana–Champaign. in other academic disciplines, in industrial research, in His e-mail address is [email protected]. NGOs and governmental research organizations. Mathe- Richard Laugesen is professor of mathematics and director of matics is vital for society to move forward. We just need graduate studies at the University of Illinois, Urbana–Champaign. to start believing our own words and putting them into His e-mail address is [email protected]. practice through our students. For permission to reprint this article, please contact: Another hurdle faced by students willing to venture [email protected]. “outside” to industry is the reluctance of many potential DOI: http://dx.doi.org/10.1090/noti1483 employers to hire a mathematician. Department heads

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of the notion of an “in-house” mathematician in every company facing technical or theoretical challenges. Turning now from diagnosing the problem to proposing solutions, we ask: how can we help our students to overcome career transition hurdles in the short term, and in the longer term create an explicit appreciation in industry and government of the need for mathematically trained personnel? We believe that at the very least, we need to begin providing to all interested students the skill sets that will enable them to transition more readily to careers outside mathematics departments and that we should educate them, in specific, hands-on ways, about the options outside mathematical departments. To be clear: we do not advocate turning mathemati- cal graduate education into These a training funnel for corpo- changes can Teaching assistant Stefan Klajbor Goderich (center) rate R&D or pushing any of helps graduate students William Linz (left) and our students away from the- together Colleen Robichaux (right) with a Python programming oretical mathematics or away assignment during the beginning-of-summer from academia. However, we unlock a huge Computational Mathematics Bootcamp. should work to change the gen- eral level of ignorance among untapped and project managers in industry know what they get academic mathematicians of by hiring a statistician or an electrical engineer. But a what mathematicians can do reservoir of mathematician? What are they good for?!1 Unless the outside universities and to career enterprise has previously employed PhD mathematicians, change the ignorance prevalent the hiring managers have little experience to draw upon. in industry of the benefits of opportunities And indeed, few of our graduates really are ready to hiring mathematicians. These walk right in and start work in a typical industrial R&D changes can together unlock for our environment. The training they need might be minor a huge untapped reservoir of compared to what they have already learned in graduate career opportunities for our graduate school, but still it creates an additional risk and burden graduate students. that industrial research units might be unwilling to bear. The good news is that to in- students. We do have allies in industry, consisting at a minimum stigate these changes, we need of those of our graduates who went there. A back-of- not radically reshape what our students do during gradu- envelope calculation2 shows that at least six hundred ate school or how we operate professionally as academics. PhD graduates in mathematics leave academia yearly. The Small but sustained tweaks in graduate training and a majority of them do not get the word “mathematician” robust effort to connect with potential employers will in their job title (indeed, the total number of “mathe- suffice. maticians” outside academia in the US is a paltry three With these principles in mind, at the University of thousand five hundred [USB]). Nonetheless, these individ- Illinois we created the Program for Interdisciplinary uals work in demanding and technical environments that and Industrial Internships at Illinois (PI4), with much- use their mathematical talent and training at a maximal appreciated support from the National Science Foundation level. These “hidden mathematicians” enable scientific under grant DMS MCTP 1345032. and technological progress at a level we are generally The main goal of the PI4 program is to expose math- unaware of. One of the long-term goals for the mathemat- ematics PhD students to alternative career paths early ical community should be to generate broad acceptance in their graduate careers so that they understand what skill sets they should acquire by the time they graduate. 1 V. I. Arnold quoted a famous physicist, Y. Zeldovich, as remark- Interested students can then prepare for a career path in ing that formulating a problem precisely is valuable because it industry or government during the middle of their PhD allows one to involve mathematicians, who, “like flies, can walk on program (rather than in a rush at the end), for example, the ceiling” [Sun04, p. 198]. This attitude is what we would like to by picking up coding skills from online MOOCs over see in potential employers! winter break or by taking a course in machine learning 2The number of new PhDs in mathematical sciences is about 1900 per year, and one third of these graduates take their first job in a from the statistics department and also doing one or nonacademic institution. Meanwhile, the number of tenure-track more internships to develop the soft skills of teamwork positions under recruitment is fewer than 1000 annually, and not and project management. The secondary goal of PI4 is all these positions get filled [AMSa], [AMSb]. to develop a cadre of supportive local employers who

March 2017 Notices of the AMS 261 COMMUNICATION have experienced first-hand the value provided to their organization by hiring a mathematics student. Program components include: in-semester topics courses such as Top Ten Algorithms for the Twenty-First Century, a beginning-of-summer Computational Boot- camp, summer working groups for junior students on exploratory topics such as “Topologically constrained problems of statistical physics, and internship placements for both junior and senior students. The two-week intensive Computational Bootcamp kicks off the summer, teaching practical techniques with Python programming to students having little or no programming experience. Each day, students: (i) learn new concepts, (ii) test their understanding on short programming exercises, (iii) present and critique student project solutions, and (iv) code the next project. Graduate students Derrek Yager (left) and Vaibhav The examples typically center around numerical linear Karve (not pictured) uncovered low-dimensional algebra, optimization, graph theory algorithms, differen- structure in New York City taxi data during a tial equations, and data fitting. It is an intensive program, scientific internship with Professor Richard Sowers with most of the students’ days spent programming. (right). Their work reduced the dimension of the Students then split into three tracks for the remainder problem from hundreds of thousands down to about of the summer: Prepare, Train, and Intern. The Prepare fifty dominant roadway combinations. cohort is typically made up of incoming students and students at the end of the first year of graduate school. Dow AgroSciences, and customer data analytics work at They work in a group under the supervision of a faculty utility company Ameren. member (sometimes local, sometimes external), and it is Interdisciplinary internships are something completely useful to think of this group as an enhanced Research new for us. We embed a student in a scientific lab on Experience for Undergraduates, an “REU++.” The students our campus, where they tackle a mathematical problem spend most of their summer learning about a subject and that supports the agenda of that lab. Almost always the then some time at the end working on questions related student works in the lab space or is colocated with the to it. This subject usually has a computational tilt, but it group in some other manner. We request that the mentors is still open-ended research in some mathematical topic, think of this student as a member of the lab, so that the such as random matrix pencils, for example. Members of student learns the techniques required and soaks up the the Train cohort tend to be more senior students, with culture of the field. We really want the student to get a one or two years of graduate school experience, who feel for what it is like to work in that discipline and how work in small groups on a focused research problem one can contribute with mathematics. in mathematics, where simulations and experimentation Internships attract must be combined with theoretical analysis. These parts students from all ar- of the summer program are useful for all mathematics What we are doing eas of mathematics, graduate students as they transition from course-takers to researchers. here addresses a not just those in tradi- The Intern stage is the apex of the program. We directly tionally applied fields. support about twelve students each summer in two tracks: critical need in the Recently, for exam- industrial and interdisciplinary internships. ple, a number theorist Industrial internships are hosted by enterprises having mathematical modeled ant colony a local R&D outlet. The financial support from the NSF behavior in the en- grant enables us to achieve two objectives. First, we can community. tomology department, work with smaller firms, such as startups or small research and a logic student an- groups at companies that are not considered traditional alyzed fMRI data as part of a larger project trying to find players in research, that would not normally be able to physiological imprints of tinnitus. support interns on their own. Second, the financial support Once the internship culture is established, students reduces the perceived risk to new industrial partners when start finding their own positions and funding. Our de- they hire the unknown quantity of a mathematics intern. partment has 210 graduate students, including 160 in By getting a foot in the door, we lay the groundwork for the PhD program. Internship numbers have grown from them to hire interns with company funds in future years. six in 2013 to a total of forty-two interns in 2016, with Internship examples include image recognition research thirty-one internships taking place over the summer and at the start-up Personify, weed resistance modeling at eleven during the fall or spring semesters.

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Intern Success Story

Byron Heersink (PhD in number theory expected 2017) did a summer internship through the PI4 program on “A Response Threshold Model for Ant Colonies,” a project supervised by faculty members Samuel Beshers (Entomology) and Lee DeVille (Mathematics). Byron developed a computational model of the division of labor in social insect colonies. There is a large literature modeling, calibrating, and checking the individual- based rules insects might use for specialization, but it is difficult to determine how these rules turn into colony-level behaviors. Byron’s simulations revealed two surprising effects in the models considered: the degree of specialization is mostly independent of the size of the colony, and the overall colony task behavior depends most strongly on relatively few workers most sensitive to each task and not on the average task Biology professor Carla Cáceres (left) mentored sensitivity of the colony. graduate student Vanessa Rivera Quiñones (right) in This project was Byron’s first experience of mathe- her lab for a summer scientific internship. This matics outside the mathematics department, and his ongoing work aims to develop biologically realistic first experience of computational modeling. He subse- SIR population models, applied initially to quently took these skills to an internship at Sandia evolutionary branching in organisms such as water National Laboratories and then to a second intern- fleas. ship at HRL Laboratories, and he is now applying for postgraduation positions in industry as well as academia. National Internship Efforts

The NSF-IPAM Mathematical Sciences Internship Work- shop Report [IPA15] proposes a national effort Internships are not slowing down the students’ time to at multiple scales to provide career and intern- degree here at Illinois. Students who aim at an industry ship placement resources for mathematics graduate or government career after doing an internship develop a students: sharpened focus that speeds up their academic progress. • National: create a network to increase intern- The type of industry careers sought by PhD students ship information exchange, data collection, at Illinois is shifting. Some still aim at the world of access, and opportunities; finance, but national labs and data science careers are • Regional: establish internship centers to build increasingly valued. The PI4 student cohorts are moving internship contacts and organize training through our PhD program, and we view the coming opportunities; diffusion of mathematical talent into the broader world • Local: encourage and enable student participa- as a highly positive development. tion in internships in mathematical sciences departments. We believe what we are doing here addresses a crit- ical need in the mathematical community: the need to A new BIG Math Network is promoting this concept (BIG = Business + Industry + Government), which has open to our graduates a variety of rewarding, creative, generated enthusiasm at national societies including and intellectually challenging mathematical careers. Many AMS, SIAM, MAA, ASA, AWM, NAM, AMATYC, INFORMS, graduate students will continue to follow a traditional and SACNAS. Supporters who want to volunteer time academic career path, but having the option to choose and effort or who simply want to stay in the loop careers in industry and governmental organizations will can keep informed at the website [BIG] and sign up at benefit all of them. tinyurl.com/BIGmathnetwork. Mathematics departments must change their culture and (some of) their practices in order to remain relevant References in the coming century. Certain departments around the [AMSa] AMS Annual Survey of the Mathematical Sciences: Aca- country are already doing so. We hope this article about demic Recruitment and Hiring, www.ams.org/ our experiences at Illinois will foster a conversation in profession/data/annual-survey/hiring, accessed 2016-10-13. many more mathematics departments about how the [AMSb] AMS Annual Survey of the Mathematical Sciences: community can move forward from the current crisis Doctorates Granted, www.ams.org/profession/data/ toward new opportunities for our PhD graduates. annual-survey/docsgrtd, accessed 2016-10-13.

March 2017 Notices of the AMS 263 COMMUNICATION

[BIG] BIG Math Network, www.siam.org/bigmathnetwork. [Coh16] Amy Cohen, Disruptions of the academic math em- ployment market, Notices of the AMS 63 (2016), no. 9, 1057–1060. [IPA15] NSF-IPAM Mathematical Sciences Internship Workshop Report, goo.gl/QYe3ie, 2015. [Sun04] Rashid Alievich Sunyaev (ed.), Zeldovich: Reminis- cences, CRC Press, 2004. [USB] United States Department of Labor, Bureau of Labor Statistics, Occupational Outlook Handbook, www.bls. gov/ooh/math/mathematicians.htm, accessed 2016- 10-13.

Photo Credits Photos in article are courtesy of University of Illinois at Urbana-Champaign. Photo of Lee Deville is courtesy of Brian Stauffer. Photo of Richard Laugesen is courtesy of Darrell Hoemann.

264 Notices of the AMS Volume 64, Number 3 BOO KREVIEW A Doubter’s Almanac

A Review by Sheldon Axler

Milo’s relationship with his family, especially his son, who A Doubter’s Almanac is also a mathematician. At this point the book also has Ethan Canin a dramatic shift of viewpoint, using a nice literary device Random House, 2016 that I will not disclose here because I enjoyed the surprise US$28.00, 551 pages and do not want to spoil it for future readers. ISBN: 978-1-4000-6826-5 Surely this novel’s appearance on bestseller lists was not due to its mathematical context. I assume that non- Ethan Canin’s novel A Doubter’s Almanac follows fictional mathematician readers were attracted by the beautifully mathematician Milo Andret through several decades, in- written portrait of a seriously flawed genius and his cluding graduate school, winning a Fields Medal, and de- family struggling with cline. The author is not a mathematician, although he was alcohol and drug prob- an engineering major at Stanford before switching his ma- lems in the midst of The novel does a jor to English. He then went on to obtain an MD degree too many lies and de- before leaving medicine for a successful career in writing. ceptions. Although I fine job of Several of his novels, including the one under review, have appreciated the well- depicting the made the New York Times bestseller list. crafted prose and The first part of the book concentrates on Milo’s life loved the mathemat- difficult creative up to a few years after getting his PhD at UC Berkeley. ical setting, I did not A topologist, he receives his Fields Medal for proving like the nasty Milo, process of the (fictional) Malosz conjecture. Now world-famous, he and I found the last 60 searches for another big open problem to tackle, finally percent of the book to mathematics. deciding on the Abendroth conjecture (also fictional) be depressing. after briefly considering the Goldbach conjecture (every All three universities at which fictional mathematician even number greater than 2 is the sum of two primes). Milo Andret was a student or faculty member are also Milo’s partial progress on the Abendroth conjecture places where I was a student or faculty member (Prince- leads to four excellent papers: one published in Annals of ton, UC Berkeley, Michigan State University). Thus I paid Mathematics, one in Acta Mathematica, and two in Inven- special attention to the representations of the mathemat- tiones. Milo’s confidence that he will crack the Abendroth ical culture of those institutions. Here are the most signif- conjecture and reap even more fame is shattered when icant departures from reality that I noticed in this book: a proof of the Abendroth conjecture is discovered by a • The book gives the following description of mathemat- teenager. ics graduate students at UC Berkeley at the time that Milo never recovers from losing the race to prove the Milo was a graduate student there: Abendroth conjecture, and he is never again mathemat- At the time, most of his classmates hoped to ically productive. The remaining 60 percent of the book find jobs at Xerox in Palo Alto or at IBM in deals with decades of decline and despair, focusing on White Plains or at one of the industry-funded think tanks that were popping up now along Sheldon Axler is professor of mathematics at San Francisco State the coast. University. His e-mail address is [email protected]. In fact, as I know because I was a graduate student at For permission to reprint this article, please contact: UC Berkeley at the same time as (the fictional) Milo, a [email protected]. large majority of our classmates wanted academic DOI: http://dx.doi.org/10.1090/noti1491 jobs.

March 2017 Notices of the AMS 265 That’s a pretty big secret to keep. Except it should not have gone that way because Milo was awarded the Fields Medal at the ICM held in War- saw. That meeting of the ICM was originally scheduled for the summer of 1982 in Warsaw. However, in De- cember 1981 martial law was declared in Poland and Solidarity was suppressed. The International Mathe- matical Union, which organizes the ICM, decided that the ICM could not be held in those conditions. Thus the ICM that was scheduled for 1982 was postponed until the summer of 1983, still in Warsaw. The Interna- tional Mathematical Union announced the winners of the Fields Medal in 1982 rather than waiting until the postponed ICM in 1983. Thus the book is inaccurate in describing an element of secrecy concerning the The main character in A Doubter’s Almanac was a announcement of Milo’s Fields Medal at the Warsaw mathematics professor at Princeton University in the ICM. 1970s, with an office in Fine Hall, the tower on the • Late in the book, Milo’s granddaughter is supposed right in this picture. Is the book accurate in suggest- to be reading Swiss Family Robinson, but her father ing that neckties were common attire among Prince- discovers that she is actually reading ton mathematicians in the 1970s? The reviewer, who Zygmund and Fefferman’s Trigonometric Se- earned his bachelor’s degree at Princeton in 1971, ries. remembers a largely necktie-free department. This is a family of geniuses, so I think it’s a reasonable stretch to have a child reading this classic and highly respected work. However, the authorship above is • The book refers to the mathematics department at wrong. The first edition of this book appeared in Princeton University as the 1935, with Antoni Zygmund as the sole author. In Department of Broken Englishes—that’s how 1959 an expanded second edition was published in they were known around campus. two volumes, again with Zygmund as the sole author. I was an undergraduate at Princeton a few years before In 2003 a third (and so far final) edition was published, the time of this remark, and I cannot recall ever hear- ing a complaint from a fellow student about the Eng- with a forward by Robert Fefferman but still with Zyg- lish language skills of any mathematics department mund as the sole author. Mathematicians working in faculty member. Fourier series refer to these volumes as “Zygmund.” • At one point when Milo is a tenure-track-but-still- I have never heard anyone call them “Zygmund and untenured faculty member in the Princeton math- Fefferman.” ematics department, he is called into a committee I noticed four more tiny errors, all less consequential meeting of the nine most senior faculty members than those discussed above. Overall, I consider the num- of the department. Here is the book’s description of ber of errors in this 551-page novel written by a nonmath- Milo’s perception of this committee: ematician about a mathematician to be remarkably low. A startlingly uniform wall of bulbous Semitic The author gets many big issues right. In particular, features, threadbare sport coats, and colorless the novel does a fine job of depicting the difficult creative ties. process of mathematics. The reader will come away with I am at a loss as to how to deal with the author’s use the understanding that even a superstar mathematician’s of the phrase “bulbous Semitic features.” Thus I will research breakthroughs come only after being stuck with just mention the inaccuracy at the end of the sentence little progress for long periods of time. above: my memory as a student at Princeton is that Mathematicians who read this book will appreciate nu- faculty rarely wore ties, and I never saw anything like merous tidbits that other readers will miss. For example, nine of them wearing ties. all the chapter titles have a mathematical meaning, start- • The Fields Medals are awarded at the once-every-four- ing with “Induction” as the title of the first chapter and years meeting of the International Congress of Mathe- ending with “Proof” as the title of the last chapter. As an- maticians (ICM). The Fields Medalists are notified in advance of their award so that they can be sure to other example, Milo’s daughter, Paulette, is named after attend the ICM, but otherwise the names of the win- Paul Erdős (the book gets the Hungarian double accute ners are kept secret until the announcement at the accent correct, in contrast to the incorrect umlaut that ICM. That’s how it goes in this novel. After Milo returns one often sees decorating this mathematician’s name). Al- from the ICM, the chair of the Princeton mathematics though the novel explains that Paul Erdős is a famous department congratulates Milo on his award and on mathematician, readers who are not mathematicians will his discretion in not leaking it even to his colleagues, probably be unaware of the startlingly unusual aspects of telling Milo: Erdős’s career that Milo may have found attractive.

