Gunther Cornelissen, Jan van Neerven Interview with Kenneth Ribet NAW 5/18 nr. 2 juni 2017 115

Gunther Cornelissen Jan van Neerven Mathematical Institute Delft Institute of Applied Mathematics Utrecht University Delft University of Technology [email protected] [email protected]

Interview Kenneth Ribet, winner Brouwer Medal 2017 “No one had ever accused me of proving a theorem before”

On April 12th, 2017, Gunther Cornelissen and Jan van Neerven interviewed Ken Ribet at the classes. I was in a programme that, believe occasion of him winning the Brouwer Medal 2017. it or not, was called the ‘maximum learn- ing programme’. The students in this pro- Professor Ribet, congratulations on win- everyone up to the same level, but instead, gramme went from class to class together ning the Brouwer Medal! We would like to they would identify specially talented stu- (this was called ‘block programming’), we ask you some questions about your scien- dents and put them in exceptionally strong got to know each other very well and we tific career and endavours. had among the best teachers in the high school, so I had a very good preparation High school: the ‘maximum learning pro- in mathematics, science, English and so- gramme’ cial studies. Foreign languages were not block programmed because some people How was your high school education? You took French and some people took Span- went to the same high school as Feynman ish, but all of the core courses were block (and Madoff, for that matter): the no-longer- programmed.” existing Far Rockaway High School. “I grew up in a part of New York City that Then was it already clear before you en- is pretty far away from the centre. Students tered high school that you had a ‘science- in New York City who were very oriented talent’? towards mathematics or science tended “Well, the New York school system identi- to apply to specialised high schools, like fied me as a ‘smart person’, but some of the Bronx High School of Science, Stuyve- the students with me in the programme sant High School, or Brooklyn Tech. I didn’t were talented in English, social studies or want to spend hours every day on the bus history. It was not focussed specifically on or the subway, so there was one local high science.” school, and that was my high school. At that time, the educational system How was the mathematics education?

had not yet developed this idea that lots Gil Cavalcanti Photo: “It was great. First of all, I learnt a lot of resources should be spent on bringing Ken Ribet at the NMC 2017, holding the Brouwer Medal from the teachers, but we also had a math 116 NAW 5/18 nr. 2 juni 2017 Interview with Kenneth Ribet Gunther Cornelissen, Jan van Neerven

team and I learnt a lot from the other stu- programme for science students at Brown be I will major in chemistry.’ And over the dents. There is one fellow who went on University, which is actually the university I summer, when I learnt about epsilons and to do research mathematics, another be- ended up attending and I was attracted to deltas, I completely changed my mind and came a specialist in mathematical prob- the university because I enjoyed that sum- decided that mathematics was going to be lems and he published many books about mer so much. We had three courses. One my subject. In fact, by the time I got to that. We were always pushing each other in computer programming where I learnt university, I decided to abandon chemistry to learn clever techniques to solve ge- how to write programmes in original For- completely.” ometry and algebra problems. I learnt a tran (typed into punch cards); another in little bit of calculus from these other stu- calculus that was taught by an engineering Your story on epsilons and deltas is quite dents even before I had my first calculus professor, who had absolutely no interest interesting. What is your opinion on the course.” in limit arguments and mathematical sub- big calculus books that nowadays precede tleties, but he taught us how to compute a decent class in analysis? Did you also take part in competitions, and it was great! That’s how you should “I think the American system is fairly rea- such as Math Olympiads? learn calculus at first, through calculations, sonable, that students learn calculus and “Math competitions weren’t so well organ- and I really liked that. Then there was a then they take a separate course in real ized in those days, but we did a few, and third course in material science — which analysis and they learn what really goes I was OK. I was not a big star but I would coincidentally is the specialty of my older on in calculus. The reason the books are sometimes get an honourable mention.” daughter — but in that course, I learnt ab- so thick is simply that the publishers want solutely nothing. to be able to sell to the largest number It is quite surprising that these kind of spe- There was another thing that was some- of universities and each university has cial schools already existed in those days. what special. We had a kind of ‘math tal- some necessary conditions that have to Nowadays it is rather common to offer ent class’, an additional math class, when be satisfied by a calculus book, so it tries special programmes to talented students. I was in my last year of high school, where to satisfy all of them. So you get these “My high school had something like 3000– we did a lot of problems from a book big books, it is a big juggernaut, with sup- 4000 students, and this programme was from around 1891 called ‘Higher Alge- plementary material, online homework, just for a very small number, something bra’ by Hall and Knight.1 It was just full guide for the instructor, guide for the stu- like 50 students per year.” of tricky problems from the competitions dent, solution manual. All of this has to in Cambridge University and I learnt a lot be prepared, so the whole thing is a very What were the typical topics and what was of strange mathematics; algebraic manip- big project.” the level? ulations, discrete mathematics, all sorts “Just the standard high school topics, but of topics.” Undergraduate years at Brown University: during the summer of 1964, before my interacting with Ireland and Rosen last year in high school I was in a summer In hindsight, has this benefited you? “Oh, I am sure. I got very excited about Then you went to Brown University. How math, and by the summer after I left high was that? Short biography of Kenneth A. Ribet school, I really understood about epsilons “Oh, I loved it. Brown — where my younger Kenneth Alan Ribet was born on 28 June and deltas for the first time, I found it re- daughter is a student now — is the same 1948 and grew up in New York, where he ally great. So I was full of enthusiasm and now as it was back then: there is lots of attended a selective programme at Far ready for university. research, but there is a big focus on the Rockaway High School, before entering Godement used to say that in France, undergraduates; on giving them lots of Brown University to study mathematics. the students in the Lycée were just stuffed attention and lots of time and opportuni- He received his PhD from Harvard, di- full of information and by the time they got ties to learn different things. The profes- rected by John Tate, in 1973. After some to university, they knew quite a lot but they sors in the Brown Math Department liked postdoc years at Princeton, he moved were very tired. In the USA, students learn me and were willing to spend lots of time to Berkeley, where he has been a pro- much less, but when they get to college, with me. The first course I took was a year- fessor ever since. they are really ready to learn. I think that long course in linear algebra and multi- He published about 60 papers and had was true for me. I was very excited about variable calculus, but it was an honours around 25 graduate students, and was going to university, I was 17 years old and course, so the linear algebra was abstract an editor of many journals and book I was in great anticipation of what I would linear algebra, which I learnt for the first series, including the acclaimed Springer learn there from these professors, whom I time and which I thought was absolutely Graduate Texts series. considered to be humungous figures.” beautiful. He is a member of the U.S. National I also liked multivariable calculus. We Academy of Science, and has received But you enrolled in chemistry. had a book called Modern Multidimension- the Fermat Prize in 1989, and, this year, “That is not exactly correct. What hap- al Calculus by Monroe 2 and I never exactly the Brouwer Medal of the Royal Dutch pened is that I had a very strong high knew what was going on, but I liked the Mathematical Society. Since 2017, he school chemistry teacher. When I was grad- fact that there was actually a formal defini- serves as President of the American uating from high school I thought: ‘You tion of what a differential was, as a linear Mathematical Society. know, this chemistry is really great; may- form on a tangent space. The book was full Gunther Cornelissen, Jan van Neerven Interview with Kenneth Ribet NAW 5/18 nr. 2 juni 2017 117 Photo: Gil Cavalcanti Photo: Ken Ribet delivering the Brouwer lecture at NMC 2017 of Greek letters and the idea was that by volving cyclotomic fields. This was my first So you are not like Andrew Wiles, who in thinking abstractly, you would gain more encounter with Kenneth Ireland. He turned the famous BBC-documentary 5 states that information. But I am not sure that really to me and said ‘Read this’ and hand- even as little kid he already came across worked.” ed me André Weil’s paper on number of the Fermat Conjecture and he wanted to solutions of equations over finite fields,3 solve that. Does the interest of Brown faculty on un- which was basically where he made the “I was not motivated by and had absolute- dergraduates mean more individual tutor- Weil conjectures. Except he was unwilling ly no ambition to solve a great problem ing, or group tutorials? to say it in print, because he was wor- or to be a great mathematician, but my “It just means more availability and less ried that he might be wrong. He wrote this ambition, locally, was to be like the pro- rushing to the next seminar or conference.” article and went around lecturing about fessors that I had in math classes. I real- it, and in public, orally, he would explain ly liked them, I respected them, I thought What was your first encounter with num- what he really had in mind, but not in the they were doing something worthy. This ber theory? article. The article is completely