266 Notices of the AMS Volume 64, Number 3 Normally, book reviewers Canin writes with such luxuriant beauty and ten- should not read other reviews a very der sympathy that even victims of Algebra II will of the book in question until follow his calculations of the heart with rapt com- after finalizing their own re- unpleasant prehension. view. In this case, I broke that The Boston Globe review of this book assumed that some rule because the public percep- and extremely of its readers have forgotten arithmetic that they learned tion of mathematics deserves unhappy in elementary school: our attention. Too often we [O]ne need not have studied calculus (or even re- see negative comments about mathematical member how to add and subtract fractions) to ap- mathematics in the media. One preciate the book. of my least favorite examples genius…who Two of the seven reviews that I read justifiably raised comes from the New York issues about the role of women in this book. Milo horri- Times editorial in 1984 en- is no role bly mistreats his lovers, his wife, and his daughter, and dorsing Walter Mondale for US they keep coming back for more. Canin’s blinkered con- president over Ronald Reagan. model ception of his female characters strains credulity. After Trying to balance its positive summarizing one incident in the book when a departmen- comments about Mondale with something negative, the tal secretary finally admits a drunken Milo into her hotel New York Times wrote: room, the reviewer for the New York Times wrote: Walter Mondale has all the dramatic flair of a At this juncture of “no” meaning “yes” (not the trigonometry teacher. only such in these pages), I took a moment to toss To check whether the reviews of this book would be the book across the room. filled with such sentiments, I chose eight large cities in Milo has one great mathematical triumph when he widespread geographic areas of the United States and proves the Malosz conjecture, but everything else about looked for reviews of this book in the main newspaper his life reeks of disaster. I cannot imagine that any reader of each of those cities. Seven of those eight newspapers would want to be Milo, not even in return for a Fields (all except the Atlanta Journal-Constitution) reviewed Medal. The author’s creation of Milo as a very unpleasant this book in 2016, showing that this book received good and extremely unhappy mathematical genius may be a exposure in popular culture. The reviews contained less literary success, but Milo is no role model. hostility to mathematics than I had expected. The worst offender was the Washington Post, whose review said that the book Photo Credit Photo of Princeton University is courtesy of Princeton Uni- is a long, complex novel about math, which versity, Office of Communications; photo by Brian Wil- sounds like the square root of tedium, but sus- son. pend your flight instinct for a moment. Ethan

March 2017 Notices of the AMS 267 From the AMS Secretary 2017Class of Fellows of the AMS

Sixty-five mathematical scientists from around the world have been named Fellows of the American Mathematical Society (AMS) for 2017. The Fellows of the American Mathematical Society program recognizes members who have made outstanding con- tributions to the creation, exposition, advancement, communication, and utilization of mathematics. Among the goals of the program are to create an enlarged class of mathematicians recognized by their peers as distinguished for their contributions to the profession and to honor excellence. The 2017 class of Fellows was honored at a dessert reception held during the Joint Mathematics Meetings in Atlanta, Georgia. Names of the individuals who are in this year’s class, their institutions, and citations appear below. The nomination period for Fellows is open each year from February 1 to March 31. For additional information about the Fellows program, as well as instructions for making nominations, visit the web page www.ams.org/profession/ ams-fellows.

Jeffrey Brock, Brown University For contributions to Kleinian groups, low-dimensional to- pology and geometry, and Teichmüller theory.

Jim Bryan, The University of British Columbia For contributions to algebraic geometry and service to the mathematical community.

Gunnar Carlsson, Stanford University For contributions to algebraic topology, particularly equi- variant stable homotopy theory, algebraic K-theory, and applied algebraic topology.

Mei-Chu Chang, University of California, Riverside For contributions to arithmetic combinatorics, analytic AMS Executive Director Catherine A. Roberts chats number theory, and algebraic geometry. with 2017 Fellows, inlcuding Scott T. Chapman of Sam Houston State University. Sagun Chanillo, Rutgers The State University of New Jer- sey, New Brunswick Dan Abramovich, Brown University For contributions to partial differential equations and For contributions to algebraic geometry and service to the geometric analysis. mathematical community. Scott T. Chapman, Sam Houston State University Guillaume Bal, Columbia University For contributions to algebra and for service to the mathe- For contributions to inverse problems and wave propaga- matical community. tion in random media. Gui-Qiang G. Chen, University of Oxford and Keble College John T. Baldwin, University of Illinois at Chicago For contributions to partial differential equations, nonlinear For contributions to model theory, exposition, and service analysis, fluid mechanics, hyperbolic conservation laws, to mathematics education. and shock wave theory.

Alexandra Bellow, Northwestern University Jungkai Alfred Chen, National Taiwan University For contributions to analysis, particularly ergodic theory For contributions to algebraic geometry and for service to and measure theory, and for exposition. the mathematical community.

Aaron Bertram, University of Utah Mihai Ciucu, Indiana University, Bloomington For contributions to algebraic geometry and for service to For contributions to combinatorics, particularly relating the mathematical community. gaps in lattice tilings to electrostatics. (Continued on next page)

268 Notices of the AMS Volume 64, Number 3 Fellows of the AMS FROM THE AMS SECRETARY

Miriam Cohen, Ben Gurion University of the Negev For contributions to Hopf algebras and their representa- tions, and for service to the mathematical community.

Donatella Danielli, Purdue University For contributions to partial differential equations and geo- metric measure theory, and for service to the mathematical community.

Moon Duchin, Tufts University For contributions to geometric group theory and Te- ichmüller theory, and for service to the mathematical community.

Yalchin Efendiev, Texas A&M University For contributions to the field of multiscale finite-element Attendants at the 2017 Fellows of the AMS Reception. methods with applications to porous-media fluid flow.

Kirsten Eisenträger, Pennsylvania State University For contributions to computational number theory and Dmitry Kleinbock, Brandeis University number-theoretic undecidability. For contributions to homogeneous dynamics and its appli- cations to number theory, especially in metric Diophantine Mark Feighn, Rutgers The State University of New Jersey, approximation. New Brunswick For contributions to geometric group theory. Toshiyuki Kobayashi, University of Tokyo For contributions to the structure and representation theory Rui Loja Fernandes, University of Illinois, Urbana-Cham- of reductive Lie groups. paign For contributions to the study of Poisson geometry and Lie Alex Kontorovich, Rutgers The State University of New algebroids, and for service to the mathematical community. Jersey, New Brunswick For contributions to analytic number theory and for math- Yan Guo, Brown University ematical exposition. For contributions to the mathematical theory of fluids and plasmas. Daniel Krashen, University of Georgia For contributions to the study of central simple algebras Piotr Hajlasz, University of Pittsburgh and local-global principles and for service to the mathe- For contributions to analysis in metric spaces, in particular matical community. the notion of Sobolev spaces in metric-measure spaces also known as Hajlasz-Sobolev spaces. Henning Krause, Universität Bielefeld For contributions to representation theory and homological Kathryn Hess, Ècole Polytechnique Fédérale de Lausanne algebra, and for service to the mathematical community. (EPFL) For contributions to homotopy theory, applications of to- Michael Krivelevich, Tel Aviv University pology to the analysis of biological data, and service to the For contributions to extremal and probabilistic combina- mathematical community. torics.

Nancy Hingston, The College of New Jersey Joseph M. Landsberg, Texas A&M University For contributions to differential geometry and the study of For contributions to differential geometry, geometry of closed geodesics. projective varieties, representation theory, and complexity theory. Yulij Ilyashenko, Cornell University and the National Research University Higher School of Economics Congming Li, University of Colorado, Boulder For contributions to dynamical systems and for service to For contributions to nonlinear partial differential equations the mathematical community. and applications.

Marius Junge, University of Illinois, Urbana-Champaign Jian-Guo Liu, Duke University For contributions to the study of operator algebras, Banach For contributions to the analysis of numerical methods spaces, harmonic analysis, and noncommutative probabil- for fluid dynamics, kinetic theory, and nonlinear partial ity, and for applications to quantum information theory. differential equations.

March 2017 Notices of the AMS 269 Fellows of the AMS FROM THE AMS SECRETARY

Xiaochun Rong, Rutgers The State University of New Jer- sey, New Brunswick For contributions to Riemannian geometry.

Daniel Ruberman, Brandeis University For contributions to low-dimensional topology.

David Savitt, Johns Hopkins University, Baltimore For contributions to number theory and service to the mathematical community.

Richard Evan Schwartz, Brown University For contributions to dynamics, geometry, and experimental mathematics and for exposition. AMS Immediate Past President Robert L. Bryant with former AMS Executive Director and Fellow William Nimish A. Shah, Ohio State University, Columbus "Bus" Jaco. For contributions to ergodic theory and homogeneous dy- namics and applications to number theory.

Ciprian Manolescu, University of California, Los Angeles Peter B. Shalen, University of Illinois at Chicago For contributions to Floer homology and the topology of For contributions to three-dimensional topology and for manifolds. exposition.

Kevin M. McCrimmon, University of Virginia Jie Shen, Purdue University For contributions to the theory of Jordan algebras, exposi- For contributions to theoretical numerical analysis, scien- tion, and service to the mathematical community. tific computing, computational fluid dynamics, and com- putational materials science. Umberto Mosco, Worcester Polytechnic Institute For contributions to analysis and partial differential equa- Zuowei Shen, National University of Singapore tions, in particular for introducing a theory of variational For contributions to approximation theory, wavelet theory, convergence. and image processing.

Allen Moy, Hong Kong University of Science and Tech- Ivan Shestakov, Universidade de São Paulo nology For contributions to nonassociative algebra and affine For contributions to representation theory of reductive algebraic geometry. groups over nonarchimedian local fields and for service to the mathematical community. Sergei Starchenko, University of Notre Dame For contributions to model theory and its applications to Isabella Novik, University of Washington geometry, analysis, number theory, and combinatorics. For contributions to algebraic and geometric combinatorics. Jason Starr, Stony Brook University Tony Pantev, University of Pennsylvania For contributions to algebraic geometry. For contributions to algebraic geometry, mathematical physics, and string theory, and for service to the mathe- Robert Strichartz, Cornell University matical community. For contributions to analysis and partial differential equa- tions, for exposition, and for service to the mathematical Julia Pevtsova, University of Washington community. For contributions to modular representation theory. Daniel B. Szyld, Temple University Ami Radunskaya, Pomona College For contributions to numerical and applied linear algebra. For contributions to mathematical oncology, immuno-dy- namics, and applications of dynamical systems to medicine, Tao Tang, Southern University of Science and Technology and for service to the mathematical community. For contributions to numerical methods for partial differ- ential equations. Lasse Rempe-Gillen, The University of Liverpool For contributions to complex dynamics and function the- ory, and for communication of mathematical research to broader audiences.

270 Notices of the AMS Volume 64, Number 3 Fellows of the AMS FROM THE AMS SECRETARY

Dinesh S. Thakur University of Rochester For contributions to the arithmetic of function fields, expo- sition, and service to the mathematical community.

Dylan Paul Thurston, Indiana University, Bloomington For contributions to low-dimensional topology and geome- try, and to cluster algebras.

Tatiana Toro, University of Washington For contributions to geometric measure theory, potential theory, and free boundary theory.

Ben Weinkove, Northwestern University For contributions to complex geometry and for service to the mathematical community.

Alexandru Zaharescu, University of Illinois, Urbana-Cham- paign and Institute of Mathematics, Romanian Academy For contributions to analytic number theory.

Ofer Zeitouni, Weizmann Institute of Science For contributions to probability theory.

—See more at: www.ams.org/profession/ams- fellows/new-fellows

—Photos courtesy of Kate Awetry/JMM 2017 photogra- pher.

March 2017 Notices of the AMS 271 NEWS

Mathematics People

moted to associate professor and the Viazovska Awarded Presidential first to serve as Chair of the Depart- Mentoring Awards ment of Mathematics. His research Salem Prize began with work on non-well-posed Given problems, which led him to the study of Besov spaces and harmonic analy- The Presidential Awards for Excel- sis. His interest in harmonic analysis lence in Science, Mathematics, and continues today. After forty years at Engineering Mentoring for fiscal year the University of Maryland, Johnson 2012 were awarded in 2015 by Presi- retired and returned to Rice as a visit- dent Obama to fourteen individuals ing professor. and one organization. Two scholars At the University of Maryland, whose work involves the mathemati- Johnson personally mentored nu- cal sciences were selected to receive merous graduate students, with the awards: Raymond L. Johnson of the largest number of students advised University of Maryland and J. Tilak in the period between 1990 and 2009. Ratnanather of Johns Hopkins Uni- During that time, fifty-three under- versity. represented minority students pur- sued their MA and/or PhD degrees. All but one of these students was Maryna Viazovska African American, and twenty-two of the students were African Ameri- Maryna Viazovska of the Berlin can women. Many of his graduate Mathematical School and Humboldt students graduated from smaller, historically black colleges and uni- University of Berlin has been awarded versities. Johnson’s mentoring plan the 2016 Salem Prize “for her break- involved regular group meetings to through work on densest sphere pack- develop a sense of community, as well ings in dimensions 8 and 24 using as course selection and counseling. methods of modular forms.” She Keenly aware of the need to familiar- earned her PhD from the University ize minority students with interdis- of Bonn in 2013 with her disserta- ciplinary environments, Johnson en- tion “Modular functions and special couraged his protégés to interact with minority graduate students across cycles.” She works in the areas of Raymond L. Johnson department lines. number theory, discrete geometry, ap- Of his fifty-three graduate stu- proximation theory, and physics. The Johnson was recognized “for his dents, twenty-three completed their prize, in memory of Raphael Salem, is tireless and highly successful mentor- PhDs in mathematics. In 2000, the awarded yearly to young researchers ing efforts with students from groups first African American woman earned for outstanding contributions to the underrepresented in mathematics.” a PhD in mathematics at the Univer- field of analysis. He was the first African American sity of Maryland; in fact, three Afri- admitted to Rice University (1964), Editor’s note: For more on the work can American women graduated at where he earned a doctorate in math- the same time. Fourteen of his PhD of Maryna Viazovska, see the feature ematics in 1969 for his dissertation, recipients currently hold academic article “A Conceptual Breakthrough in “A priori estimates and unique con- appointments at major US institu- Sphere Packing” by Henry Cohn in the tinuation theorems for second order tions of higher education, and three February 2017 Notices. parabolic equations.” He joined the are tenured professors. Johnson was —Elly Gustafsson, faculty at Maryland in 1980, becoming honored with the AAAS Lifetime Men- Institute for Advanced Study the first African American to be pro- tor Award in 2007.

272 Notices of the AMS Volume 64, Number 3 Mathematics People NEWS

vide the accommodation that allows tion for the Advancement of Science them to participate in professional for launching “dramatic education meetings. His success is clearly docu- and research changes leading to an in- mented: In 1991, Ratnanather and one crease in the number of female doctor- other graduate student were the only ates in the field of mathematics.” She deaf and hard-of-hearing individuals has mentored eighty-two students— pursuing studies in the auditory sci- eighty of whom are women—in earn- ences. In 2011, there were seven deaf ing their PhD degrees in mathematics. and hard-of-hearing faculty mem- Twenty-three of her students are Afri- bers in the auditory sciences, with can American and five are Latino, and more than fifteen currently pursuing most of them are now affiliated with graduate degrees in that field. He universities and colleges. Radunskaya established the Hearing-Impaired As- is president-elect of the Association sociation for Research in Otolaryngol- for Women in Mathematics and di- ogy in 1992. rector of the Enhancing Diversity in Ratnanather has personally Graduate Education (EDGE) program J. Tilak Ratnanather mentored thirteen deaf or hard-of- in mathematics. She earned her PhD in hearing students in both STEM and mathematics from Stanford University Ratnanather was recognized for medical school programs (five of and has been honored with several his work “creating a system to sup- whom are pursuing careers in medi- awards for teaching and mentoring, port deaf and hard-of-hearing indi- cine). His mentoring work includes as well as being named the AWM Fal- viduals in STEM.” He was the first twenty hearing female students, coner Lecturer in 2010. congenitally deaf person to earn an all of whom have pursued doctoral undergraduate degree in mathemat- degrees in STEM. He continues to —From an AAAS announcement ics from University College, London, think about how new ideas in educa- in 1985, and in 1989 he became the tion and technology can broaden ac- first ever congenitally deaf individual cess by people with hearing loss. Mnev Awarded in the world to be graduated with a doctorate in mathematics, which he —From a National Science Lichnerowicz Prize earned from Oxford University. His Foundation announcement interest in postdoctoral research in the auditory sciences brought him to Baltimore, Maryland, and the Johns Radunskaya Hopkins University School of Medi- cine. Ratnanather’s research interests Receives AAAS include computational anatomy ap- plications in neurodevelopmental and Mentor Award neurodegeneration disorders, as well as cochlear micromechanics and fluid mechanics. There is a simple and powerful objective to Ratnanather’s mentoring programs: to provide opportunities for education and research in science, technology, engineering, and math- ematics (STEM) for deaf and hard- Pavel Mnev of-hearing individuals who may not have otherwise been exposed to STEM Pavel Mnev of Notre Dame University and to achieve this objective through has been awarded the André Lichnero- extensive and involved networking wicz Prize in Poisson geometry for so that his protégés can later serve as his work, which is “at the interface mentors themselves. of Poisson geometry, topology, and The lack of accommodations for mathematical physics.” Mnev received the deaf and hard of hearing at an- Ami Radunskaya his PhD in 2008 from the Steklov Insti- nual meetings of the Association for tute of Mathematics in St. Petersburg Research in Otolaryngology led him Ami Radunskaya of Pomona College under the direction of Ludwig Fadeev to begin his work to recruit deaf and has been honored with the 2016 Men- and has held positions at the Uni- hard-of-hearing students and pro- tor Award of the American Associa- versity of Zurich and the Max Planck

March 2017 Notices of the AMS 273 Mathematics People NEWS

Institute. His research interests are in mathematical physics, in particular Rizell Receives Mathematical Society in the interactions of quantum field Wallenberg of Japan Prizes theory with topology, homological/ homotopical algebra, and supergeom- Fellowship The Mathematical Society of Japan etry. The award is given for outstand- (MSJ) awarded the following prizes ing work by a young mathematician in at the MSJ Autumn Meeting in 2016. Poisson geometry. The 2016 Autumn Prize was awarded to Shigeyuki Morita, emeri- tus professor at the University of —From a University of Notre Dame Tokyo and Tokyo Institute of Technol- announcement

Burkhardt and Swan Receive Castelnuovo Award

Georgios Dimitroglou Rizell

Georgios Dimitroglou Rizell of Uppsala University has been awarded a Wallenberg Academy Fellowship for his contributions to the develop- Shigeyuki Morita ment of symplectic geometry. Ac- ogy, for his outstanding contributions cording to the prize citation, “As a to work on cohomology theory of Wallenberg Academy Fellow he will mapping class groups and outer auto- continue to build upon the theory Hugh Burkhardt and Malcolm Swan morphism groups of free groups. The by investigating and classifying so- MSJ Autumn Prize and the MSJ Spring Hugh Burkhardt and Malcolm called Lagrangian submanifolds; these Prize are the most prestigious prizes Swan of the Shell Centre for Math- are subspaces whose properties are awarded by the MSJ to its members. ematical Education at Nottingham important for understanding the am- The 2016 Analysis Prizes were University have been awarded the bient symplectic space.” Wallenberg awarded to Soichiro Katayama of 2016 inaugural Emma Castelnuovo Fellows receive five-year grants of 5–9 Osaka University for studies on null structure in systems of nonlinear Award for Excellence in the Practice million Swedish krona (approximately of Mathematics Education “in recog- hyperbolic partial differential equa- US$542,000–976,000), depending on nition of their more than thirty-five tions; to Shigeaki Koike of Tohoku the field, and may apply for an ad- years of development and implemen- University for work on the theory of ditional five years. The program was p tation of innovative, influential work L -viscosity solutions for fully nonlin- established by the Knut and Alice Wal- in the practice of mathematics educa- ear elliptic and parabolic partial dif- tion, including the development of lenberg Foundation in close coopera- ferential equations; and to Tomohiro curriculum and assessment materi- tion with five learned academies and Sakamoto of the Tokyo Institute of Technology for studies on nonequilib- als, instructional design concepts, sixteen Swedish universities to give rium stochastic dynamical systems by teacher preparation programs, and the most promising young researchers exact solutions. educational system changes.” The a work situation that enables them to The 2016 Geometry Prizes were Castelnuovo Award is given by the focus on their projects and address awarded to Teruhiko Soma of Tokyo International Commission on Math- difficult research questions over an Metropolitan University for a series ematical Instruction (ICMI) to recog- extended period of time. of works on 3-manifold theory and nize outstanding achievements in the to Shigeharu Takayama of the Uni- practice of mathematics education. —From a Wallenberg Academy versity of Tokyo for the algebro-geo- —From an ICMI announcement announcement metric study of birationality of pluri-

274 Notices of the AMS Volume 64, Number 3 Mathematics People NEWS canonical maps of algebraic varieties named recipients of the NZMS Re- in order to identify the mean value of of general type. search Award. Bryant was honored a set of numbers. Runners-up were The 2016 Takebe Katahiro Prizes for “work developing mathematical, Crystal Frommert, Jemal Graham, were awarded to Norihisa Ikoma of statistical and computational tools and Maria Hernandez. Kanazawa University for work on vari- for evolutionary biology, and work ational and nonvariational approaches drawing on evolutionary biology to —From a National Museum of for nonlinear elliptic problems; to develop new theories in mathemat- Mathematics announcement Takefumi Nosaka of Kyushu Univer- ics.” Krauskopf was honored for “out- sity for work on algebraic topology of standing contributions to dynamical quandles and low-dimensional mani- systems, especially bifurcation theory folds; and to Makoto Yamashita of and its application to diverse physical W. Wistar Comfort Ochanomizu University for operator phenomena.” algebraic studies on quantum groups. Gaven Martin of Massey Univer- (1933–2016) The Takebe Katahiro Prize is awarded sity received the Kalman Prize for Best to young researchers who have ob- Paper for his article with T. H. Mar- tained outstanding results. shall “Minimal co-volume hyperbolic The 2016 Takebe Katahiro Prizes lattices, II: Simple torsion in a Klei- for Encouragement of Young Re- nian group,” Annals of Mathematics searchers were awarded to Ken Abe of 176(2012). [See Martin's cover story Kyoto University for work on the anal- in the December 2016 Notices.] ysis of the Navier-Stokes equations Alexander Melnikov of Massey by maximum norm; to Yoshihiro University received the Early Career Abe of Kobe University for detailed Award for “highly original contribu- estimates on cover times and local tions to the theory of computability times of random walks on graphs; in algebra and topology.” to Takahiro Oba of Tokyo Institute Naomi Gendler of the University of Technology for work on contact of Auckland was awarded the Aitken manifolds and their Stein fillings; to Prize for best contributed talk by a William Wistar Comfort Ryo Kanda of Osaka University for student at the annual NZMS Collo- work on atom spectra of Grothendieck quium for her talk “Pulse dynamics of William Wistar “Wis” Comfort, categories; to Yu Kitabeppu of Kyoto fibre lasers with saturable absorbers.” long-time associate secretary of the University for work on geometry of AMS, was known to many for his gal- spaces with Ricci curvature bounded —From an NZMS announcement lantry, dry wit, and humility. He was below; and to Yuta Wakasugi of a widely published mathematician Nagoya University for studies on the and scholar whose teaching career asymptotic behavior of solutions to Jackson Awarded spanned five decades. A formidable damped wave equations. The Takebe runner and racquet sports athlete, Wis Prize is intended for young math- 2016 Rosenthal was beloved by his peers for his sense ematicians who are deemed to have Prize of fairness. As a lifelong Quaker, he begun promising careers in research could be counted on to speak for the by obtaining significant results. Traci Jackson of Oak Valley Middle marginalized or overlooked.

School, San Diego, California, has been Wis held academic positions at —From an MSJ announcement awarded the 2016 Rosenthal Prize Harvard University, the University for Innovation in Math Teaching for of Rochester, and the University of her lesson “Creating color combos: Massachusetts at Amherst. He came Visual modeling of proportional re- Prizes of the to Wesleyan in July 1967 and from lationships!” In the lesson, students 1982 onwards was the Edward Burr New Zealand explore proportional reasoning by Van Vleck Professor of Mathematics. mixing colored solutions, creating dif- Mathematical ferent color combinations to visualize He served as department chair three Society ratios. Jackson received a cash award times before retiring in December of US$25,000. The second place award 2007. The New Zealand Mathematical So- went to Dena Lordi of Diamond Bar He continued to publish mathe- ciety (NZMS) has announced several High School in Diamond Bar, Califor- matical works in the weeks leading up awards for 2016. nia, for her lesson “Where can I find to his death; more will be published David Bryant of the University a weightless stick?” In this lesson, posthumously. Wis mentored more of Otago and Bernd Krauskopf of students trace the changing balance than twenty-five graduate students the University of Auckland have been point on a scale as weights are added and has about 150 publications with

March 2017 Notices of the AMS 275 Mathematics People NEWS fifty-seven coauthors from at least in Peru, Mexico, Italy, and the Nether- seventeen countries. Robert Seeley lands. He loved to sing, bike, run, and Wis worked principally in general (1932–2016) ski cross-country. Visitors to his home topology, with a specialty in topologi- in Newton enjoyed both his hospital- cal groups and cardinal invariants. He ity and the marvels of carpentry he installed there to complement the enjoyed infinitary combinatorics and furniture he built. their applications to topological struc- tures. He held various editorial and —Ethan Bolker, advocacy positions for mathematical University of Massachusetts Boston journals. He had an extensive record of ser- vice to the AMS, including ten years, Photo Credits from 1973 to 1983, as associate sec- Photo of Maryna Viazovska is by retary for the Eastern Section. He also Daniil Yevtushynsky, courtesy of served on the AMS Council, as well as Maryna Viazovska. Photo of Raymond L. Johnson is by several committees. He was the topol- Robert Seeley Malcolm Johnson. ogy editor for the Proceedings from Robert Seeley, professor emeritus of Photo of Tilak Ratnanather is cour- 1971 to 1974, and in the last year of mathematics at the University of Mas- tesy of Will Kirk, Johns Hopkins that period served as managing editor. sachusetts at Boston, was a pioneer University. He was named a Fellow of the AMS in in the theory of pseudo-differential Photo of Ami Radunskaya is courtesy the 2013 inaugural class. operators. He earned his PhD at the of AAAS. Born on April 19, 1933, in Bryn Massachusetts Institute of Technology Photo of Pavel Mnev is by Barbara Mawr, Pennsylvania, Wis Comfort with Alberto Calderón and taught at Johnston. graduated from Haverford College Harvey Mudd College and Brandeis Photo of Hugh Burkhardt and Mal- (Pennsylvania) in 1954. His roots at University before settling at the Uni- colm Swan is courtesy of Hugh Haverford run deep. His father, How- versity of Massachusetts. He was as Burkhardt. interested in the beginners as in the Photo of Georgios Dimitroglou Rizell ard Comfort II, was head of the Clas- math majors, so he gladly taught the is courtesy of Markus Marcetic. sics Department, and his grandfather, full range of courses. When students Photo of Shigeyuki Morita is courtesy William Wistar Comfort, for whom were ready for advanced work beyond of the Department of Information he was named, was a noted Quaker what was available in the curriculum, Science, Nihon University. scholar and president of Haverford he cheerfully and regularly supervised Photo of William Wistar Comfort is from 1917 to 1940. Wis received the independent study. In retirement he courtesy of Martha Comfort. PhD degree from the University of taught mathematics in prisons. Photo of Robert Seeley is courtesy of Washington (Seattle) in 1958, where Seeley was one of the first Fellows Charlotte Seeley. his thesis director was Edwin Hewitt. of the AMS and a member-at-large of He married Mary Constance Lyon in the AMS Council from 1972 to 1974. March 1957, and the couple produced One of the great mathematical in- ventions of the 1960s was the theory two children, Martha Wistar Comfort of pseudo-differential operators. This and Howard Comfort III, and enjoyed theory made possible both the Atiyah- fifty-nine years of marriage before Singer Index Theorem and “microlo- Mary Connie passed in May 2016. cal” techniques with which one could Wis Comfort’s principal postretire- apply the tools of symplectic geometry ment avocation was Dixieland trom- and dynamical systems to problems in bone, and he was affiliated with many partial differential equation theory. groups in Connecticut and Maine. His The origins of microlocal analysis rich, deep singing voice delighted were the papers on “singular integral many, and he worked assiduously at operators” of Calderón and Zygmund in the late 1950s, but it was Seeley’s his music, living and breathing the work in the next decade that turned old-time tunes. these ideas into the modern theory of pseudo-differential operators. —Martha Wistar Comfort Bob Seeley was a deep, thoughtful, kind family man, a mainstay of the Quaker community in Cambridge, and a world traveler who spent sabbaticals

276 Notices of the AMS Volume 64, Number 3 NEWS

Mathematics Opportunities

Project NExT 2017–2018 Intensive Research Programs MAA Project NExT (New Experiences in Teaching) is a year- at CRM long professional development program of the Mathemati- cal Association of America (MAA) for new or recent PhDs The Centre de Recerca Matemàtica (CRM) invites pro- in the mathematical sciences. The program is designed to posals for research programs consisting of one to five connect new faculty with master teachers and leaders in months of intensive research in any branch of math- the mathematics community and address the three main ematics or the mathematical sciences. Proposals are aspects of an academic career: teaching, research, and sought for programs to be organized preferably after service. Recent program sessions have included: August 2018. Preliminary proposals are due Febru- ary 28, 2017; final proposals should be submitted be- Getting your research and grant writing off to a good • fore April 28, 2017. For guidelines and instructions, see start www.crm.cat/en/Host/SciEvents/IRP/Pages/ Innovative teaching and assessment methods and why • CallApplication.aspx. they work

Finding your niche in the profession • —From a CRM announcement •Attracting and retaining underrepresented students •Balancing teaching, research, and service demands •Starting an undergraduate research program •Preparing for tenure MAA Project NExT Fellows join an active community of faculty who have gone on to become award-winning teachers, innovators on their campuses, active members of the MAA, and leaders in the profession. MAA Project NExT welcomes and encourages applications from new and recent PhDs in postdoctoral, tenure-track, and visiting positions. We particularly encourage applicants from un- derrepresented groups (including women and minorities). The deadline for applications for the 2017 cohort of MAA Project NExT Fellows is April 15, 2017. For applications and further information, see projectnext.maa.org.

—From an MAA announcement

Call for Nominations for Graham Wright Award The Graham Wright Award for Distinguished Service of the Canadian Mathematical Society (CMS) recognizes individu- als who have made sustained and significant contributions to the Canadian mathematical community and, in particu- lar, to the CMS. The deadline for nominations is March 31, 2017. See cms.math.ca/Prizes/dis-nom.

—From a CMS announcement

March 2017 Notices of the AMS 277 New Publications Offered by the AMS To subscribe to email notification of new AMS publications, please go to www.ams.org/bookstore-email.

Algebra and Algebraic lattice Heisenberg algebras; Y. Qi and J. Sussan, Categorification at prime roots of unity and hopfological finiteness; B. Elias, Geometry Folding with Soergel bimodules; L. T. Jensen and G. Williamson, The p-canonical basis for Hecke algebras. Contemporary Mathematics, Volume 683 Categorification and March 2017, approximately 363 pages, Softcover, ISBN: 978- 1-4704-2460-2, 2010 Mathematics Subject Classification: 81R50, Higher Representation 17B10, 20C08, 14F05, 18D10, 17B50, 17B55, 17B67, AMS members Theory US$88.80, List US$111, Order code CONM/683 Anna Beliakova, Universität Zürich, Switzerland, and Categorification in Aaron D. Lauda, University of Geometry, Topology, Southern California, Los Angeles, and Physics CA, Editors Anna Beliakova, Universität The emergent mathematical philosophy Zürich, Switzerland, and of categorification is reshaping our view of modern mathematics by uncovering a hidden layer of structure in mathematics, Aaron D. Lauda, University of revealing richer and more robust structures capable of describing Southern California, Los Angeles, more complex phenomena. Categorified representation theory, or CA, Editors higher representation theory, aims to understand a new level of structure present in representation theory. Rather than studying The emergent mathematical philosophy actions of algebras on vector spaces where algebra elements act by of categorification is reshaping our view of modern mathematics linear endomorphisms of the vector space, higher representation by uncovering a hidden layer of structure in mathematics, theory describes the structure present when algebras act on revealing richer and more robust structures capable of describing categories, with algebra elements acting by functors. The new level more complex phenomena. Categorification is a powerful tool of structure in higher representation theory arises by studying for relating various branches of mathematics and exploiting the the natural transformations between functors. This enhanced commonalities between fields. It provides a language emphasizing perspective brings into play a powerful new set of tools that essential features and allowing precise relationships between deepens our understanding of traditional representation theory. vastly different fields. This volume exhibits some of the current trends in higher This volume focuses on the role categorification plays in geometry, representation theory and the diverse techniques that are being topology, and physics. These articles illustrate many important employed in this field with the aim of showcasing the many trends for the field including geometric representation theory, applications of higher representation theory. homotopical methods in link homology, interactions between The companion volume (Contemporary Mathematics, Volume 684) higher representation theory and gauge theory, and double affine is devoted to categorification in geometry, topology, and physics. Hecke algebra approaches to link homology. Contents: I. Losev, Rational Cherednik algebras and The companion volume (Contemporary Mathematics, Volume 683) categorification; O. Dudas, M. Varagnolo, and E. Vasserot, is devoted to categorification and higher representation theory. Categorical actions on unipotent representations of finite classical This item will also be of interest to those working in geometry and groups; J. Brundan and N. Davidson, Categorical actions and topology. crystals; A. M. Licata, On the 2-linearity of the free group; M. Ehrig, Contents: B. Webster, Geometry and categorification; Y. Li,A C. Stroppel, and D. Tubbenhauer, The Blanchet-Khovanov geometric realization of modified quantum algebras; T. Lawson, algebras; G. Lusztig, Generic character sheaves on groups over R. Lipshitz, and S. Sarkar, The cube and the Burnside category; 푘[휖]/(휖푟); D. Berdeja Suárez, Integral presentations of quantum S. Chun, S. Gukov, and D. Roggenkamp, Junctions of surface

278 Notices of the AMS Volume 64, Number 3 New AMS-Distributed Publications operators and categorification of quantum groups; R. Rouquier, A publication of the Société Mathématique de France, Marseilles Khovanov-Rozansky homology and 2-braid groups; I. Cherednik (SMF), distributed by the AMS in the U.S., Canada, and Mexico. and I. Danilenko, DAHA approach to iterated torus links. Orders from other countries should be sent to the SMF. Members Contemporary Mathematics, Volume 684 of the SMF receive a 30% discount from list. Astérisque, Number 383 March 2017, approximately 268 pages, Softcover, ISBN: 978- 1-4704-2821-1, 2010 Mathematics Subject Classification: 81R50, October 2016, 120 pages, Softcover, ISBN: 978-2-85629-844-2, 57M25, 14F05, 18D10, 58J28, 17B81, 20C08, 17B55, 17B67, AMS 2010 Mathematics Subject Classification: 16E35, 16W70, 18A25, members US$88.80, List US$111, Order code CONM/684 18D10, 18D35, 18F20, 32B20, 32C05, 32C38, 32S60, 46E35, 58A03, AMS members US$41.60, List US$52, Order code AST/383

Representation New AMS-Distributed Theory—Current Trends and Publications Perspectives Henning Krause, University of Bielefeld, Germany, Peter Algebra and Algebraic Littelmann, University of Cologne, Germany, Gunter Malle, Geometry University of Kaiserlautern, Germany, Karl-Hermann Neeb, University of Erlangen- Subanalytic Sheaves Nuernberg, Germany, and Christoph Schweigert, and Sobolev Spaces University of Hamburg, Stéphane Guillermou, Université Germany, Editors de Grenoble I, Saint-Martin From April 2009 until March 2016, the German Science d’Hères, Gilles Lebeau, Université Foundation generously supported the Priority Program SPP 1388 Nice Sophia Antipolis, France, in Representation Theory. The core principles of the projects Adam Parusiński, Université realized in the framework of the priority program have been Nice Sophia Antipolis, France, categorification and geometrization, which are also reflected in the contributions to this volume. Pierre Schapira, Université Paris 6, Jussieu, France, Apart from the articles by former postdocs supported by the priority program, the volume contains a number of invited and Jean-Pierre Schneiders, research and survey articles. This volume covers current research Université de Liège, Belgique topics from the representation theory of finite groups, of algebraic groups, of Lie superalgebras, of finite dimensional algebras, and of Sheaves on manifolds are perfectly suited to treat local problems, infinite dimensional Lie groups. but many spaces that one naturally encounter, especially in analysis, are not of a local nature. The subanalytic topology (in Graduate students and researchers in mathematics interested in the sense of Grothendieck) on real analytic manifolds allows the representation theory will find this volume inspiring. It contains authors to partially overcome this difficulty and to define, for many stimulating contributions to the development of this broad example, sheaves of functions or distributions with temperate and extremely diverse subject. growth but not to make the growth precise. A publication of the European Mathematical Society (EMS). In this volume, the authors introduce the linear subanalytic Distributed within the Americas by the American Mathematical topology, a refinement of the preceding one, and construct various Society. objects of the derived category of sheaves on the subanalytic site EMS Series of Congress Reports, Volume 11 with the help of the Brown representability theorem. In particular, January 2017, 773 pages, Hardcover, ISBN: 978-3-03719-171-2, they construct the Sobolev sheaves. These objects have the nice property that the complexes of their sections on open subsets with 2010 Mathematics Subject Classification: 14Mxx, 16Gxx, 17Bxx, Lipschitz boundaries are concentrated in degree zero and coincide 18Exx, 20Gxx, 22Exx; 58Cxx, 81Txx, AMS members US$94.40, List with the classical Sobolev spaces. US$118, Order code EMSSCR/11 Another application of this topology is that it allows the authors to functorially endow regular holonomic D-modules with filtrations (in the derived sense). In the course of the text, the authors also obtain some results on subanalytic geometry and make a detailed study of the derived category of filtered objects in symmetric monoidal categories.