explicit, was the time of the Vietnam war, where a “In my second year I took my first ab- calculating numbers, using Gauss sums lot of people that I knew in business, like stract algebra course, which I also loved. and Jacobi sums. The article expresses my parents’ friends, were doing things that By chance, just by walking in the math each idea in several different ways. I could were tangentially related to the war effort department, in the mail room I met a pro- follow it all line by line. It seemed pret- and they were seeking profit. And these fessor who started interrogating me. ‘What ty interesting and I liked the fact that people at university were more like monks, are you studying, what are you doing, this Brown professor was interested in very serene and motivated by the love of what did you just learn?’ I said I was in me. Then he gave me another paper, the knowledge. They seemed to me to be real- this abstract algebra course, so he asked: sequel, called ‘Jacobi sums as Grössen­ ly people to emulate.” ‘What is an example of a cyclic Galois charaktere’ by Weil.4 I read it and sort group?’ So I said, you take an extension of understood it. But then there was all Was this in particular true for the mathe- of finite fields, but he said: ‘No, no, I want this number theory, and I took a course in matics professors, or was it more general? number fields, characteristic zero.’ I think number theory and that got me interested “Well, I liked the academic atmosphere in I might have come up with an example in- in arithmetic.” general, but the math professors were the 118 NAW 5/18 nr. 2 juni 2017 Interview with Kenneth Ribet Gunther Cornelissen, Jan van Neerven

ones that I saw and they were very much anti-authority sentiment. The exams were He told me to continue on, and I went unlike the adults I saw when I grew up, two sessions of three hours each, and I to the Antwerp conference in 1972 where I who were in business. My first professor, came into the exams, dashed off answers met Serre.8 I told Serre what I was working Frank Moore Stewart, who taught me linear on paper and left after half an hour; two on and what I had done so far. But as you algebra and multivariable calculus, looked times. It is clear that my answers were can see from the Serre–Tate correspond- like some farmer, but he was in pursuit of lacking in detail. Apparently, the Harvard ence, Serre already knew about me; Tate knowledge and he had spent a great deal faculty wanted to fail me, because grading had written to him. ‘This kid Ribet is com- of time with one of the historically black by the strict criteria, I didn’t have a passing ing to Antwerp; be nice to him, he is very colleges in the South. Brown University had score, but Tate intervened and said that shy.’ When I met Serre he was standing on a partnership with Tougaloo College in Mis- this guy is really good and you can see his head, demonstrating J. Frank Adams’s sissippi, which I had never heard of before from his answers that he knows what is method of drinking beer while standing on I went to Brown. There was an exchange going on, so we’ll pass him. your head. He had a big crowd of admir- program, where a few Brown students who I spent the next two years taking ers around him and I didn’t know when I go to Mississippi and a few Mississippi stu- all kinds of courses, from Tate and John could possibly ask him about my thesis, so dents would go to Brown, and some faculty Coates. It seems there was an endless I decided to just interrupt him with: ‘Excuse members would go to the other university. supply of visitors: Swinnerton-Dyer gave me, professor Serre, I want to ask you some This guy had spent some time teaching at a course on modular forms, and Birch questions about abelian varieties.’ Every- Mississippi and he was deeply committed would come sometimes, and Serre must body who was there started laughing. But to equality and pacifism. I was very im- have been there, although I don’t remem- Serre asked me who I was, and when he pressed and I had never met anyone like ber that. I learnt a lot. Finally, I came into remembered who I was, he started talking that before coming to university.” Tate’s office in my third year, and I asked to me. He told me how to continue my the- when to start research; I had no idea how sis, what to do, and how it would work out. Graduate years at Harvard: interacting with to do it. Tate said ‘What are you doing?’ I actually found this very depressing, I was Tate and Serre; the Antwerp conference and I said I was in this course, and this working on something, working, working, course, et cetera, and he said: ‘Maybe you working, and Serre, in thirty seconds could Then you went to Harvard. should go to fewer courses and work on see the entire sweep of what I was doing “The two professors that mentored me, this problem.’ He gave me some problem and tell me what to do next — literally while Michael Rosen and Kenneth Ireland,6 just that involved the Birch–Swinnerton-Dyer standing on his head. It was in some sense told me what to do. ‘This is your subject. conjecture and elliptic curves with com- very discouraging. What am I ever going to You are going to study arithmetic geom- plex multiplication. I couldn’t figure out do in mathematics and will anybody care etry. Apply to these universities: Harvard, what he had in mind, it just didn’t make about me when there are these other peo- Princeton, Brandeis and Chicago.’ So I ap- any sense to me. I came back to him and ple who can do the thing ten times better plied to these four universities, and I guess said that I didn’t like this problem. You im- and thirty times faster.” they planned to write me very good letters agine most advisors are barely willing to of recommendation, and I got into all four. give a student one problem, but Tate just Postdoc at Princeton, interacting with Katz I reported back that I got these four enve- gave me another problem. He and I were and work with Deligne lopes back and they are offering me admis- reading a paper of Serre on the Galois ac- sion and they said to me: ‘Go to Harvard tion on division points of elliptic curves. “Nonetheless, I continued writing my the- and work with Tate’, and that was it. So I We had a seminar where we were working sis, handed it in, and graduated.9 During said: ‘OK.’ through the manuscript line by line and the year when I wrote my thesis, which was I showed up at Harvard in September taking turns giving lectures. I understood my fourth year at Harvard, I really had the 1969 and I knocked on John Tate’s door the paper pretty well, so Tate said: ‘Why impression that no one would ever care and I said: ‘I am Ken Ribet, your new stu- don’t you do certain abelian varieties?’ about what I was writing and that I would dent.’ He seemed rather taken aback by (later called ‘of GL2-type’), as an encore. get a job in some really obscure university, this, but he never said anything negative, I said that sounds like a good thing and not in a major centre. When I applied for he said: ‘OK.’ ” I started working and learnt something by jobs, I wrote to about sixty universities, mimicking some arguments in Serre’s book hoping that one of them would pay some Had you already done your qualifying ex- on Abelian ,-adic representations and ellip- attention to me. It was really quite a shock ams? tic curves.7 I thought I was doing a com- that in January of the year I was writing my “Oh, that was a real embarrassment. Name- putation, and I came to Tate and told him thesis, very early in the hiring cycle, I got ly, at the end of my first year at Harvard, what I had computed and he said: ‘Well, a phone call from Princeton offering me a there were written exams everyone had to that’s a very nice theorem.’ No one had job. It was the craziest thing that ever hap- take and this was at a time when the whole ever accused me of proving a theorem be- pened to me. Like everybody who came university was barely hanging together. fore. It was a transformative moment when to Princeton as a postdoc — at the time There were people on strike, the USA was I actually had been told that I had done my title was ‘Lecturer’, the phrase ‘Post- bombing Cambodia, people demonstrating some research, because I had no idea what doc’ was not used yet in mathematics — I in the streets and classes being closed research was or how you would go about was terrified that the senior faculty would down. There was a big anti-war effort and doing it, so it was very empowering. find out that they hired someone who was Gunther Cornelissen, Jan van Neerven Interview with Kenneth Ribet NAW 5/18 nr. 2 juni 2017 119

completely stupid. Everybody had that divisible by 2n and a whole class where he and then that would get you started. There fear. I showed up and my mentor was Nick couldn’t decide. He did one calculation for was also a lot of drinking in the evening Katz, who was wonderful and very sup- one totally real quadratic field, where he after dinner, with beer and wine and ping- portive. He asked me some question that saw it wasn’t divisible by 2n (by 4 in that pong. The food and wine at that confer- I was able to answer during my first year, case) and he wondered what happened in ence were just excellent.” and this is how I got into this collaboration general. I did a few more examples and with Deligne about totally real fields.” gave a lecture about it at Orsay. Hendrik This gregariousness seems very much in Lenstra immediately saw what the answer line with the reputation of the algebra What was it like to work with Deligne? was supposed to be. Getting that hint from group at Antwerp. “I did not really work with Deligne. Katz Lenstra, I worked and worked and worked “At the time I was still a graduate student. told me you had to prove the irreducibility and compared Eisenstein series and theta What I also remember from the conference of some moduli space in characteristic p series and actually proved that the surmise was that after every talk, students would and Katz had already found in his Antwerp of Lenstra was correct. So my second sig- gather around Nick Katz in front of the article that this amounted to the surjec- nificant contribution to that joint paper building. He would explain to them what tivity of some Galois representation, which had to do with powers of 2 in the 2-adic the lecture was really about. The most el- in the case of elliptic curves was proved L-function. So I did two different things for ementary lectures I would understand but by looking at the image of some inertia that article.” the more advanced things required some group around supersingular points; a meth- help. Listening to Katz was just great.” od that did not work for higher-dimension- Antwerp again, and Grothendieck al abelian varieties. Katz just asked me a Was this the conference where Grothendieck very narrow question about some Galois The Antwerp conference was organ- refused to attend because of the confer- representation, where he translated all the ised by Willem Kuyk, the mathematical ence being funded by NATO? difficulties into one place where I was sup- great-grandfather of one of us, a Dutch “Somehow I managed to miss the first posed to be knowledgeable, and asked: mathematician that had gone to Belgium day of the conference by mis-timing some ‘How do you prove it has a big image?’ I to found the mathematics department trains. I came to Europe a week or two said, the way we usually do this is using at Antwerp. On the web page of William before the conference and travelled with Frobenius elements, and I made some ar- Stein, we found an interesting conference some friends, and the morning the confer- gument using Honda-Tate theory showing picture 10, in which you identified many ence began I was in Switzerland. that the Frobenius elements fill the thing participants and to which you contributed At the first day proceedings, Grothen­ up, so I solved the problem and did it liter- a little note about the alcohol consump- dieck was making a big stink, protesting ally overnight. Katz said ‘Holy cow, this is tion at that conference. and interrupting. The second day, when the missing piece!’ and asked me to write “There was wine served at lunch and din- I came, Grothendieck was sitting in front to Deligne, who replied that now we could ner. I often sat with Bombieri who used the of the building where the lectures were write a joint paper. He already had a man- following ploy: as we sat down, we would held, with a huge bouquet of helium filled uscript, a letter to Serre, outlining what to pour out the first bottle; so we had an balloons. I knew who Grothendieck was do, but with this missing piece. Working empty bottle and we got a second bottle, because he came to Harvard when I was with Deligne meant that I had to under- stand this manuscript. So I went to Paris and spent the first year there (1975–1976) at the IHES and mostly during that year I tried to understand the manuscript. I saw Deligne every day but I didn’t talk to him all that often, since I didn’t have all that much to say. My main contribution to that paper was the irreducibility in character- istic p, and then there was another point, which had to do with powers of 2. The idea was that the p-adic L-function was integral (a p-adic power series), except that when p = 2 and you look at the Eisenstein series, the constant term is an L-value divided by 2n where n is the degree. It would be very natural to think that L-values for the 2-adic L-function were divisible by 2n, but after appropriate symmetrisation, Deligne could prove in complete generality only that they were divisible by n - 1 and had some gen-

2 http://wstein.org/antwerp_photo Photo: eral situations where he knew they were Conference photo from the Antwerp conference, 1972 120 NAW 5/18 nr. 2 juni 2017 Interview with Kenneth Ribet Gunther Cornelissen, Jan van Neerven

a graduate student, but I don’t think he and the Bulletin. I could take them home Three theorems of Ken Ribet recognised me. So the only direct conver- and read them. Nowadays the University is A central role in Ribet’s work is played sation I ever had with Grothendieck was subscribed to these publications as insti- by so-called ‘Galois representations’, a that I went up to him and asked him: ‘Ex- tutional members. That means that when way to look at extensions of number cuse me, why are you sitting here with bal- you are at the university, you can read all fields through the eyes of representa- loons?’ He looked at me and said: ‘Because these publications on your laptop or tab- tion theorists, connecting them to al- they are nice.’ That was the only sentence let. A benefit of electronic copies is that gebraic geometry through automorphic he ever spoke directly to me. you can do searches. Young people are forms and Shimura varieties. No, I take it back. He spoke other sen- less and less married to physical copies. Ribet rose to fame when he published tences to me, because when I was at Har- You can decide not to be a member and what is now known as the ‘Herbrand–Ri- vard, he was already on his ‘Survival’ kick. still have the electronic versions available bet’ theorem 11, a result on class groups When visiting universities, he would lec- through the university. So basically the of cyclotomic fields that was spectac- ture in mathematics only if he was allowed problem which the AMS is attempting to ular not just as a statement, but also to give a general lecture about the evils counter is the attitude that young people from the point of view of the methods of doing mathematics. I went to both his might have of ‘what’s in it for me?’ When I that he used; a whole new generation algebraic geometry lecture, which was on was a young faculty member I joined the of number theorists copied successful- the definition of étale cohomology, and in society because everyone did, it was the ly his algebro-geometric ideas to solve the evening to his public lecture, as did thing to do. When I came to Berkeley I also purely number theoretical problems, all math graduate students, and he was joined the Faculty Club. Once you pay dues and the paper made a lasting impact on going on that we should go till the soil to become a member of the club, you get a the field of Iwasawa theory. and not waste our time doing mathemat- discount on you purchases. But many will Ribet proved the f-conjecture 12, effec- ics. I asked: ‘Excuse me, I have spent three say that it is not worth it. Mathematicians tively implying that, through a method years now learning higher mathematics, are very good at calculating the benefits!” of Frey, proving the Taniyama–Shimura including arithmetical algebraic geometry. conjecture would solve Fermat’s Last I think I am pretty good at it and that I What are the main challenges that the Theorem (which Andrew Wiles subse- would be able to make a contribution, AMS is facing? quently did). whereas if I go and till the soil as a gar- “Many challenges have been identified in In a famous work with Deligne 13, he dener, I won’t be any better than any other the strategic plan, but an obvious one is showed congruences between special person, probably a lot worse. What do you to make sure that its electronic products values of L-series of real quadratic say to that?’ He gave an answer, but speak- are still relevant in an age when Goog- fields (known to be rational by work of ing to the public. I don’t think his answer le does everything for free. For instance Siegel), implying the existence of p-adic was all that satisfactory.” MathSciNet, the online review service for L-functions, which seemed very inac- mathematical research papers, is constant- cessible at that time. Again, the method Reaching out to young people ly adding more capabilities. Certainly it is of Deligne and Ribet changed the land- very important to the Society to make sure scape on how to approach such ques- The NAW would like to reach more young that MathSciNet stays ahead. People often tions: suddenly, analytic statements people. As President of the AMS, which don’t realise everything the Society is of- were approached through modern al- suggestions do you have? fering, such as the remote access button gebraic geometry and a very detailed “The new editor of the Notices of the in MathSciNet which is unknown to most study of moduli spaces. AMS, Frank Morgan, has devoted a tremen- users. We do regular surveys among our dous amount of energy to the journal. He members who then request all sorts of fea- has worked to make the articles shorter tures that we already have.” the other, and many people think that the and more visually appealing. In particu- Societies are a good compromise: they lar, articles tend to have short bibliogra- You have said elsewhere that the impor- are non-profit organisations and they are phies — or none at all. He has worked to tance of printed journals is declining. really serving the mathematical communi- move routine material out of the Notices “Fields Medallist Timothy Gowers will tell ty. The AMS has a Gold Open Access ver- on the grounds that it can be found eas- you that the future of publishing in jour- sion of its Transactions and Proceedings ily online. Frank has edited many of the nals is overlay journals: you publish on journals and charges $ 2000 per article. submissions quite heavily; the practice of arXiv and all the journal does is to anoint That is what it actually costs to take an editing submissions is common in the lit- certain of these articles. That’s it, very lit- article and process it. There are all sorts erary world but rare in the mathematical tle work needs to be done. An argument of arguments at to whether or not these community. can be made that commercial publishers, are real costs. The AMS makes money on When I was a student, graduate stu- including one in this country, are exploiting some of its publishing activities, but this dents would be members for free; grad- the mathematical community by extracting money is turned back to the mathemati- uate schools would nominate them. Then a huge profit from a product that is ba- cal community through the many servic- it was very natural to continue once they sically powered by free labour from the es that the AMS provides to the profes- had a real job. The real benefit was that I mathematical community. Overlay journals sion. At the end of every year the balance would get my own copies of the Notices are one extreme, commercial publishers should be even.” Gunther Cornelissen, Jan van Neerven Interview with Kenneth Ribet NAW 5/18 nr. 2 juni 2017 121