March 2017 Notices of the AMS 279 New AMS-Distributed Publications Analysis Differential Equations

Arithmétique 푝-adique Topics on des Formes de Hilbert Compressible Fabrizio Andreatta, Universita Navier-Stokes di Milano, Italy, Stéphane Equations Bijakowski, Université Paris 13, Villetaneuse, France, Adrian Didier Bresch, Université de Iovita, Concordia University, Savoie LAMA, Le Bourget-du-Lac, Montreal, Canada, Payman L. France, Editor; Antonin Kassaei, McGill University, Novotný, Université du Sud Montreal, Canada, Vincent Toulon-Var, La Garde, France, Pilloni, École Normale Supérieure Raphaël Danchin, Université de Lyon, France, Benoît Stroh, Paris-est, France, and Misha Université Paris 13, Villetaneuse, Perepelitsa, University of France, Yichao Tian, Chinese Houston, Texas Academy of Sciences, Beijing, This issue includes contributions from the session États de la China, and Liang Xiao, University Recherche: Topics on Compressible Navier-Stokes Equations of Connecticut, Storrs, CT that was held from May 21–25, 2012 at the Laboratoire de Mathématiques in Le Bourget du Lac, France. This volume is devoted to the study of Hilbert 푝-adic modular This national training session provided the opportunity to forms. It contains classicality theorems for overconvergent forms gather four internationally renowned specialists (D. Bresch, which generalize on the first hand Coleman criterion, which can be A. Novotný, R. Danchin, and M. Perepetlisa) and allow them to applied in big weights, and on the second hand Buzzard-Taylor present the major actual mathematical developments related criterion, which can be applied in weight one. The authors deduce to the well-posedness character problem for the compressible applications to the Artin and Fontaine-Mazur conjectures. They Navier-Stokes equations to non-subject specialists. conclude by constructing Hecke varieties for Hilbert modular forms. For the sake of unity, this special issue includes only the contributions dedicated to the non-degenerate viscosities case, A publication of the Société Mathématique de France, Marseilles aiming to present a self-contained contribution on the subject: (SMF), distributed by the AMS in the U.S., Canada, and Mexico. global weak-solutions à la Leray, intermediate solutions à la Hoff Orders from other countries should be sent to the SMF. Members and strong solutions in critical spaces à la Fujita-Kato. of the SMF receive a 30% discount from list. A publication of the Société Mathématique de France, Marseilles Astérisque, Number 382 (SMF), distributed by the AMS in the U.S., Canada, and Mexico. October 2016, 266 pages, Softcover, ISBN: 978-2-85629-843-5, Orders from other countries should be sent to the SMF. Members 2010 Mathematics Subject Classification: 37A20, 37D25, 37D30, of the SMF receive a 30% discount from list. 37A50, 37C40, AMS members US$65.60, List US$82, Order code Panoramas et Synthèses, Number 50 AST/382 November 2016, 135 pages, Softcover, ISBN: 978-2-85629-847-3, 2010 Mathematics Subject Classification: 35Q30, 76N10, 35Q35, AMS members US$48, List US$60, Order code PASY/50

280 Notices of the AMS Volume 64, Number 3 New AMS-Distributed Publications

The Monge-Ampère magnetic wells in dimension three) are intended for seasoned Equation and Its researchers. This item will also be of interest to those working in analysis. Applications A publication of the European Mathematical Society (EMS). Alessio Figalli, ETH Zürich, Distributed within the Americas by the American Mathematical Society. Switzerland EMS Tracts in Mathematics, Volume 27 The Monge-Ampère equation is one of January 2017, 394 pages, Hardcover, ISBN: 978-3-03719-169-9, the most important partial differential 2010 Mathematics Subject Classification: 35P15, 35P20, 49R05, equations, appearing in many problems in analysis and geometry. This monograph 81Q10, 81Q20, AMS members US$62.40, List US$78, Order code is a comprehensive introduction to the EMSTM/27 existence and regularity theory of the Monge-Ampère equation and some selected applications; the main goal is to provide the reader with a wealth of results and techniques he or she can draw from to General Interest understand current research related to this beautiful equation. The presentation is essentially self-contained, with an appendix that contains precise statements of all the results used from different areas (linear algebra, convex geometry, measure theory, Pristine Landscapes nonlinear analysis, and PDEs). in Elementary This book is intended for graduate students and researchers Mathematics interested in nonlinear PDEs: explanatory figures, detailed proofs, and heuristic arguments make this book suitable for self-study Titu Andreescu, University and also as a reference. of Texas at Dallas, Cristabel A publication of the European Mathematical Society (EMS). Mortici, Valahia University, Distributed within the Americas by the American Mathematical Targoviste, Romania, and Society. Marian Tetiva, Gh. Rosca Zurich Lectures in Advanced Mathematics, Volume 22 Codreanu National College, January 2017, 210 pages, Softcover, ISBN: 978-3-03719-170-5, Barlad, Romania 2010 Mathematics Subject Classification: 35J96; 35B65, 35J60, 35J66, 35B45, 35B50, 35D05, 35D10, 35J65, 53A15, 53C45, AMS This book takes familiar ideas and extends them to a rich variety of problems. The intended audience is the ambitious high school members US$33.60, List US$42, Order code EMSZLEC/22 or college student. The topics covered span algebra, geometry, number theory, and even a few elements of mathematical analysis. Each chapter explores specific themes and ideas that underlie the aforementioned subject areas. The “landscapes” presented Bound States of the provide a “view” into areas that are not typically encountered Magnetic Schrödinger in great depth in standard coursework but nonetheless have Operator profound implications. This item will also be of interest to those working in math Nicolas Raymond, Université de education. Rennes, France A publication of XYZ Press. Distributed in North America by the American Mathematical Society. This book is a synthesis of recent XYZ Series, Volume 22 advances in the spectral theory of the magnetic Schrödinger operator. It can December 2016, 280 pages, Hardcover, ISBN: 978-0-9968745-7-1, be considered a catalog of concrete 2010 Mathematics Subject Classification: 00A05, 00A07, 97U40, examples of magnetic spectral asymptotics. 97D50, AMS members US$47.96, List US$59.95, Order code Since the presentation involves many notions of spectral theory XYZ/22 and semiclassical analysis, it begins with a concise account of concepts and methods used in the book and is illustrated by many elementary examples. Assuming various points of view (power series expansions, Feshbach–Grushin reductions, WKB constructions, coherent states decompositions, normal forms) a theory of magnetic harmonic approximation is then established which allows, in particular, accurate descriptions of the magnetic eigenvalues and eigenfunctions. Some parts of this theory, such as those related to spectral reductions or waveguides, are still accessible to advanced students while others (e.g., the discussion of the Birkhoff normal form and its spectral consequences or the results related to boundary

March 2017 Notices of the AMS 281 New AMS-Distributed Publications

112 Combinatorial A publication of Delta Stream Media, an imprint of Natural Math. Distributed in North America by the American Mathematical Problems from Society. the AwesomeMath Natural Math Series, Volume 6 Summer Program December 2016, 96 pages, Softcover, ISBN: 978-1-945899-01-0, AMS members US$12, List US$15, Order code NMATH/6 Vlad Matei, University of Wisconsin, Madison, and Elizabeth Reiland, Johns Number Theory Hopkins University, Baltimore, MD

This book aims to give students a chance to begin exploring some Autour des Motifs introductory to intermediate topics in combinatorics, a fascinating and accessible branch of mathematics centered around (among Asian–French Summer other things) counting various objects and sets. School on Algebraic The book includes chapters featuring tools for solving counting Geometry and Number problems, proof techniques, and more to give students a broad Theory: Volume III foundation to build on. The only prerequisites are a solid background in arithmetic, some basic algebra, and a love for Takeshi Saito, University of learning mathematics. Tokyo, Japan, Laurent Clozel, This item will also be of interest to those working in math Université Paris-Sud 11, Orsay, education. France, and Jörg Wildeshaus, A publication of XYZ Press. Distributed in North America by the Université Paris 13, Villetaneuse, American Mathematical Society. France XYZ Series, Volume 21 This volume contains the third part of the lecture notes of December 2016, 196 pages, Hardcover, ISBN: 978-0-9968745-2-6, the Asian–French Summer School on Algebraic Geometry and 2010 Mathematics Subject Classification: 00A05, 00A07, 97U40, Number Theory, which was held at the Institut des Hautes Études 97D50, AMS members US$47.96, List US$59.95, Order code Scientifiques (Bures-sur-Yvette) and the Université Paris-Sud XI XYZ/21 (Orsay) in July 2006. This summer school was devoted to the theory of motives and its recent developments and to related topics, notably Shimura varieties and automorphic representations. Math Education This item will also be of interest to those working in algebra and algebraic geometry. A publication of the Société Mathématique de France, Marseilles Avoid Hard Work! (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members ... And Other Encouraging of the SMF receive a 30% discount from list. Mathematical Problem- Panoramas et Synthèses, Number 49 Solving Tips for the Young, November 2016, 131 pages, Softcover, ISBN: 978-2-85629-846-6, the Very Young, and the 2010 Mathematics Subject Classification: 14F42, 14C35, 14D10, Young at Heart 19E15, 19F27, AMS members US$41.60, List US$52, Order code PASY/49 Maria Droujkova, James Tanton, and Yelena McManaman

Avoid Hard Work gives a playful view on ten powerful problem-solving techniques. These techniques were first published by the Mathematical Association of America to help high school students with advanced math courses. These techniques and sample problems have been adapted for much younger children. This item will also be of interest to those working in general interest.

282 Notices of the AMS Volume 64, Number 3 MATHEMATICS CALENDAR

This section contains new announcements of worldwide meet- 21 – 24 Computational Issues in Oil Field Applications Tutorials ings and conferences of interest to the mathematical public, Location: Institute for Pure and Applied Mathematics (IPAM), including ad hoc, local, or regional meetings, and meetings UCLA, Los Angeles, CA. and symposia devoted to specialized topics, as well as an- URL: www.ipam.ucla.edu/programs/workshops/ nouncements of regularly scheduled meetings of national or computational-issues-in-oil-field-applications- international mathematical organizations. New announce- ments only are published in the print Mathematics Calendar tutorials featured in each Notices issue. An announcement will be published in the Notices if it contains 23 – April 7 Noncommutative Geometry Festival in Shanghai a call for papers and specifies the place, date, subject (when Location: Fudan University, Shanghai, China. applicable). A second announcement will be published only URL: www.fdias.fudan.edu.cn/en/non-commutative if there are changes or necessary additional information. As- -geometry terisks (*) mark those announcements containing revised information. 24 – 25 8th Western Conference on Mathematical Finance In general, print announcements of meetings and con- ferences carry only the date, title and location of the Location: University of Washington, Seattle, WA. event. URL: depts.washington.edu/amath/wcmf The complete listing of the Mathematics Calendar is available at: www.ams.org/meetings/calendar/mathcal April 2017 All submissions to the Mathematics Calendar should be done online via: www.ams.org/cgi-bin/mathcal/mathcal- *8 – 8 Midwest Mini-conference on Stochastic Processes and Math- submit.pl ematical Finance Any questions or difficulties may be directed to mathcal@ams. Location: North Dakota State University, Fargo, North Dakota. org. URL: www.ndsu.edu/pubweb/~isengupt/miniconf.html

10 – 11 2-Representation Theory Workshop February 2017 Location: Uppsala University, Sweden. 27 – March 10 KK-theory, Gauge Theory, and Topological Phases URL: www2.math.uu.se/%7Emazor/workshop.html Location: Lorentz Center, Leiden, NL. URL: www.lorentzcenter.nl/lc/web/2017/858/info 18 – October 14 Thematic Semester: Dynamics and Geometry .php3?wsid=858&venue=Oort Location: Universities of Angers, Brest, Nantes, Rennes (FRANCE). March 2017 URL: https://www.lebesgue.fr/content/sem2017 6 – 7 ModCompShock: Spring School in Multiscale Modelling Location: RWTH Aachen University, Department of Mathemat- 24 – May 3 Simons Lecture Series in Mathematics ics, Pontdriesch 14/16, 52062 Aachen, Germany. Location: Department of Mathematics, Massachusetts Institute URL: https://www.modcompshock.de of Technology, Cambridge, MA. URL: math.mit.edu/news/seminars/simons.php 9 – 11 42nd Annual Arkansas Spring Lecture Series in the Math- ematical Sciences: Geometry and the Equations Defining Projective 27 – 29 Redbud Topology Conference Varieties Location: University of Arkansas, Fayetteville, Arkansas. Location: University of Arkansas, Fayetteville, Arkansas, USA. URL: comp.uark.edu/~mattclay/Redbud URL: fulbright.uark.edu/departments/math/research /spring-lecture-series/index.php May 2017

10 – 11 The Topology of Real Algebraic Varieties: Deterministic and 1 – 5 Ninth Discrete Geometry and Algebraic Combinatorics Confer- Random Aspects—Shanks Workshop on Real Algebraic Geometry ence Location: Vanderbilt University, Nashville, Tennessee, USA. Location: South Padre Island, Texas. URL: my.vanderbilt.edu/ragworkshop URL: www.utrgv.edu/discgeo

18 – 18 Analytic Methods in Algebraic Geometry Day 3 – 7 4th International Intuitionistic Fuzzy Sets and Contemporary Location: Northwestern University, Mathematics Department, Mathematics Conference—IFSCOM2017 2033 Sheridan Rd, Evanston, IL. Location: Mersin, Turkey. URL: www.math.northwestern.edu/~tosatti/gaga.html URL: ifscom.com

March 2017 Notices of the AMS 283 Mathematics Calendar

4 – 8 4th International Workshop on Nonlinear and Modern Mathe- 26 – 30 Nonlinear Analysis in Rome matical Physics Location: Rome, Italy. Location: University Putra Malaysia, Kuala Lumpur, Malaysia. URL: www3.nd.edu/~conf/nar17 URL: einspem.upm.edu.my/nmmp2017 28 – 30 XVIII Congress of the Portuguese Association of Operational 11 – 13 International Conference on Mathematics and Mathematics Research Education (ICMME-2017) Location: Polytechnic Institute of Viana do Castelo, School of Location: Harran University, Sanlıurfa, Turkey. Business Sciences (ESCE), based in Valenca, Viana do Castelo URL: theicmme.com district, Portugal. 16 – 18 Geometric and Combinatorial Methods in Group Theory URL: http://apdio.pt/en/web/io2017/home Location: University of Illinois at Urbana-Champaign, IL, USA. URL: math.uiuc.edu/~jsapir2/Sapir60 July 2017 22 – 24 2nd Petra International Conference on Mathematics 3 – 7 12th Biennial International Conference on Sampling Theory and Location: Al-Hussein Bin Talal University￿Ma’an, Jordan. Applications (SampTA 2017) URL: ahu.edu.jo/mathconf/Location.aspx Location: Tallinn University of Technologoy/Tallinn University, Tallinn, Estonia. 22 – 26 International Conference on Elliptic and Parabolic Problems URL: www.sampta2017.ee Location: Hotel Serapo, Gaeta, Italy. URL: www.math.uzh.ch/gaeta2017 5 – 7 The 2017 International Conference of Applied and Engineering 22 – 27 Constructive Nonsmooth Analysis and Related Topics Mathematics Location: Euler Institute, Saint Petersburg, Russia. Location: Imperial College London, London, UK. URL: www.pdmi.ras.ru/EIMI/2017/CNSA/index.html URL: www.iaeng.org/WCE2017/ICAEM2017.html

30 – June 2 10th Chaotic Modeling and Simulation International 17 – 21 ACA 2017—The 23rd International Conference on Applica- Conference tions of Computer Algebra Location: Casa Convalescència of the Universitat Autònomea, Location: Jerusalem College of Technology, Jerusalem, Israel. Barcelona, Spain. URL: www.aca2017.jct.ac.il URL: www.cmsim.org

31 – June 4 ET’nA 2017—Encounter in Topology ’n Algebra 22 – 30 The International Conference and PhD-Master Summer Location: Scuola Superiore di Catania, Catania, Italy. School ”Groups and Graphs, Metrics and Manifolds” URL: ivanmartino.com/ETNA2017.php Location: Yekaterinburg, Russia. URL: g2.imm.uran.ru/g2m2 June 2017 6 – 10 Geometry and Algebra of PDEs 23 – 29 7th PhD Summer School in Discrete Mathematics Location: UiT the Arctic University of Norway, Tromsø, Nor- Location: Rogla, Slovenia. way. URL: https://conferences.famnit.upr.si/event/2 URL: serre.mat-stat.uit.no/pdes2017/index.html 24 – 27 All Kinds of Mathematics Remind me of You 8 – 10 Geometry and Physics, a conference dedicated to the memory Location: Faculdade de Ciencias Universidade de Lisboa, Lis- of William Thurston bon, Portugal. Location: Institut de Recherche Mathématique Avancée, Uni- URL: cameron17.campus.ciencias.ulisboa.pt versity of Strasbourg (France). URL: www-irma.u-strasbg.fr/article1576.html 24 – 28 International Conference on Mathematical Modelling in 19 – July 28 Math-to-Industry Boot Camp II Applied Sciences, ICMMAS’17 Location: University of Minnesota, Minneapolis, MN. Location: SPbPU, Saint Petersburg, Russia. URL: www.ima.umn.edu/2016-2017/SW6.19-7.28.17 URL: icmmas.alpha-publishing.net

26 – 28 Conference on Classical and Geophysical Fluid Dynamics: 24 – August 11 Summer Northwestern Analysis Program Modeling, Reduction, and Simulation Location: Northwestern University, Evanston, IL. Location: Virginia Tech, Blacksburg, Virginia, USA. URL: www.math.northwestern.edu/SNAP2017 URL: www.math.vt.edu/GFD_conference2017/index .html September 2017 26 – 29 The Sixth Biennial Conference of the Society for Mathemat- 7 – 9 Geometry, Dynamics and Physics ics and Computation in Music. Location: Institut de Recherche Mathématique Avancée (Uni- Location: Faculty of Sciences, Universidad Nacional Autonoma versity of Strasbourgg, France). de Mexico, Mexico City, Mexico. www-irma.u-strasbg.fr/article1609.html URL: www.mcm2017.org URL:

284 Notices of the AMS Volume 64, Number 3 Mathematics Calendar

October 2017 4 – 6 Optimization of Infinite Dimensional Non-Smooth Distributed Parameter Systems Location: TU-Darmstadt, Darmstadt, Germany. THE HONG KONG UNIVERSITY OF URL: www3.mathematik.tu-darmstadt.de/index.php? SCIENCE AND TECHNOLOGY id=3150 School of Science Head of the Department of Mathematics 6 – 8 10th International Symposium on Biomathematics and Ecology The School of Science of the Hong Kong University of Science and Technology (HKUST) is seeking applications from outstanding academicians to lead the Department of Mathematics. Education and Research (BEER) Opened in October 1991, HKUST is a research-intensive university dedicated to the advancement Location: Illinois State University, Normal, IL USA. of learning and scholarship, with special emphasis on postgraduate education, and close collaboration with business and industry. The School of Science, in which the Department of URL: symposium.beer Mathematics is located, is also home to world-class Departments of Physics, Chemistry and Life Science. Its faculty is international in background and the offi cial language of both administration and instruction at HKUST is English. 7 – 8 The 37th Southeastern-Atlantic Regional Conference on Differ- Reporting to the Dean of Science, the Head of the Department is expected to provide leadership ential Equations (SEARCDE) for the Department, oversee faculty recruitment activities, guide and monitor resource allocation, and be responsible for the Department’s academic advancement in both teaching and research. He/ Location: Kennesaw State University, Kennesaw, Georgia. she is also expected to devise strategies to promote and facilitate collaborative, interdisciplinary URL: conference.kennesaw.edu/searcde research with individuals in other Departments within the School of Science as well as in the Schools of Engineering, Business and Humanities and Social Science. Applicants should have an outstanding record of scholarship achievement, consistent with December 2017 an appointment as Full Professor with tenure. They should have proven leadership abilities, experience leading collaborative research programs and demonstrated managerial skills. Qualifi ed 3 – 8 73rd Annual Deming Conference on Applied Statistics individuals should also have a broad appreciation of the research and educational opportunities in Location: Tropicana Casino Resort Atlantic City, New Jersey, modern mathematics and possess outstanding communication and interpersonal skills. HKUST salaries are highly competitive in the world market; within this context, the level of USA. compensation will be commensurate with qualifi cations and experience. Generous fringe benefi ts URL: www.demingconference.com will also be provided. Application packages, including a curriculum vitae, a vision statement as well as the names, addresses, phone numbers and email addresses of at least three referees should be sent to: Offi ce 9 – 12 SIAM Conference on Analysis of Partial Differential Equations of the Dean of Science (Re: Head of the Department of Mathematics), The Hong Kong University (PD17) of Science and Technology, Clear Water Bay, Kowloon, Hong Kong (or by email: [email protected]). Review of applications will begin immediately and will continue until the position is fi lled. Location: Hyatt Regency Baltimore Inner Harbor, Baltimore, For further information about HKUST, the School of Science and the Department of Mathematics, Maryland, USA. please visit the following websites: HKUST - http://www.ust.hk URL: www.siam.org/meetings/pd17 School of Science – http://science.ust.hk Department of Mathematics - http://www.math.ust.hk

March 2017 Notices of the AMS 285 Classified Advertising Positions available, items for sale, services available, and more

KOREA

KOREA INSTITUTE FOR ADVANCED STUDY (KIAS) Assistant Professor & Research Fellow in Pure and Applied Mathematics

The School of Mathematics at the Korea Institute for Advanced other times based on the availability of Study (KIAS) invites applicants for the positions at the level of positions. The starting date of the ap- KIAS Assistant Professor and Postdoctoral Research Fellow in pointment is negotiable. Applications pure and applied mathematics. must include a complete vitae with a cover KIAS, founded in 1996, is committed to the excellence of re- letter, a list of publications, a research search in basic sciences (mathematics, theoretical physics, and plan, and three letters of recommenda- computational sciences) through high-quality research programs tion. (All documents should be in English.) and a strong faculty body consisting of distinguished scientists All should be sent by post and visiting scholars. or e-mail to: Ms. Sojung Bae Applicants are expected to have demonstrated exceptional ([email protected]) School of Mathe- research potential, including major contributions beyond or matics Korea Institute for Advanced Study through the doctoral dissertation. (KIAS) 85 Hoegiro (Cheongnyangni-dong The annual salary starts from 49,500,000 Korean Won (ap- 207-43), Dongdaemun-gu, Seoul 02455, proximately US$40,000 at current exchange rate) for Research Republic of Korea. Fellows, and 56,500,000 Korean Won for KIAS Assistant Pro- 000004 fessors, respectively. In addition, individual research funds of 10,000,000–13,000,000 Korean Won are available per year. The initial appointment for the position is for two years and is re- newable for up to two additional years, depending on research performance and the needs of the research program at KIAS. Applications will be reviewed twice a year, May 20 and Novem- ber 20, and selected applicants will be notified in a month after the review. In exceptional cases, applications can be reviewed

Suggested uses for classified advertising are positions available, books or lecture notes for sale, books being sought, exchange or rental of houses, and typing services. The publisher reserves the right to reject any advertising not in keeping with the publication's standards. Acceptance shall not be construed as approval of the accuracy or the legality of any advertising. The 2017 rate is $3.50 per word with a minimum two-line headline. No discounts for multiple ads or the same ad in consecutive issues. For an additional $10 charge, announcements can be placed anonymously. Correspondence will be forwarded. Advertisements in the “Positions Available” classified section will be set with a minimum one-line headline, consisting of the institution name above body copy, unless additional headline copy is specified by the advertiser. Headlines will be centered in boldface at no extra charge. Ads will appear in the language in which they are submitted. There are no member discounts for classified ads. Dictation over the telephone will not be accepted for classified ads. Upcoming deadlines for classified advertising are as follows: April 2017—February 3, 2017; May 2017—March 9, 2017; June/July 2017—May 9, 2017; August 2017—June 6, 2017; September 2017—July 7, 2017; October 2017—August 4, 2017; November 2017—September 5, 2017; December 2017—Sep- tember 28, 2017. US laws prohibit discrimination in employment on the basis of color, age, sex, race, religion, or national origin. “Positions Available” advertisements from institutions outside the US cannot be published unless they are accompanied by a statement that the institution does not discriminate on these grounds whether or not it is subject to US laws. Details and specific wording may be found on page 1373 (vol. 44). Situations wanted advertisements from involuntarily unemployed mathematicians are accepted under certain conditions for free publication. Call toll-free 800-321-4AMS (321-4267) in the US and Canada or 401-455-4084 worldwide for further information. Submission: Promotions Department, AMS, P.O. Box 6248, Providence, Rhode Island 02904; or via fax: 401-331-3842; or send email to [email protected]. AMS location for express delivery packages is 201 Charles Street, Providence, Rhode Island 02904. Advertisers will be billed upon publication.

286 Notices of the AMS Volume 64, Number 3 MEETINGS & CONFERENCES OF THE AMS MARCH TABLE OF CONTENTS

The Meetings and Conferences section of The most up-to-date meeting and confer- necessary to submit an electronic form, the Notices gives information on all AMS ence information can be found online at: although those who use L ATEX may submit meetings and conferences approved by www.ams.org/meetings/. abstracts with such coding, and all math press time for this issue. Please refer to Important Information About AMS displays and similarily coded material the page numbers cited on this page for Meetings: Potential organizers, (such as accent marks in text) must more detailed information on each event. speakers, and hosts should refer to be typeset in LATEX. Visit www.ams.org/ Invited Speakers and Special Sessions are page 75 in the January 2017 issue of the cgi-bin/abstracts/abstract.pl/. Ques- listed as soon as they are approved by the Notices for general information regard- tions about abstracts may be sent to abs- cognizant program committee; the codes ing participation in AMS meetings and [email protected]. Close attention should be listed are needed for electronic abstract conferences. paid to specified deadlines in this issue. submission. For some meetings the list Abstracts: Speakers should submit ab- Unfortunately, late abstracts cannot be may be incomplete. Information in this stracts on the easy-to-use interactive accommodated. A issue may be dated. Web form. No knowledge of LTEX is

MEETINGS IN THIS ISSUE

–––––––– 2017 –––––––– –––––––– 2018 , cont'd. –––––––– April 14–15 Portland, Oregon p. 301 March 10–12 Charleston, South Carolina p. 288 April 21–22 Boston, Massachusetts p. 302 April 1–2 Bloomington, Indiana p. 289 June 11–14 Shanghai, People's Republic April 22–23 Pullman, Washington p. 290 of China p. 302 September 29–30 Newark, Delaware p. 302 May 6–7 New York, New York p. 291 October 6–7 Fayetteville, Arkansas p. 302 October 20–21 Ann Arbor, Michigan p. 302 July 24–28 Montréal, Quebec, Canada p. 298 October 27–28 San Francisco, California p. 303 September 9–10 Denton, Texas p. 298

September 16–17 Buffalo, New York p. 299 –––––––– 2019 –––––––– January 16–19 Baltimore, Maryland p. 303 September 23–24 Orlando, Florida p. 299 March 29–31 Honolulu, Hawaii p. 303 November 4–5 Riverside, California p. 300 –––––––– 2020 –––––––– –––––––– 2018 –––––––– January 15–18 Denver, Colorado p. 303 January 10–13 San Diego, California p. 300

March 24–25 Columbus, Ohio p. 301 –––––––– 2021 –––––––– January 6–9 Washington, DC p. 303 April 14–15 Nashville, Tennesse p. 301

See www.ams.org/meetings/ for the most up-to-date information on these conferences.

ASSOCIATE SECRETARIES OF THE AMS Central Section: Georgia Benkart, University of Wisconsin- Southeastern Section: Brian D. Boe, Department of Mathemat- Madison, Department of Mathematics, 480 Lincoln Drive, ics, University of Georgia, 220 D W Brooks Drive, Athens, GA Madison, WI 53706-1388; e-mail: [email protected]; 30602-7403, e-mail: [email protected]; telephone: 706- telephone: 608-263-4283. 542-2547. Eastern Section: Steven H. Weintraub, Department of Math- Western Section: Michel L. Lapidus, Department of Mathemat- ematics, Lehigh University, Bethlehem, PA 18015-3174; e-mail: ics, University of California, Surge Bldg., Riverside, CA 92521- [email protected]; telephone: 610-758-3717. 0135; e-mail: [email protected]; telephone: 951-827-5910.

MARCH 2017 NOTICES OF THE AMS 287 MEETINGS & CONFERENCES

Meetings & Conferences of the AMS

IMPORTANT INFORMATION REGARDING MEETINGS PROGRAMS: AMS Sectional Meeting programs do not appear in the print version of the Notices. However, comprehensive and continually updated meeting and program informa- tion with links to the abstract for each talk can be found on the AMS website. See www.ams.org/meetings/. Final programs for Sectional Meetings will be archived on the AMS website accessible from the stated URL .

Chamberlain Jr., Aubrey Kemp, and Leslie Meadows, Charleston, South Georgia State University. Advances in Long-term Behavior of Nonlinear Disper- Carolina sive Equations, Brian Pigott, Wofford College, and Sarah Raynor, Wake Forest University. College of Charleston Advances in Nonlinear Waves: Theory and Applications, March 10–12, 2017 Constance M. Schober, University of Central Florida, and Andrei Ludu, Embry Riddle University. Friday – Sunday Algebras, Lattices, Varieties, George F. McNulty, Uni- Meeting #1126 versity of South Carolina, and Kate S. Owens, College of Southeastern Section Charleston. Associate secretary: Brian D. Boe Analysis and Control of Fluid-Structure Interactions Announcement issue of Notices: January 2017 and Fluid-Solid Mixtures, Justin T. Webster, College Program first available on AMS website: January 2017 of Charleston, and Daniel Toundykov, University of Issue of Abstracts: Volume 38, Issue 2 Nebraska-Lincoln. Analysis, Control and Stabilization of PDE’s, George Deadlines Avalos, University of Nebraska-Lincoln, and Scott Hansen, For organizers: Expired Iowa State University. For abstracts: Expired Bicycle Track Mathematics, Ron Perline, Drexel Uni- The scientific information listed below may be dated. versity. For the latest information, see www.ams.org/amsmtgs/ Coding Theory, Cryptography, and Number Theory, Jim sectional.html. Brown, Shuhong Gao, Kevin James, Felice Manganiello, and Gretchen Matthews, Clemson University. Invited Addresses Commutative Algebra, Bethany Kubik, University of Pramod N. Achar, Louisiana State University, Represen- Minnesota Duluth, Saeed Nasseh, Georgia Southern Uni- tations of algebraic groups via algebraic topology. versity, and Sean Sather-Wagstaff, Clemson University. Hubert L. Bray, Duke University, The Geometry of Spe- Computability in Algebra and Number Theory, Val- cial and General Relativity. entina Harizanov, The George Washington University, Alina Chertock, North Carolina State University, Nu- Russell Miller, Queens College and Graduate Center - City merical Methods for Chemotaxis and Related Models. University of New York, and Alexandra Shlapentokh, East Special Sessions Carolina University. Active Learning in Undergraduate Mathematics, Data Analytics and Applications, Scott C. Batson, Draga Vidakovic, Georgia State University, Harrison Lucas A. Overbey, and Bryan Williams, Space and Naval Stalvey, University of Colorado, Boulder, and Darryl Warfare Systems Center Atlantic.

288 Notices of the AMS Volume 64, Number 3 MEETINGS & CONFERENCES

Factorization and Multiplicative Ideal Theory, Jim Coykendall, Clemson University, and Evan Houston and Bloomington, Indiana Thomas G. Lucas, University of North Carolina, Charlotte. Indiana University Fluid-Boundary Interactions, M. Nick Moore, Florida State University. April 1–2, 2017 Frame Theory, Dustin Mixon, Air Force Institute of Saturday – Sunday Technology, John Jasper, University of Cincinnati, and Meeting #1127 James Solazzo, Coastal Carolina University. Central Section Free-boundary Fluid Models and Related Problems, Mar- Associate secretary: Georgia Benkart celo Disconzi, Vanderbilt University, and Lorena Bociu, Announcement issue of Notices: February 2017 North Carolina State University. Program first available on AMS website: February 23, 2017 Geometric Analysis and General Relativity, Hubert L. Issue of Abstracts: Volume 38, Issue 2 Bray, Duke University, Otis Chodosh, Princeton University, Deadlines and Greg Galloway and Pengzi Miao, University of Miami. For organizers: Expired Geometric Methods in Representation Theory, Pramod N. For abstracts: Expired Achar, Louisiana State University, and Amber Russell, Butler University. The scientific information listed below may be dated. For the latest information, see www.ams.org/amsmtgs/ Geometry and Symmetry in Integrable Systems, An- sectional.html. nalisa Calini, Alex Kasman, and Thomas Ivey, College of Charleston. Invited Addresses Graph Theory, Colton Magnant, Georgia Southern Ciprian Demeter, Indiana University, Decouplings and University, and Zixia Song, University of Central Florida. applications: a journey from continuous to discrete. Sarah Koch, University of Michigan, Title to be an- Knot Theory and its Applications, Elizabeth Denne, nounced. Washington & Lee University, and Jason Parsley, Wake Richard Evan Schwartz, Brown University, Modern Forest University. scratch paper: Graphical explorations in geometry and Nonlinear Waves: Analysis and Numerics, Anna Ghaz- dynamics (Einstein Public Lecture in Mathematics). aryan, Miami University, St é phane Lafortune, College Special Sessions of Charleston, and Vahagn Manukian, Miami University. If you are volunteering to speak in a Special Session, you Numerical Methods for Coupled Problems in Computa- should send your abstract as early as possible via the ab- tional Fluid Dynamics, Vincent J. Ervin and Hyesuk Lee, stract submission form found at www.ams.org/cgi-bin/ Clemson University. abstracts/abstract.pl. Oscillator Chain and Lattice Models in Optics, the Power Grid, Biology, and Polymer Science, Alejandro Aceves, Algebraic and Enumerative Combinatorics with Ap- plications, Saúl A. Blanco, Indiana University, and Kyle Southern Methodist University, and Brenton LeMesurier, Peterson, DePaul University. College of Charleston. Analysis and Numerical Computations of PDEs in Fluid Recent Trends in Finite Element Methods, Michael Mechanics, Gung-Min Gie, University of Louisville, and Neilan, University of Pittsburgh, and Leo Rebholz, Clem- Makram Hamouda and Roger Temam, Indiana University. son University. Analysis of Variational Problems and Nonlinear Partial Di fferential Equations, Nam Q. Le and Peter Sternberg, Representation Theory and Algebraic Mathematical Indiana University. Physics, Iana I. Anguelova, Ben Cox, and Elizabeth Juris- Automorphic Forms and Algebraic Number Theory, Pat- ich, College of Charleston. rick B. Allen, University of Illinois at Urbana-Champaign, Riemann-Hilbert Problem Approach to Asymptotic and Matthias Strauch, Indiana University Bloomington. Problems in Integrable Systems, Orthogonal Polynomials Commutative Algebra, Ela Celikbas and Olgur Celik- bas, West Virginia University. and Other Areas, Alexander Tovbis, University of Central Computability and Inductive Definability over Struc- Robert Jenkins Florida, and , University of Arizona. tures, Siddharth Bhaskar, Lawrence Valby, and Alex Rigidity Theory and Inversive Distance Circle Packings, Kruckman, Indiana University. John C. Bowers, James Madison University, and Philip L. Dependence in Probability and Statistics, Richard C. Bowers, The Florida State University. Bradley and Lanh T. Tran, Indiana University.

March 2017 Notices of the AMS 289 MEETINGS & CONFERENCES

Differential Equations and Their Applications to Biol- Topics in Extremal, Probabilistic and Structural Graph ogy, Changbing Hu, Bingtuan Li, and Jiaxu Li, University Theory, John Engbers, Marquette University, and David of Louisville. Galvin, University of Notre Dame. Discrete Structures in Conformal Dynamics and Ge- Topological Mathematical Physics, E. Birgit Kaufmann ometry, Sarah Koch, University of Michigan, and Kevin and Ralph M. Kaufmann, Purdue University, and Emil Pilgrim and Dylan Thurston, Indiana University. Prodan, Yeshiva University. Extremal Problems in Graphs, Hypergraphs and Other Combinatorial Structures, Amin Bahmanian, Illinois State University, and Theodore Molla, University of Illinois Urbana-Champaign. Pullman, Washington Financial Mathematics and Statistics, Ryan Gill, Univer- Washington State University sity of Louisville, Rasitha Jayasekera, Butler University, and Kiseop Lee, Purdue University. April 22–23, 2017 Fusion Categories and Applications, Paul Bruillard, Saturday – Sunday Pacific Northwest National Laboratory, and Julia Plavnik and Eric Rowell, Texas A&M University. Meeting #1128 Harmonic Analysis and Partial Differential Equations, Lucas Chaffee, Western Washington University, William Western Section Green, Rose-Hulman Institute of Technology, and Jarod Associate secretary: Michel L. Lapidus Hart, University of Kansas. Announcement issue of Notices: February 2017 Homotopy Theory, David Gepner, Purdue University, Program first available on AMS website: March 9, 2017 Ayelet Lindenstrauss and Michael Mandell, Indiana Uni- Issue of Abstracts: Volume 38, Issue 2 versity, and Daniel Ramras, Indiana University-Purdue University Indianapolis. Deadlines Model Theory, Gabriel Conant, University of Notre For organizers: Expired Dame, and Philipp Hieronymi, University of Illinois Ur- For abstracts: February 28, 2017 bana Champaign. Multivariate Operator Theory and Function Theory, The scientific information listed below may be dated. Hari Bercovici, Indiana University, Kelly Bickel, Bucknell For the latest information, see www.ams.org/amsmtgs/ University, Constanze Liaw, Baylor University, and Alan sectional.html. Sola, Stockholm University. Network Theory, Jeremy Alm and Keenan M.L. Mack, Invited Addresses Illinois College. Michael Hitrik, University of California, Los Angeles, Nonlinear Elliptic and Parabolic Partial Differential Spectra for non-self-adjoint operators and integrable dy- Equations and Their Various Applications, Changyou namics. Wang, Purdue University, and Yifeng Yu, University of Andrew S. Raich California, Irvine. , University of Arkansas, Title to be Probabilistic Methods in Combinatorics, Patrick Bennett announced. and Andrzej Dudek, Western Michigan University. Daniel Rogalski, University of California, San Diego, Probability and Applications, Russell Lyons and Nick Title to be announced. Travers, Indiana University. Randomness in Complex Geometry, Turgay Bayraktar, Special Sessions Syracuse University, and Norman Levenberg, Indiana If you are volunteering to speak in a Special Session, you University. should send your abstract as early as possible via the ab- Representation Stability and its Applications, Patricia stract submission form found at www.ams.org/cgi-bin/ Hersh, North Carolina State University, Jeremy Miller, abstracts/abstract.pl. Purdue University, and Andrew Putman, University of Notre Dame. Analysis on the Navier-Stokes equations and related Representation Theory and Integrable Systems, Eugene PDEs (Code: SS 9A), Kazuo Yamazaki, University of Mukhin, Indiana University,Purdue University Indianapo- Rochester, and Litzheng Tao, University of California, lis, and Vitaly Tarasov, Indiana University, Purdue Uni- Riverside. versity Indianapolis. Clustering of Graphs: Theory and Practice (Code: SS Self-similarity and Long-range Dependence in Stochastic 18A), Stephen J. Young and Jennifer Webster, Pacific Processes, Takashi Owada, Purdue University, Yi Shen, University of Waterloo, and Yizao Wang, University of Northwest National Laboratory. Cincinnati. Combinatorial and Algebraic Structures in Knot Theory Spectrum of the Laplacian on Domains and Manifolds, (Code: SS 5A), Sam Nelson, Claremont McKenna College, Chris Judge and Sugata Mondal, Indiana University. and Allison Henrich, Seattle University.

290 Notices of the AMS Volume 64, Number 3 MEETINGS & CONFERENCES

Combinatorial and Computational Commutative Al- gebra and Algebraic Geometry (Code: SS 21A), Hirotachi New York, New York Abo, Stefan Tohaneanu, and Alexander Woo, University of Idaho. Hunter College, City University of New Commutative Algebra (Code: SS 3A), Jason Lutz and York Katharine Shultis, Gonzaga University. Fixed Point Methods in Differential and Integral Equa- May 6–7, 2017 tions (Code: SS 1A), Theodore A. Burton, Southern Illinois Saturday – Sunday University in Carbondale. Geometric Measure Theory and it’s Applications (Code: Meeting #1129 SS 20A), Kevin R. Vixie, Washington State University. Eastern Section Geometry and Optimization in Computer Vision (Code: Associate secretary: Steven H. Weintraub SS 15A), Bala Krishnamoorthy, Washington State Univer- Announcement issue of Notices: March 2017 sity, and Sudipta Sinha, Microsoft Research, Redmond, Program first available on AMS website: March 29, 2017 WA. Issue of Abstracts: Volume 38, Issue 2 Inverse Problems (Code: SS 2A), Hanna Makaruk, Los Deadlines Alamos National Laboratory (LANL), and Robert Owcza- For organizers: Expired rek, University of New Mexico, Albuquerque & Los Alamos. For abstracts: March 21, 2017 Mathematical & Computational Neuroscience (Code: SS 12A), Alexander Dimitrov, Washington State University The scientific information listed below may be dated. Vancouver, Andrew Oster, Eastern Washington University, For the latest information, see www.ams.org/amsmtgs/ and Predrag Tosic, Washington State University. sectional.html. Mathematical Modeling of Forest and Landscape Change (Code: SS 11A), Demetrios Gatziolis, US Forest Service, and Invited Addresses Nikolay Strigul, Washington State University Vancouver. Jeremy Kahn, City University of New York, Title to be Microlocal Analysis and Spectral Theory (Code: SS 17A), announced. Michael Hitrik, University of California, Los Angeles, and Fernando Coda Marques, Princeton University, Title to Semyon Dyatlov, Masssachusetts Institute of Technology. be announced. Noncommutative algebraic geometry and related topics James Maynard, Magdalen College, University of Ox- (Code: SS 16A), Daniel Rogalski, University of California, ford, Title to be announced (Erdo˝s Memorial Lecture). San Diego, and James Zhang, University of Washington. Kavita Ramanan, Brown University, Title to be an- Partial Differential Equations and Applications (Code: nounced. SS 8A), V. S. Manoranjan, C. Moore, Lynn Schreyer, and Hong-Ming Yin, Washington State University. Special Sessions Recent Advances in Applied Algebraic Topology (Code: If you are volunteering to speak in a Special Session, you SS 14A), Henry Adams, Colorado State University, and should send your abstract as early as possible via the ab- Bala Krishnamoorthy, Washington State University. stract submission form found at www.ams.org/cgi-bin/ Recent Advances in Optimization and Statistical Learn- abstracts/abstract.pl. ing (Code: SS 19A), Hongbo Dong, Bala Krishnamoorthy, Haijun Li, and Robert Mifflin, Washington State Univer- Analysis and Numerics on Liquid Crystals and Soft Mat- sity. ter (Code: SS 16A), Xiang Xu, Old Dominion University, and Recent Advances on Mathematical Biology and Their Wujun Zhang, Rutgers University. Applications (Code: SS 7A), Robert Dillon and Xueying Applications of Modular Forms (Code: SS 35A), Cormac Wang, Washington State University. O’Sullivan and Karen Taylor, Bronx Community College, Several Complex Variables and PDEs (Code: SS 10A), City University of New York. Andrew Raich and Phillip Harrington, University of Applications of Network Analysis, in Honor of Charlie Arkansas. Suffel’s 75th Birthday (Code: SS 18A), Michael Yatauro, Analytic Number Theory and Automorphic Forms (Code: Pennsylvania State University-Brandywine. SS 6A), Steven J. Miller, Williams College, and Sheng-Chi Asymptotic Properties of Discrete Dynamical Systems Liu, Washington State University. (Code: SS 29A), Ann Brett, Johnson and Wales University, Theory and Applications of Linear Algebra (Code: SS and M. R.S. Kulenovic´ and O. Merino, University of Rhode 4A), Judi McDonald and Michael Tsatsomeros, Washing- Island. ton State University. Automorphic Forms and Arithmetic (Code: SS 22A), Jim Undergraduate Research Experiences in the Classroom Brown, Clemson University, and Krzysztof Klosin, Queens (Code: SS 13A), Heather Moon, Lewis-Clark State College. College, City University of New York.