New developments in number theory cohomology classes are just mind-blowing. During the interview, we asked Ken Ri- Certainly, if I were starting out and could sit bet to describe how the AMS reaches out Let’s talk about recent developments in for two years and think about nothing but to young people (a continuing point of in- the field. Is there something that gets you learning a new subject, that would have terest, also for KWG and NAW), and this is really excited? If you would have two years to be it.” the answer that the executive director of to study only something mathematical, the AMS, Dr. Catherine Roberts, later pro- what would it be? Then being president of the AMS is not the vided by email: “Well, it would clearly be perfectoid spac- best strategy, is it? Do you still have time es.” 14 for research? “There are many ways that the American “You know, I have done lots of different Mathematical Society supports young peo- I actually wrote that down as next subject! things in my life and the AMS is very in- ple interested in Mathematics. Frank Mor- Two of our Utrecht students attended the teresting, learning about the society and gan has devoted a tremendous amount Arizona Winter School about that, I guess anticipating in its activities and this is of energy in transforming the Notices of you were not there? some way in which I grow, even if it is the AMS. I think his attention to short, in- “No, but their inventor Peter Scholze was not learning perfectoid spaces. One of the teresting articles with lots of illustrations in Berkeley for a semester fairly recent- things with being president is that I teach and his addition of ‘Lecture Samplers’ and ly and he gave a course. I have tried to only half as much, and even when I am the ‘Graduate Student Section’ make this follow the beginning of the course and I teaching and I have to go away for an AMS publication, freely available online, more was just completely blown away by (1) the activity, a graduate student is there ready interesting and accessible to students. beauty of the subject and (2) the amazing to replace me; this is part of the agreement Our members have established the Epsi- command of an entire landscape that he between my university and the AMS.” lon Fund, which supports many math pro- has. He was lecturing to many people who grams and summer math camps for high were experts in different corners, like (,z C) Chances are that you will meet the Presi- school students. We provide travel funds -modules and different ways of express- dent of the United States? to students interested in attending our ing Galois representations in somewhat “Every year there is a ceremony for the Na- conferences. Last year, at the Joint Math- concrete terms and Scholze understands tional Medals of Science. Presumably I get ematical Meeting, we ran Mathemati-Con all of this in his new framework in some invited if a mathematician wins. If I would jointly with the Mathematical Association illuminating way that completely surprised find myself in a room with Donald Trump, I of America. This event was free and open the original perpetrators of the subject, would try to have an affable conversation to families and celebrated mathematics who were sitting in the audience shaking without much substance. I would tell him with engaging presentations as well as their heads. Obviously, as you see from that I admire his Twitter style and would the national finals of our popular quiz pro- the joint work of Scholze and Jared Wein- ask him if he is having a good time. I prob- gram, Who Wants to be a Mathematician? stein,15 when you take modular curves and ably wouldn’t have more than three min- These are just a few of the programs I can you add lots of p-power level, in the end utes to talk to him!” think of that might be of interest to young you are dealing with a perfectoid space, people, and I encourage those interested and these recent developments where you Professor Ribet, thank you very much for in mathematics to explore our website associate Galois representations to mod-p the interview! at www.ams.org.” s

Notes and references 1 H. S. Hall and S. R. Knight, Higher Alge- 7 J.-P. Serre, Abelian l-adic representa- 11 K. A. Ribet, A modular construction of bra 1887, Macmillan and Co. (Free copy tions and elliptic curves, McGill Uni- unramified p-extensions of Q()np , In- easily found online.) versity lecture notes written with the vent. Math. 34(3) (1976), 151–162. 2 M. Monroe, Modern Multidimensional collaboration of Willem Kuyk and John 12 K. A. Ribet, On modular representations Calculus, Wiley, 1963. Labute, W. A. Benjamin, 1968. of Gal(QQr /) arising from modular forms, 3 A. Weil, Numbers of solutions of equa- 8 Modular functions of one variable. I– Invent. Math. 100(2) (1990), 431–476. tions in finite fields, Bull. A. M. S. 55(5) IV, Proceedings of the International 13 P. Deligne, Pierre and K. A. Ribet, Val- (1949), 497–508. Summer School, University of Antwerp, ues of abelian L-functions at negative RUCA, July 17 – August 3, 1972, edit- 4 A. Weil, Jacobi sums as ‘Grössencharak- integers over totally real fields, Invent. ed by Willem Kuyk, Lecture Notes in tere’, Trans. A. M. S. 3 (1952), 487–495. Math. 59(3) (1980), 227–286. Mathematics, Vol. 320, 349, 350, 476, 14 Introduced by Peter Scholze. See, for 5 Fermat’s Last Theorem, BBC Horizon Springer, 1973–1975. Documentary by Simon Singh, 1995– example: B. Bhatt, What is ... a perfec- 9 K. A. Ribet, Galois Action on Divi- 1996; also a book by Simon Singh first toid space? Notices of the A. M. S. 61(9) sion-points of Abelian Varieties with published by Fourth Estate. (2014), 1082–1084. Many Real Multiplications, PhD thesis, 15 P. Scholze and J. Weinstein, Moduli of 6 Known, amongst others, for their book Harvard, 1973. A Classical Introduction to Modern p-divisible groups, Cambridge J. Math. 10 http://wstein.org/antwerp_photo. Number Theory, Springer GTM 84. 1(2) (2013), 145–237.