March 2017 Notices of the AMS 291 MEETINGS & CONFERENCES

Banach Space Theory and Metric Embeddings (Code: SS Hydrodynamic and Wave Turbulence (Code: SS 11A), 10A), Mikhail Ostrovskii, St John’s University, and Beata Tristan Buckmaster, Courant Institute of Mathematical Randrianantoanina, Miami University of Ohio. Sciences, New York University, and Vlad Vicol, Princeton Bases in Hilbert Function Spaces (Code: SS 30A), Azita University. Mayeli, City University of New York, and Shahaf Nitzan, Infinite Permutation Groups, Totally Disconnected Lo- Georgia Institute of Technology. cally Compact Groups, and Geometric Group Theory (Code: Cluster Algebras in Representation Theory and Com- SS 4A), Delaram Kahrobaei, New York City College of binatorics (Code: SS 6A), Alexander Garver, Université Technology and Graduate Center-CUNY, and Simon Smith, du Quebéc à Montréal and Sherbrooke, and Khrystyna New York City College of Technology-CUNY. Serhiyenko, University of California at Berkeley. Invariants in Low-dimensional Topology (Code: SS 24A), Cohomologies and Combinatorics (Code: SS 15A), Re- Abhijit Champanerkar, College of Staten Island and The becca Patrias, Université du Québec à Montréal, and Oliver Graduate Center, City University of New York, and Anas- Pechenik, Rutgers University. tasiia Tsvietkova, Rutgers University, Newark. Common Threads to Nonlinear Elliptic Equations and Invariants of Knots, Links and 3-manifolds (Code: SS Systems (Code: SS 14A), Florin Catrina, St. John’s Univer- 26A), Moshe Cohen, Vassar College, Ilya Kofman Kofman, sity, and Wenxiong Chen, Yeshiva University. College of Staten Island and The Graduate Center, City Uni- Commutative Algebra (Code: SS 1A), Laura Ghezzi, versity of New York, and Adam Lowrance, Vassar College. New York City College of Technology-CUNY, and Jooyoun Mathematical Phylogenetics (Code: SS 34A), Megan Hong, Southern Connecticut State University. Owen, City University of New York, and Katherine St. Computability Theory: Pushing the Boundaries (Code: John, Lehman College, City University of New York, and SS 9A), Johanna Franklin, Hofstra University, and Russell American Museum of Natural History. Miller, Queens College and Graduate Center, City Univer- Model Theory: Algebraic Structures in “Tame” Model sity of New York. Theoretic Contexts (Code: SS 28A), Alfred Dolich, Kings- Computational and Algorithmic Group Theory (Code: borough Community College and The Graduate Center, SS 7A), Denis Serbin and Alexander Ushakov, Stevens City University of New York, Michael Laskowski, Univer- Institute of Technology. sity of Maryland, and Mahmood Sohrabi, Stevens Institute Cryptography (Code: SS 3A), Xiaowen Zhang, College of Technology. of Staten Island and Graduate Center-CUNY. Nonlinear and Stochastic Partial Differential Equations: Current Trends in Function Spaces and Nonlinear Analy- Theory and Applications in Turbulence and Geophysical sis (Code: SS 2A), David Cruz-Uribe, University of Alabama, Flows (Code: SS 8A), Nathan Glatt-Holtz, Tulane University, Jan Lang, The Ohio State University, and Osvaldo Mendez, Geordie Richards, Utah State University, and Xiaoming University of Texas at El Paso. Wang, Florida State University. Differential and Difference Algebra: Recent Develop- Numerical Analysis and Mathematical Modeling (Code: ments, Applications, and Interactions (Code: SS 12A), Omár SS 32A), Vera Babenko, Ithaca College. León-Sanchez, McMaster University, and Alexander Levin, Operator Algebras and Ergodic Theory (Code: SS 17A), The Catholic University of America. Genady Grabarnik and Alexander Katz, St John’s Uni- Dynamical Systems (Code: SS 31A), Marian Gidea, Ye- versity. shiva University, W. Patrick Hooper, City College of New Qualitative and Quantitative Properties of Solutions to York and the City University of New York, and Anatole Partial Differential Equations (Code: SS 20A), Blair Davey, Katok, Pennsylvania State University. The City College of New York-CUNY, and Nguyen Cong Ehrhart Theory and its Applications (Code: SS 25A), Phuc and Jiuyi Zhu, Louisiana State University. Dan Cristofaro-Gardiner, Harvard University, Quang- Recent Developments in Automorphic Forms and Rep- Nhat Le, Brown University, and Sinai Robins, University resentation Theory (Code: SS 21A), Moshe Adrian, Queens of Saõ Paulo. College-CUNY, and Shuichiro Takeda, University of Mis- Euler and Related PDEs: Geometric and Harmonic Meth- souri. ods (Code: SS 27A), Stephen C. Preston, Brooklyn College, Representation Spaces and Toric Topology (Code: SS City University of New York, and Kazuo Yamazaki, Kazuo 13A), Anthony Bahri, Rider University, and Daniel Ramras Yamazaki, University of Rochester. and Mentor Stafa, Indiana University-Purdue University Finite Fields and their Applications (Code: SS 23A), Ri- Indianapolis. cardo Conceicao and Darren Glass, Gettysburg College, Topological Dynamics (Code: SS 33A), Alica Miller, and Ariane Masuda, New York City College of Technology, University of Louisville. City University of New York. Geometric Function Theory and Related Topics (Code: SS Accommodations 19A), Sudeb Mitra, Queens College and Graduate Center- Participants should make their own arrangements di- CUNY, and Zhe Wang, Bronx Community College-CUNY. rectly with the hotel of their choice. Special discounted Geometry and Topology of Ball Quotients and Related rates were negotiated with the hotels listed below. Rates Topics (Code: SS 5A), Luca F. Di Cerbo, Max Planck Insti- quoted do not include the New York City hotel tax (14.75 tute, Bonn, and Matthew Stover, Temple University. percent) and a US$3.50 per room, per night occupancy fee.

292 Notices of the AMS Volume 64, Number 3 MEETINGS & CONFERENCES

Participants must state that they are with the American your reservation. The deadline for reservations at this rate Mathematical Society (AMS) Meeting at Hunter College is April 3, 2017. to receive the discounted rate. The AMS is not responsible Hyatt Place New York/Midtown-South, 52 West 36th for rate changes or for the quality of the accommodations. Street, New York, NY 10018; 212-239-9100; https:// Hotels have varying cancellation and early checkout newyorkmidtown.place.hyatt.com/en/hotel/home. penalties; be sure to ask for details. html. Rates are US$199 per night for a room with one The Benjamin, 125 East 50th Street, New York NY, king bed and US$229 for a room with two queen beds, 10022; 866.222.2365; www.thebenjamin.com/ Rates are these rates are applicable for single or double occupancy. US$279 per night for a guest room with one queen bed, Queen rooms also include sofa sleeper. To book please call this rate is applicable for single or double occupancy; each 1-888-492-8847 or visit Hyatt.com and use G-MATH as additional person US$40. To book call 1-866-AFFINIA. A the group/corporate number. Amenities include compli- US$30 Facility Fee has been waived and internet access has mentary Wi-Fi, refrigerator and single serve coffee maker been provided complimentarily in guest rooms. Amenities in room, fitness room, and complimentary hot breakfast. include a kitchenette with a mini fridge in-room, fitness Also located on property are Gallery Menu, serving made center, Rest and Renew program for better sleep, room to order fresh food, and Coffee to Cocktails, serving Star- service, and on property James Beard Award-winning bucks® specialty coffees & teas, premium beers, wines & restaurant, The National, serving breakfast, lunch, and cocktails. Self-parking is located at Meyers Parking Garage dinner. Valet parking is available at a daily rate of US$65 at 12 West 36th Street. This property is located approxi- for a sedan and US$70 for an SUV, per day. This property is mately 2.3 miles from campus. Check-in is at 3:00 pm, located approximately 1 mile from campus. Check-in is at check-out is at 12:00 pm. Cancellation and early check-out 3:00 pm, check-out is at 12:00 pm. Cancellation and early policies vary and penalties exist at this property; be sure check-out policies vary and penalties exist at this prop- to check when you make your reservation. The deadline erty; be sure to check when you make your reservation. for reservations at this rate is April 3, 2017. The deadline for reservations at this rate is April 7, 2017. Hyatt Centric Times Square, 135 West 45th Street, New Fifty NYC, 155 East 50th Street At Third Avenue New York, NY 10036; 646-364-1234; https://timessquare. York, NY 10022; 212.751.5710; www.affinia.com/ centric.hyatt.com/en/hotel/home.html. Rates are fifty. Rates are US$235 per night for a guest room with US$319 per night for a room with one king bed or for a one queen bed or two double beds, this rate is applicable room with two queen beds, these rates are applicable for for single or double occupancy; each additional person single or double occupancy. Amenities include complimen- US$40. To book call 1-866-AFFINIA. A US$25 Facility Fee tary Wi-Fi, fitness room, business center, and Timeless: A has been waived and internet access has been provided Marilyn Monroe Spa. Also located on property are T45, a complimentarily in guest rooms. Amenities include a cof- bistro-style restaurant, and Bar 54, a rooftop bar located fee maker and a mini refrigerator in-room, the Club Room on the 54th level. Valet parking is available, inclusive of lounge, fitness center, and room service is available. Valet SUVs and oversized vehicles, for a flat fee of US$65 per Parking is available for US$55 for cars, and US$65 for day. This property is located approximately 1.7 miles from SUVs and minivans, per day. In-and-out privileges are not campus. Check-in is at 4:00 pm, check-out is at 11:00 am. included. This property is located approximately 1 mile Cancellation and early check-out policies vary and penal- from the campus. Check-in is at 3:00 pm, check-out is at ties exist at this property; be sure to check when you make 12:00 pm. Cancellation and early check-out policies vary your reservation. The deadline for reservations at this rate and penalties exist at this property; be sure to check when is April 5, 2017. you make your reservation. The deadline for reservations Lexington New York City, 511 Lexington Avenue at this rate is April 7, 2017. (at 48th Street), New York, NY, 10017, 1-800-448-4471 Hyatt Herald Square, 30 W 31st St, New York, NY 1000, or 212-755-4400; https://www.lexingtonhotelnyc. 212-330-1234; https://newyorkheraldsquare.hyatt. com/. Rates are US$179 per night for a single or double com/en/hotel/home.html. Rates are US$219 for a room occupancy room with a king bed and complimentary Wi- with one king bed and US$249 for a room with two queen Fi. Amenities include in-room laptop safe, fitness center, beds, for single or double occupancy. To book please call business center, Starbucks coffeehouse and on-site bar, 1-800-233-1234 or visit Hyatt.com and reference AM the Mixing Room, and bistro Raffles serving breakfast, MATHEMATICAL SOC. Additional charges of US$10 per lunch, and dinner. Valet parking or off-site parking is avail- person will apply for triple or quad occupancy. Ameni- able for US$65 per day, oversized vehicles will be US$75 ties include complimentary Wi-Fi, a coffee maker and a per day. This property is located approximately 1.2 miles mini refrigerator in-room, fitness center, room service, from the campus. Check-in is at 4:00 pm, check-out is at and on-property dining at the Den, Espresso Bar, and Up 12:00 pm. Cancellation and early check-out policies vary on 20. Valet parking is available at a daily rate of US$60 and penalties exist at this property; be sure to check when for a standard car and US$70 for an SUV, per day. This you make your reservation. The deadline for reservations property is located approximately 2.2 miles from campus. at a reduced rate is April 5, 2017. Cancellation and early check-out policies vary and penal- Millennium Broadway Hotel New York, 145 ties exist at this property; be sure to check when you make West 44th Street, New York, NY; 212-768-4400;

March 2017 Notices of the AMS 293 MEETINGS & CONFERENCES https://www.millenniumhotels.com/en/new-york/ Renaissance New York Hotel 57, 130 East 57th St. millennium-broadway-hotel-new-york/#about. New York, NY 10022; 207-443-9741; www.marriott. Rates are US$279 per night for a Superior guest room with com/hotels/travel/nycbr-renaissance-new-york- a king or two double beds or US$319 for a Premier guest hotel-57/. To book a room at this rate directly call room with a king or queen bed. Either rate is for single or Marriott Reservations at 1-800-468-3571 and identify the double occupancy. Premier rooms offer complimentary room block as the American Mathematical Society Spring continental breakfast, and afternoon hors d’oeuvres with Meeting at Hunter College at the Renaissance. Rates are wine or beer. Complimentary internet access in guest US$199 per night for a cozy guest room with one double rooms has been included in this rate. Amenities include bed, single/double occupancy. Complimentary internet fitness center, business center, and on-site restaurant, access in guest rooms is included in this rate. Amenities Charlotte, serving breakfast. Valet parking is available, include on-site business center, fitness center, on-site bar, inquire about rates when making a reservation. This prop- Opia Lounge and Bar, and Opia Restaurant offering French erty is located approximately 1.8 miles from the campus. cuisine for breakfast, lunch, and dinner. Valet parking or Check-in is at 4:00 pm, check-out is at 11:00 am. Cancella- off-site parking is available for US$55 per day, oversized tion and early check-out policies vary and penalties exist vehicles will be US$65 per day. This property is located at this property; be sure to check when you make your approximately 0.5 miles from the campus. Cancellation reservation. The deadline for reservations at this rate is and early check-out policies vary and penalties exist April 5, 2017. at this property; be sure to check when you make your NYLO, 2178 Broadway (at W 77th Street), New York, reservation. The deadline for reservations at this rate is NY 10024; 212-362-1100; http://www.nylohotels.com/ April 5, 2017. nyc. Rates are US$269 on 5/5 for a standard room with a queen or king bed and US$289 on 5/6 for a standard room Food Services with a queen or king bed this rate is applicable for single On Campus: Most campus dining will be closed during or double occupancy. To book please call 1-800-509-7598. the dates of this meeting. There is a Starbucks located in Amenities include complimentary Wi-Fi, fitness center, in the West Lobby which will be open on Saturday between room coffee maker, and complimentary bottled water in 7:30 am–2:30 pm. guest rooms. Also located on property are Serafina, serv- Off Campus: New York City offers an endless array of din- ing Northern Italian cuisine for breakfast, lunch, or dinner, ing options throughout the city. For more information on for dine in or take-out; Red Farm serving modern Chinese dining throughout NYC and for more information on visit- cuisine and dim sum for lunch, brunch, or dinner; and ing New York in general please visit, www.nycgo.com/. LOCL Bar, a lounge. This property is located approximately Some dining options nearby to the 68th Street Campus 1.8 miles from campus. Check-in is at 4:00 pm, check-out include: is at 11:00 am. Cancellation and early check-out policies Burger King, 226 E 86th Street (3rd & 2nd Avenue), vary and penalties exist at this property; be sure to check (212) 452-4675, Saturday 6:00 am–10:00 pm and Sunday, when you make your reservation. The deadline for reserva- 7:00 am–10:00 pm; fast-food burgers. tions at this rate is April 5, 2017. Dunkin Donuts/Baskin Robbins, 1225 1st Avenue (66th & Park Central, 870 7th Avenue New York, NY 10019; 67th Street), (212) 734-5465, open 24 hours; coffee, donuts, 212-247-8000; www.parkcentralny.com/hotel. Rates sandwiches, ice cream. for classic guest rooms (one king bed or two double beds) Fig and Olive, 808 Lexington Ave. (Between 62nd & 63rd at this property will be discounted by 15 percent off of Streets), (212) 207-4555, Saturday 11:00 am–11:00 pm and the best available rate at the time of booking. A Facility Sunday, 11:00 am–10:00 pm; French/Spanish fusion. Fee of US$31 has been waived for this property and allows Gourmet Bagel, 874 Lexington Avenue (65th & 66th Street), guests Wi-Fi access through the hotel’s guest’s rooms and (212) 879-5200, Saturday and Sunday, 6:00 am–6:00 pm; ba- public spaces for up to five devices, unlimited local and gels, sandwiches, deli fare. domestic long distance calls, and use of the fitness center. Hale & Hearty Soups, 849 Lexington Ave (64th & 65th Street), Amenities include on-site business center, fitness center, (212) 517-7600, Saturday 11:00 am–5:30 pm and Sunday, Park Lounge cocktail lounge, and “grab-and-go”Central 11:00 am–4:30 pm; Brooklyn based chain with from-scratch Market Cafe. The full service restaurant, Redeye Grill, is soups, sandwiches, and salads. located just steps outside this property, serving lunch Hunter Delicatessen, 966 Lexington Ave (70th & 71st Street), and dinner. Valet parking or off-site parking is available (212) 439-7758, Saturday and Sunday, 7:00 am–5:00 pm; salad for US$65 per day, oversized vehicles will be US$75 per bar and made-to-order sandwiches. day. This property is located approximately 1.2 miles J G Melon, 1291 3rd Ave. (at 74th St.), (212) 744-0585, Sat- from the campus. Check-in is at 4:00 pm, check-out is at urday 11:30 am– 4:00 am and Sunday, 11:30 am–midnight, 12:00 pm. Cancellation and early check-out policies vary cash only ; burgers, traditional American fare. and penalties exist at this property; be sure to check when Lenwich/Lenny's Sandwiches, 1269 1st Avenue (68th & you make your reservation. The deadline for reservations 69th Street), (212) 288-5171, Saturday and Sunday, 8:00 am– at this rate is April 10, 2017. 7:00 pm; sandwiches and salads.

294 Notices of the AMS Volume 64, Number 3 MEETINGS & CONFERENCES

Mariella Pizza, 965 Lexington Ave (70th & 71st Street), (212) prospective authors. If you have a book project that you 249-2065, Saturday and Sunday, 11:00 am–9:00 pm; pizza by wish to discuss with the AMS, please stop by the book the slice and pie, pasta and sandwiches. exhibit. McDonald's, 1286 1st Avenue (69th & 70th Street), (212) 249- 3551, open 24 hours; fast-food burgers. Erd˝os Memorial Lecture PJ Bernstein Deli, 1215 3rd Ave (70th & 71st Street), (212) There will be an Erdös Memorial Lecture given by James 879-0914, Saturday and Sunday, 8:00 am– 9:00 pm; kosher- Maynard, Magdalen College. This lecture will take place in style deli restaurant. W615, in the Hunter West Building at 5:10 pm on Saturday. Starbuck's Coffee, 1128 3rd Avenue (65th & 66th Street), There will be a reception for participants on Saturday (212) 472-6535, Saturday and Sunday, 5:30 am–9:30 pm; cof- evening immediately following the Erd˝os lecture, in the fee and light fare. Faculty Dining Room and adjacent terrace, located on the Subway, 1269 1st Avenue (68th & 69th Street), (212) 628- 8th Floor of the Hunter West Building. Please watch for 6358, Saturday 8:00 am–11:30 pm and Sunday, 9:00 am– more details on this event on the AMS website and at the 4:30 pm; fast-food sandwich shop. registration desk on-site at the meeting. The AMS thanks Tasti D-Lite, 1221 3rd Avenue (70th & 71st Street), (212) 288- our hosts for their gracious hospitality. 7088, Saturday and Sunday, 11:00 am–11:00 pm; soft serve frozen dessert. Special Needs It is the goal of the AMS to ensure that its conferences Registration and Meeting Information are accessible to all, regardless of disability. The AMS will Advance Registration: Advance registration for this meet- strive, unless it is not practicable, to choose venues that ing opened on January 19th. Advance registration fees are fully accessible to the physically handicapped. are US$59 for AMS members, US$85 for nonmembers, If special needs accommodations are necessary in order and US$10 for students, unemployed mathematicians, for you to participate in an AMS Sectional Meeting, please and emeritus members. Participants may cancel registra- communicate your needs in advance to the AMS Meetings tions made in advance by e-mailing [email protected]. The Department by: deadline to cancel is the first day of the meeting. - Registering early for the meeting On-site Information and Registration: The registration - Checking the appropriate box on the registration desk, AMS book exhibit, and coffee service will be located form, and in the West Lobby, in the atrium of the main entrance on - Sending an e-mail request to the AMS Meetings Depart- 68th Street. The Invited Addresses and Erd˝os Memorial ment at [email protected] or [email protected]. Lecture will be held in W615, on the 6th Floor in the Hunter AMS Policy on a Welcoming Environment West Building. Special Sessions and Contributed Paper Sessions will take place in several buildings on campus, The AMS strives to ensure that participants in its activities within walking distance of each other and connected by enjoy a welcoming environment. In all its activities, the indoor skybridges. These buildings will include Hunter AMS seeks to foster an atmosphere that encourages the West Building, Hunter East Building, and Hunter North free expression and exchange of ideas. The AMS supports Building. Please look for additional information about spe- equality of opportunity and treatment for all participants, cific session room locations on the web and in the printed regardless of gender, gender identity, or expression, race, program. For further information on building locations, color, national or ethnic origin, religion or religious belief, a campus map is available at www.hunter.cuny.edu/ age, marital status, sexual orientation, disabilities, or visitorscenter/repository/images/68thStreet. veteran status. jpg. The registration desk will be open on Saturday, Local Information and Maps May 6, 7:30 am– 4:00 pm and Sunday, May 7, 8:00 am– This meeting will take place on the 68th Street Cam- 12:00 pm. The same fees apply for on-site registration, as pus of Hunter College. A campus map can be found at for advance registration. Fees are payable on-site via cash, www.hunter.cuny.edu/visitorscenter/repository/ check, or credit card. images/68thStreet.jpg. Information about the Hunter Department of Mathematics and Statistics can be found Other Activities at math.hunter.cuny.edu/. Please visit the Hunter Col- Book Sales: Stop by the on-site AMS bookstore to review lege website at www.hunter.cuny.edu. Please watch the the newest publications and take advantage of exhibit dis- AMS website at www.ams.org/meetings/sectional/ counts and free shipping on all on-site orders! AMS mem- sectional.html for additional information on this meet- bers receive 40 percent off list price. Nonmembers receive ing. a 25 percent discount. Not a member? Ask a representative about the benefits of AMS membership. Complimentary Parking coffee will be served courtesy of AMS Membership Ser- There are many parking garages nearby to the Hunter vices. campus. Participants are encouraged to use mobile apps AMS Editorial Activity: An acquisitions editor from and Web searches to determine parking availablity and the AMS book program will be present to speak with the most economical pricing for the day of their trip.

March 2017 Notices of the AMS 295 MEETINGS & CONFERENCES

The College did indicate at the time of publication that Newark Liberty International Airport (EWR), is located the following garages are close in proximity to the main in Newark, New Jersey across the Hudson River and campus: 254 East 68th Street (3rd & 2nd Avenue), 1315 approximately 16 miles from midtown Manhattan and 2nd Avenue (69th & 70th Street), and 700 Park Avenue offers service from over 23 carriers. (69th & 70th Street). To travel from EWR to the Hunter campus there are many ground transportation options to choose from. Travel Taxi fare is approximately a US$50–US$75 metered fare The main campus of Hunter College is located in the heart plus bridge and tunnel tolls and gratuity, and should take of Manhattan on 68th Street. between 45–60 minutes travel time depending on traffic. By Air: There are three major airports servicing the NYC There may also be a US$5 surcharge for travel into New area: John F. Kennedy International Airport (JFK) or La- York state, during certain days and times. Please note, Guardia Airport (LGA), both in Queens, or Newark Liberty when traveling to the airport from Midtown Manhattan, International Airport (EWR) in neighboring New Jersey. service is via New York City’s regulated yellow taxis. John F. Kennedy International Airport (JFK), New York's Metered fares range US$69–US$75, plus a US$17.50 sur- largest airport, is located in Jamaica, in the borough of charge in addition to tolls and gratuity. To travel by train, Queens, approximately 15 miles from midtown Manhattan AirTrain links to the airport via NJ Transit and Amtrak's and offers service from over 80 carriers. Newark (or EWR) train station. Travel by train will take To travel from JFK to the Hunter campus there are approximately 45 to 90 minutes to Midtown Manhattan, many ground transportation options to choose from. Taxi requiring a transfer from the AirTrain line to the NJ Tran- fare is a US$52.50 flat fare (non metered) plus bridge and tunnel tolls and gratuity, and should take between 30 sit line or Amtrak. Shuttle buses are also available from minutes and 60 minutes travel time depending on traf- the following vendors: NYC Airporter (https://www. fic. For more information, call 212-NYC-TAXI. AirTrain nycairporter.com/), Go Airlink NYC (https://www. JFK links the airport to the subway for a fare of US$5. goairlinkshuttle.com/) and SuperShuttle (https:// Connecting from the JFK AirTrain at a JFK Airport MTA www.supershuttle.com/locations/newyorkcity- Station, expect that journey should take 60 to 75 minutes jfk-lga/). For more information on JFK International to arrive in midtown Manhattan. For more information on Airport and other ground transportation options, please the subway as well as city bus options, please visit trip- visit www.panynj.gov/airports/JFK.html. planner.mta.info. Shuttle buses are also available from By Train: New York City has two main rail stations in the following vendors: NYC Airporter (https://www. Midtown: Grand Central Terminal (on the east side) and nycairporter.com/), Go Airlink NYC (https://www. Penn Station (on the west side). Grand Central Terminal is goairlinkshuttle.com/) and SuperShuttle (https:// the main terminal for Metro-North Railroadservices (www. www.supershuttle.com/locations/newyorkcity- mta.info/mnr) serving NYC and Connecticut suburbs, as jfk-lga/). For more information on JFK International well as multiple subway lines. Penn Station is served by Airport and other ground transportation options, please the Long Island Railroad (www.mta.info/lirr), Amtrak visit www.panynj.gov/airports/JFK.html. (https://www.amtrak.com/home), and NJ Transit (www. LaGuardia Airport (LGA), New York's 2nd largest njtransit.com), as well as multiple subway lines. airport, is located in Jackson Heights in the borough of By Bus: There are a number bus lines that travel to Queens, approximately 8 miles from midtown Manhattan New York City from around the United States and parts of and offers service from over 20 carriers. Canada. Options include BoltBus (https://www.boltbus. To travel from LGA to the Hunter campus there are com/), Megabus (us.megabus.com/) and Greyhound. For many ground transportation options to choose from. more information on Greyhound please call 800-231-2222 Taxi fare is approximately a US$29–US$37 metered fare or visit https://www.greyhound.com/. plus bridge and tunnel tolls and gratuity, and should By Car: Please note that driving directions are con- take between 20–25 minutes travel time depending on stantly updated due to construction and road closures. traffic. For more information, call 212-NYC-TAXI. Two From Queens/ Long Island: Take the Long Island express city buses serve LaGuardia; the M60 and Q70. Expressway west to the Queens Midtown Tunnel. Make The Q70 goes nonstop to Jackson Heights/Roosevelt a right onto Third Avenue and proceed north to 69th Avenue, a major subway hub in Queens with five lines. Street. Make a left turn on 69th Street. Proceed to Lexing- The M60 runs to Harlem and connects to all the major ton Avenue and make a left. The College is located at the subway lines in Manhattan. For more information on city bus options, please visit tripplanner.mta.info. Shuttle intersection of East 68th Street and Lexington Avenue. buses are also available from the following vendors: From Bronx / Westchester: Take the New York Thru- NYC Airporter (https://www.nycairporter.com/), Go way (I-87) to the Major Deegan Expressway. Continue to Airlink NYC (https://www.goairlinkshuttle.com/) the 3rd Avenue Bridge, then to the FDR Drive, then to the and SuperShuttle (https://www.supershuttle.com/ 71st Street exit. Continue straight to Lexington Avenue locations/newyorkcity-jfk-lga/). For more informa- and make a left. Continue to East 68th Street. The Col- tion on LaGuardia Airport and other ground transporta- lege is located at the intersection of East 68th Street and tion options, please visit laguardiaairport.com/. Lexington Avenue.

296 Notices of the AMS Volume 64, Number 3 MEETINGS & CONFERENCES

Via the Lincoln Tunnel: Take the Lincoln Tunnel. After degrees Fahrenheit and the average low is approximately exiting the tunnel, take 10th Avenue north to 65th Street. 60 degrees Fahrenheit. Visitors should be prepared for Turn right and cross Central Park. After exiting Central inclement weather and check weather forecasts in advance Park, continue along East 65th Street to Park Avenue. Turn of their arrival. left on Park Avenue. Go three blocks and make a right turn on 68th Street. The College will be at the end of the block. Social Networking Via the George Washington Bridge: Take the George Attendees and speakers are encouraged to tweet about Washington Bridge to the Harlem River Drive. Continue the meeting using the hashtags #AMSmtg. to the FDR Drive, then to the 71st Street exit. Continue straight to Second Avenue. Make a left turn and travel on Information for International Participants Second Avenue to 69th Street. Make a right on 69th Street Visa regulations are continually changing for travel to the and travel to Lexington Avenue. Make a left turn on Lex- United States. Visa applications may take from three to ington. The College is located at the intersection of East four months to process and require a personal interview, 68th Street and Lexington Avenue. as well as specific personal information. International Car Rental: Hertz is the official car rental company for participants should view the important information about the meeting. To make a reservation accessing our special traveling to the US found at https://travel.state. meeting rates online at www.hertz.com, click on the box gov/content/travel/en.html. If you need a preliminary “Enter a discount or promo code,” and type in our conven- conference invitation in order to secure a visa, please send tion number (CV): CV#04N30007. You can also call Hertz your request to [email protected]. directly at 800-654-2240 (US and Canada) or 1-405-749- If you discover you do need a visa, the National Acad- 4434 (other countries). At the time of reservation, the meeting rates will be automatically compared to other emies website (see above) provides these tips for success- Hertz rates and you will be quoted the best comparable ful visa applications: rate available. * Visa applicants are expected to provide evidence that For directions to campus, inquire at your rental car they are intending to return to their country of residence. counter. Therefore, applicants should provide proof of “binding” or sufficient ties to their home country or permanent Local Transportation residence abroad. This may include documentation of New York City has excellent public transportation in the the following: forms of buses and subways. For more information on - family ties in home country or country of legal per- the MTA or to plan your trip please visit www.mta.info/. manent residence Taxis and other car services are also available throughout - property ownership the city. - bank accounts Subway Directions: The 6 train stops directly under the - employment contract or statement from employer College at the 68th Street Station. There is an entrance to the College in the Subway station. Turn right upon exit- stating that the position will continue when the employee ing the turnstile and the entrance will be directly in front returns; of you. * Visa applications are more likely to be successful if The F train is another option, using the East 63rd Street done in a visitor’s home country than in a third country; and Lexington Avenue stop. After exiting the station, walk * Applicants should present their entire trip itinerary, north on Lexington Avenue to East 68th Street. The Col- including travel to any countries other than the United lege is located at the intersection of East 68th Street and States, at the time of their visa application; Lexington Avenue. * Include a letter of invitation from the meeting orga- Bus Directions: The crosstown M66 bus goes east on 68th nizer or the US host, specifying the subject, location and Street, and west on 67th Street. Hunter College is located dates of the activity, and how travel and local expenses at the intersection of East 68th Street and Lexington will be covered; Avenue. The M98, M101, M102, and M103 go south on * If travel plans will depend on early approval of the Lexington Avenue and north on 3rd Avenue. Hunter Col- visa application, specify this at the time of the application; lege is located at the intersection of East 68th Street and * Provide proof of professional scientific and/or Lexington Avenue. educational status (students should provide a university Weather transcript). Weather in May in New York City is typically pleasant and This list is not to be considered complete. Please visit mild. The average high temperature is approximately 75 the websites above for the most up-to-date information.

March 2017 Notices of the AMS 297 MEETINGS & CONFERENCES

should send your abstract as early as possible via the ab- Montréal, Quebec stract submission form found at www.ams.org/cgi-bin/ abstracts/abstract.pl. Canada Algebraic Combinatorics of Flag Varieties (Code: SS 11A), Martha Precup, Northwestern University, and Ed- McGill University ward Richmond, Oklahoma State University. Banach Spaces and Applications (Code: SS 9A), Pavlos July 24–28, 2017 Motakis, Texas A&M University, and Bönyamin Sari, Uni- Monday – Friday versity of North Texas. Combinatorics and Representation Theory of Reflec- Meeting #1130 tion Groups: Real and Complex (Code: SS 14A), Elizabeth The second Mathematical Congress of the Americas (MCA Drellich, Swarthmore College, and Drew Tomlin, Hendrix 2017) is being hosted by the Canadian Mathematical Soci- College. ety (CMS) in collaboration with the Pacific Institute for the Commutative Algebra (Code: SS 10A), Jonathan Mon- Mathematical Sciences (PIMS), the Fields Institute (FIELDS), tano, University of Kansas, and Alessio Sammartano, Le Centre de Recherches Mathématiques (CRM), and Purdue University. the Atlantic Association for Research in the Mathematical Differential Equation Modeling and Analysis for Complex Sciences (AARMS). Bio-systems (Code: SS 8A), Pengcheng Xiao, University of Associate secretary: Brian D. Boe Evansville, and Honghui Zhang, Northwestern Polytechni- Announcement issue of Notices: To be announced cal University. Program first available on AMS website: To be announce- Dynamics, Geometry and Number Theory (Code: SS dIssue of Abstracts: To be announced 1A), Lior Fishman and Mariusz Urbanski, University of Deadlines North Texas. For organizers: Expired Fractal Geometry and Ergodic Theory (Code: SS 5A), For abstracts: March 31, 2017 Mrinal Kanti Roychowdhury, University of Texas Rio Grande Valley. Homological Methods in Commutative Algebra (Code: Denton, Texas SS 15A), Peder Thompson, Texas Tech University, and University of North Texas Ashley Wheeler, University of Arkansas. Invariants of Links and 3-Manifolds (Code: SS 7A), Miec- September 9–10, 2017 zyslaw K. Dabkowski and Anh T. Tran, The University Saturday – Sunday of Texas at Dallas. Lie algebras, Superalgebras, and Applications (Code: SS Meeting #1131 3A), Charles H. Conley, University of North Texas, Dimitar Central Section Grantcharov, University of Texas at Arlington, and Natalia Associate secretary: Georgia Benkart Rozhkovskaya, Kansas State University. Announcement issue of Notices: June 2017 Noncommutative and Homological Algebra (Code: SS Program first available on AMS website: July 27, 2017 4A), Anne Shepler, University of North Texas, and Sarah Issue of Abstracts: Volume 38, Issue 3 Witherspoon, Texas A&M University. Deadlines Nonlocal PDEs in Fluid Dynamics (Code: SS 12A), Chan- For organizers: Expired ghui Tan, Rice University, and Xiaoqian Xu, Carnegie For abstracts: July 18, 2017 Mellon University. Numbers, Functions, Transcendence, and Geometry The scientific information listed below may be dated. (Code: SS 6A), William Cherry, University of North Texas, For the latest information, see www.ams.org/amsmtgs/ Mirela Çiperiani, University of Texas Austin, Matt Papa- sectional.html. nikolas, Texas A&M University, and Min Ru, University Invited Addresses of Houston. Real-Analytic Automorphic Forms (Code: SS 2A), Olav K Mirela Çiperiani, University of Texas at Austin, Title Richter, University of North Texas, and Martin Westerholt- to be announced. Raum, Chalmers University of Technology. Adrianna Gillman, Rice University, Title to be an- Topics Related to the Interplay of Noncommutative Al- nounced. gebra and Geometry (Code: SS 13A), Richard Chandler, Kevin Pilgrim, Indiana University, Title to be announced. University of North Texas at Dallas, Michaela Vancliff, Special Sessions University of Texas at Arlington, and Padmini Veerapen, If you are volunteering to speak in a Special Session, you Tennessee Technological University.

298 Notices of the AMS Volume 64, Number 3 MEETINGS & CONFERENCES

Nonlinear Partial Differential Equations Arising from Life Science (Code: SS 8A), Junping Shi, College of William Buffalo, New York and Mary, and Xingfu Zou, University of Western Ontario. State University of New York at Buffalo Nonlinear Wave Equations, Inverse Scattering and Ap- plications. (Code: SS 1A), Gino Biondini, University at September 16–17, 2017 Buffalo-SUNY. Saturday – Sunday Structural and Chromatic Graph Theory (Code: SS 10A), Hong-Jian Lai, Rong Luo, and Cun-Quan Zhang, West Meeting #1132 Virginia University, and Yue Zhao, University of Central Eastern Section Florida. Associate secretary: Steven H. Weintraub p-adic Aspects of Arithmetic Geometry (Code: SS 3A), Announcement issue of Notices: June 2017 Ling Xiao, University of Connecticut, and Hui June Zhu, Program first available on AMS website: August 3, 2017 University at Buffalo-SUNY. Issue of Abstracts: Volume 38, Issue 3 Deadlines Orlando, Florida For organizers: February 16, 2017 For abstracts: July 25, 2017 University of Central Florida, Orlando September 23–24, 2017 The scientific information listed below may be dated. Saturday – Sunday For the latest information, see www.ams.org/amsmtgs/ sectional.html. Meeting #1133 Southeastern Section Invited Addresses Associate secretary: Brian D. Boe Inwon Kim, University of California at Los Angeles, Announcement issue of Notices: June 2017 Title to be announced. Program first available on AMS website: August 10, 2017 Govind Menon, Brown University, Title to be announced. Issue of Abstracts: Volume 38, Issue 4 Bruce Sagan, Michigan State University, Title to be an- Deadlines nounced. For organizers: February 23, 2017 Special Sessions For abstracts: August 1, 2017 If you are volunteering to speak in a Special Session, you should send your abstract as early as possible via the ab- The scientific information listed below may be dated. stract submission form found at www.ams.org/cgi-bin/ For the latest information, see www.ams.org/amsmtgs/ abstracts/abstract.pl. sectional.html.

Advanced Techniques in Graph Theory (Code: SS 9A), Invited Addresses Sogol Jahanbekam and Paul Wenger, Rochester Institute Christine Heitsch, Georgia Institute of Technology, of Technology. Title to be announced. CR Geometry and Partial Differential Equations in Jonathan Kujawa, University of Oklahoma, Title to be Complex Analysis (Code: SS 5A), Ming Xiao, University of announced. Illinois at Urbana-Champaign, and Yuan Yuan, Syracuse Christopher D Sogge, Johns Hopkins University, Title University. to be announced. Cohomology, Deformations, and Quantum Groups: A Session Dedicated to the Memory of Samuel D. Schack Special Sessions (Code: SS 6A), Miodrag Iovanov, University of Iowa, Mihai If you are volunteering to speak in a Special Session, you D. Staic, Bowling Green State University, and Alin Stancu, should send your abstract as early as possible via the ab- Columbus State University. stract submission form found at www.ams.org/cgi-bin/ Geometric Group Theory (Code: SS 4A), Joel Louwsma, abstracts/abstract.pl. Niagara University, and Johanna Mangahas, University at Buffalo-SUNY. Algebraic Curves and their Applications (Code: SS 3A), High Order Numerical Methods for Hyperbolic PDEs and Lubjana Beshaj, The University of Texas at Austin. Applications (Code: SS 2A), Jae-Hun Jung, University at Applied Harmonic Analysis: Frames, Samplings and Buffalo-SUNY, Fengyan Li, Rensselaer Polytechnic Insti- Applications (Code: SS 6A), Dorin Dutkay, Deguang Han, tute, and Li Wang, University at Buffalo-SUNY. and Qiyu Sun, University of Central Florida. Infinite Groups and Geometric Structures: A Session in Commutative Algebra: Interactions with Algebraic Honor of the Sixtieth Birthday of Andrew Nicas (Code: SS Geometry and Algebraic Topology (Code: SS 1A), Joseph 7A), Hans Boden, McMaster University, and David Rosen- Brennan, University of Central Florida, and Alina Iacob thal, St. John’s University. and Saeed Nasseh, Georgia Southern University.

March 2017 Notices of the AMS 299 MEETINGS & CONFERENCES

Fractal Geometry, Dynamical Systems, and Their Ap- Radunovic, University of California, Riverside. plications (Code: SS 4A), Mrinal Kanti Roychowdhury, Applied Category Theory (Code: SS 4A), John Baez, University of Texas Rio Grande Valley. University of California, Riverside. Graph Connectivity and Edge Coloring (Code: SS 5A), Combinatorial Representation Theory (Code: SS 5A), Colton Magnant, Georgia Southern University. Vyjayanthi Chari, University of California, Riverside, and Nonlinear Waves and Dispersive Equations (Code: SS Maria Monks Gillespie and Monica Vazirani, University of California, Davis. 7A), Benjamin Harrop-Griffiths, New York University, Combinatorial aspects of the polynomial ring (Code: Jonas Lührmann, Johns Hopkins University, and Dana SS 1A), Sami Assaf and Dominic Searles, University of Mendelson, University of Chicago. Southern California. Structural Graph Theory (Code: SS 2A), Martin Rolek, Ring Theory and Related Topics (Celebrating the 75th Zixia Song, and Yue Zhao, University of Central Florida. Birthday of Lance W. Small) (Code: SS 2A), Jason Bell, University of Waterloo, Ellen Kirkman, Wake Forest Uni- versity, and Susan Montgomery, University of Southern Riverside, California California. University of California, Riverside

November 4–5, 2017 San Diego, California Saturday – Sunday San Diego Convention Center and San Meeting #1134 Diego Marriott Hotel and Marina Western Section January 10–13, 2018 Associate secretary: Michel L. Lapidus Wednesday – Saturday Announcement issue of Notices: September 2017 Meeting #1135 Program first available on AMS website: September 21, Joint Mathematics Meetings, including the 124th Annual 2017 Meeting of the AMS, 101st Annual Meeting of the Math- Issue of Abstracts: Volume 38, Issue 4 ematical Association of America (MAA), annual meetings of the Association for Women in Mathematics (AWM) and Deadlines the National Association of Mathematicians (NAM), and the For organizers: April 14, 2017 winter meeting of the Association of Symbolic Logic (ASL), For abstracts: September 12, 2017 with sessions contributed by the Society for Industrial and Applied Mathematics (SIAM). The scientific information listed below may be dated. Associate secretary: Georgia Benkart For the latest information, see www.ams.org/amsmtgs/ Announcement issue of Notices: October 2017 Program first available on AMS website: To be announced sectional.html. Issue of Abstracts: Volume 39, Issue 1 Invited Addresses Deadlines Paul Balmer, University of California, Los Angeles, Title For organizers: April 1, 2017 to be announced. For abstracts: To be announced Pavel Etingof, Massachusetts Institute of Technology, Title to be announced. Call for Proposals Monica Vazirani, University of California, Davi, Com- The AMS solicits proposals for AMS Special Sessions at the binatorics, Categorification, and Crystals. 2018 Joint Mathematics Meetings to be held Wednesday, January 10 through Saturday January 13, 2018, in Special Sessions San Diego, CA. Each proposal must include: 1. the name, affiliation, and e-mail address of each or- If you are volunteering to speak in a Special Session, you ganizer, with one organizer designated as the contact should send your abstract as early as possible via the ab- person for all communication about the session; stract submission form found at www.ams.org/cgi-bin/ 2. the title and a brief (no longer than one or two para- abstracts/abstract.pl. graphs) description of the topic of the proposed special session; Algebraic and Combinatorial Structures in Knot Theory 3. the primary two-digit MSC number for the topic: See (Code: SS 3A), Patricia Cahn, Smith College, and Sam Nel- www.ams.org/mathscinet/msc/msc2010.html. son, Claremont McKenna College. 4. a sample list of speakers whom the organizers plan to Analysis and Geometry of Fractals (Code: SS 6A), Erin invite. (It is not necessary to have received confirmed Pearse, California Polytechnic State University, and Goran commitments from these potential speakers.)

300 Notices of the AMS Volume 64, Number 3 MEETINGS & CONFERENCES

Organizers are strongly encouraged to consult the AMS Recent Advances in Approximation Theory and Opera- Manual for Special Session Organizers at: www.ams.org/ tor Theory (Code: SS 1A), Jan Lang and Paul Nevai, The meetings/specialsessionmanual.html. Ohio State University. Proposals for AMS Special Sessions should be sent by e-mail to AMS Associate Secretary Georgia Benkart (ben- [email protected]) by April 7, 2017. Late proposals will Nashville, Tennessee not be considered. No decisions will be made on Special Session proposals until after the submission deadline Vanderbilt University has passed. April 14–15, 2018 Special Sessions will be allotted between 5 and 10 hours Saturday – Sunday in which to schedule speakers. To enable maximum move- ment of participants between sessions, organizers must Meeting #1138 schedule session speakers for either (a) a 20-minute talk Southeastern Section with 5-minute discussion and 5-minute break or (b) a Associate secretary: Brian D. Boe 45-minute talk with 5-minute discussion and 10-minute Announcement issue of Notices: To be announced break. Any combination of 20-minute and 45-minute Program first available on AMS website: To be announced talks is permitted, but all talks must begin and end at the Issue of Abstracts: To be announced scheduled time. The number of Special Sessions in the AMS program Deadlines at the Joint Mathematics Meetings is limited and because For organizers: To be announced of the large number of high-quality proposals, not all can For abstracts: To be announced be accepted. Please be sure to submit as detailed a pro- posal as possible for review by the Program Committee. Organizers of proposals will be notified whether their Portland, Oregon proposal has been accepted by May 12, 2017. Additional Portland State University instructions and the session’s schedule will be sent to the contact organizers of the accepted sessions shortly after April 14–15, 2018 that deadline. Saturday – Sunday Meeting #1137 Columbus, Ohio Western Section Associate secretary: Michel L. Lapidus Ohio State University Announcement issue of Notices: To be announced March 17–18, 2018 Program first available on AMS website: To be announced Saturday – Sunday Issue of Abstracts: To be announced Meeting #1136 Deadlines Central Section For organizers: To be announced Associate secretary: Georgia Benkart For abstracts: To be announced Announcement issue of Notices: To be announced Program first available on AMS website: To be announced The scientific information listed below may be dated. Issue of Abstracts: To be announced For the latest information, see www.ams.org/amsmtgs/ sectional.html. Deadlines Invited Addresses For organizers: To be announced Sándor Kovács, University of Washington, Seattle, Title For abstracts: To be announced to be announced. The scientific information listed below may be dated. Elena Mantovan, California Institute of Technology, For the latest information, see www.ams.org/amsmtgs/ Title to be announced. sectional.html. Special Sessions Special Sessions If you are volunteering to speak in a Special Session, you If you are volunteering to speak in a Special Session, you should send your abstract as early as possible via the ab- should send your abstract as early as possible via the ab- stract submission form found at www.ams.org/cgi-bin/ stract submission form found at www.ams.org/cgi-bin/ abstracts/abstract.pl. abstracts/abstract.pl. Inverse Problems (Code: SS 2A), Hanna Makaruk, Los Probability in Convexity and Convexity in Probability Alamos National Laboratory (LANL), and Robert Owcza- (Code: SS 2A), Elizabeth Meckes, Mark Meckes, and Elisa- rek, University of New Mexico, Albuquerque & Los Alamos. beth Werner, Case Western Reserve University. Pattern Formation in Crowds, Flocks, and Traffic (Code: SS 1A), J. J. P. Veerman, Portland State University, Alethea

March 2017 Notices of the AMS 301 MEETINGS & CONFERENCES

Barbaro, Case Western Reserve University, and Bassam Bamieh, UC Santa Barbara. Newark, Delaware University of Delaware Boston, September 29–30, 2018 Massachusetts Saturday – Sunday Meeting #1141 Northeastern University Eastern Section April 21–22, 2018 Associate secretary: Steven H. Weintraub Saturday – Sunday Announcement issue of Notices: To be announced Program first available on AMS website: To be announced Meeting #1139 Issue of Abstracts: To be announced Eastern Section Associate secretary: Steven H. Weintraub Deadlines Announcement issue of Notices: To be announced For organizers: To be announced Program first available on AMS website: To be announced For abstracts: To be announced Issue of Abstracts: To be announced Deadlines Fayetteville, For organizers: To be announced For abstracts: To be announced Arkansas The scientific information listed below may be dated. University of Arkansas For the latest information, see www.ams.org/amsmtgs/ sectional.html. October 6–7, 2018 Special Sessions Saturday – Sunday If you are volunteering to speak in a Special Session, you Meeting #1142 should send your abstract as early as possible via the ab- stract submission form found at www.ams.org/cgi-bin/ Southeastern Section abstracts/abstract.pl. Associate secretary: Brian D. Boe Announcement issue of Notices: To be announced Arithmetic Dynamics (Code: SS 1A), Jacqueline M. Program first available on AMS website: To be announced Anderson, Bridgewater State University, Robert Bene- Issue of Abstracts: To be announcedDeadlines detto, Amherst College, and Joseph H. Silverman, Brown For organizers: To be announced University. For abstracts: To be announced Shanghai, People’s Ann Arbor, Michigan Republic of China University of Michigan, Ann Arbor Fudan University October 20–21, 2018 Saturday – Sunday June 11–14, 2018 Monday – Thursday Meeting #1143 Central Section Meeting #1140 Associate secretary: Georgia Benkart Associate secretary: Steven H. Weintraub Announcement issue of Notices: To be announced Announcement issue of Notices: To be announced Program first available on AMS website: To be announced Program first available on AMS website: To be announced Issue of Abstracts: To be announced Issue of Abstracts: To be announced

Deadlines Deadlines For organizers: To be announced For organizers: To be announced For abstracts: To be announced For abstracts: To be announced

302 Notices of the AMS Volume 64, Number 3 MEETINGS & CONFERENCES

Program first available on AMS website: To be announced San Francisco, Issue of Abstracts: To be announced Deadlines California For organizers: To be announced San Francisco State University For abstracts: To be announced October 27–28, 2018 Saturday – Sunday Denver, Colorado Meeting #1144 Colorado Convention Center Western Section Associate secretary: Michel L. Lapidus January 15–18, 2020 Announcement issue of Notices: August 2018 Wednesday – Saturday Program first available on AMS website: To be announced Joint Mathematics Meetings, including the 126th Annual Issue of Abstracts: To be announced Meeting of the AMS, 103rd Annual Meeting of the Math- ematical Association of America (MAA), annual meetings Deadlines of the Association for Women in Mathematics (AWM) and For organizers: To be announced the National Association of Mathematicians (NAM), and the For abstracts: To be announced winter meeting of the Association of Symbolic Logic (ASL), with sessions contributed by the Society for Industrial and Applied Mathematics (SIAM) Associate secretary: Michel L. Lapidus Baltimore, Maryland Announcement issue of Notices: To be announced Baltimore Convention Center, Hilton Program first available on AMS website: November 1, 2019 Baltimore, and Baltimore Marriott Inner Issue of Abstracts: To be announced Harbor Hotel Deadlines For organizers: April 1, 2019 January 16–19, 2019 For abstracts: To be announced Wednesday – Saturday Joint Mathematics Meetings, including the 125th Annual Meeting of the AMS, 102nd Annual Meeting of the Math- Washington, District ematical Association of America (MAA), annual meetings of the Association for Women in Mathematics (AWM) and the National Association of Mathematicians (NAM), and the of Columbia winter meeting of the Association of Symbolic Logic (ASL), Walter E. Washington Convention Center with sessions contributed by the Society for Industrial and Applied Mathematics (SIAM). January 6–9, 2021 Associate secretary: Steven H. Weintraub Wednesday – Saturday Announcement issue of Notices: October 2018 Joint Mathematics Meetings, including the 127th Annual Program first available on AMS website: To be announced Meeting of the AMS, 104th Annual Meeting of the Math- Issue of Abstracts: To be announced ematical Association of America (MAA), annual meetings of the Association for Women in Mathematics (AWM) and Deadlines the National Association of Mathematicians (NAM), and the For organizers: April 2, 2018 winter meeting of the Association of Symbolic Logic (ASL), For abstracts: To be announced with sessions contributed by the Society for Industrial and Applied Mathematics (SIAM). Associate secretary: Brian D. Boe Honolulu, Hawaii Announcement issue of Notices: October 2020 Program first available on AMS website: November 1, 2020 University of Hawaii at Manoa Issue of Abstracts: To be announced

March 22–24, 2019 Deadlines Friday – Sunday For organizers: April 1, 2020 Central Section & Western Section For abstracts: To be announced Associate secretaries: Georgia Benkart and Michel L. Lapidus Announcement issue of Notices: To be announced

March 2017 Notices of the AMS 303 THE BACK PAGE

"I hope that seeing the excitement of solving this problem will make young mathematicians realize that there are lots and lots of other problems in mathematics which are going to be just as challenging in the future." —Sir Andrew Wiles

Andrew Wiles Stats: Earliest indexed publication: 1977 Total publications: 26 Total author/related publications: 38 Total citations: 1,645 PhD University of Cambridge 1979 Advisor: John Henry Coates According to the MathSciNet® database, as of December 9, 2016, Andrew Wiles has 21 students and 178 descendants. Two most cited works: Modular elliptic curves and Fermat's last theorem, With Richard Taylor, Ring-theoretic properties of cer-

Ann. of Math. (1995). tain Hecke algebras, Ann. of Math (1995).

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QUESTIONABLE MATHEMATICS The Boston Globe (April 2003) reported on data suggesting that "15 percent of US youngsters are se- verely overweight or obese, defined as having a body-mass index that is greater than the index for 95 percent of their peers." Submitted by Richard Stanley

What crazy things happen to you? Readers are invited to submit original short amusing stories, math jokes, cartoons, and other material to: [email protected].

304 Notices of the AMs VoluMe 64, NuMber 3 IN THE NEXT ISSUE OF NOTICES

APRIL 2017…

Full announcements of the 2017 Leroy P. Steele The AWM Research Symposium 2017 Lecture Prizes, Ruth Lyttle Satter Prize in Mathematics, Sampler Frank and Brennie Morgan Prize for Outstanding The Symposium (April 8–9, 2017 at UCLA) Research in Mathematics by an Undergraduate will showcase the research of women in the Student, Leonard Eisenbud Prize for Mathematics mathematical professions. Some of the plenary and Physics, Joseph L. Doob Prize, Levi L. Conant speakers and members of the committee have Prize, Cole Prize in Number Theory, Bôcher offered to share a sneak-peek of their presenta- Memorial Prize, which were presented at the tions with Notices. 123rd Annual Meeting of the AMS in Atlanta, Georgia, in January 2017.

A MERICAN M ATHEMATICAL S OCIETY A Literary Spotlight on Women in Mathematics Change Is Possible Pioneering Women in American Stories of Women and Minorities Mathematics in Mathematics The Pre-1940 PhD’s Patricia Clark Kenschaft, Montclair State University, Judy Green, Marymount University, Arlington, VA, and Upper Montclair, NJ Jeanne LaDuke, DePaul University, Chicago, IL The role of minority and women mathematicians in devel- What a service Judy Greene and Jeanne LaDuke have done oping our American mathematical community is an impor- the mathematics community! Approximately thirty years tant but previously under-told story. Pat Kenschaft, in her of research have produced a detailed picture of graduate highly readable and entertaining style, fills this knowledge mathematics for women in the United States before 1940. gap. This valuable book should be in your personal library! ... The book is well-organized and well-written, and I rec- —Donald G. Saari, University of California, Irvine ommend it heartily to all. An entertaining look at previously under-told stories of —AWM Newsletter mathematicians who defied stereotypes and overcame barriers to success. History of Mathematics, Volume 34; 2009; 345 pages; Hardcover; ISBN: 978-0-8218-4376-5; List US$86; 2005; 212 pages; Softcover; ISBN: 978-0-8218-3748-1; List AMS members US$68.80; Order code HMATH/34 US$34; AMS members US$27.20; Order code CHANGE Women's History Month Visit bookstore.ams.org to learn about these titles and more! American Mathematical Society Distribution Center 35 Monticello Place, Pawtucket, RI 02861 USA

AMERICAN MATHEMATICAL SOCIETY

New from the AMS Now available for pre order

Pushing Limits - ! From West Point to Berkeley and Beyond Ted Hill, Georgia Tech, Atlanta and Cal Poly, San Luis Obispo A fascinating journey from pure adventurism...through West Point and the Vietnam War to the highest intellectual accomplishments. At the center is a beautiful portrayal of the tedious, but highly rewarding road from graduate school to becoming a substantial research mathematician. A joy to read. —David Gilat, Professor Emeritus, School of Mathematical Sciences, Tel Aviv University Ted Hill is the Indiana Jones of mathematics...A West Point graduate, [he] served in Vietnam, swam with sharks in the Caribbean, and has resolutely defied unreasoned authority. With this same love of adventure, he has confronted the sublime challenges of mathematics. Whether it’s discovering intellectual treasures or careening down jungle trails, this real life Dr. Jones has done it all. —Michael Monticino, Professor of Mathematics and Special Assistant to the President, U. North Texas This book is co-published with the Mathematical Association of America. 2017; approximately 334 pages; Hardcover; ISBN: 978-1-4704-3584-4; List US$45; AMS members US$36; Order code MBK/103

Game Theory, Alive Anna R. Karlin, University of Washington, Seattle and Yuval Peres, Microsoft Research, Redmond, WA By focusing on theoretical highlights and presenting exciting connections between game theory and other fields, this broad overview emphasizes game theory’s real-world applications and mathematical foundations. 2017; approximately 396 pages; Hardcover; ISBN: 978-1-4704-1982-0; List US$75; AMS members US$60; Order code MBK/101 It’s About Time Elementary Mathematical Aspects of Relativity Roger Cooke, University of Vermont, Burlington This book explores a selection of topics from special and general relativity, accompanied by mathematical explanations accessible to those at the intermediate undergraduate level. 2017; 403 pages; Hardcover; ISBN: 978-1-4704-3483-0; List US$75; AMS members US$60; Order code MBK/102 Algebra in Action A Course in Groups, Rings, and Fields Shahriar Shahriari, Pomona College, Claremont, CA Aimed at undergraduates who are new to abstract algebra, this text is a readable, student-friendly, and some- what sophisticated introduction to groups, rings, and fields. Pure and Applied Undergraduate Texts, Volume 27; 2016; approximately 655 pages; Hardcover; ISBN: 978-1-4704-2849-5; List US$115; AMS members US$92; Order code AMSTEXT/27

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