A Conversation with Manjul Bhargava Stephen Abbott n my way to the Math- Fest 2011 meeting in Lexington, Kentucky, OI had the pleasure of meeting Manjul Bhargava, this year’s invited Hedrick lecturer, on my shuttle bus from the airport. Although I knew something of his reputation, I did a quick Internet search at my hotel when I arrived and was inundated with anecdotes about the extraordinary talents of this young mathematician. A math prodigy, he graduated sec- ond in his class from Harvard, received his doctorate at Princeton working under Andrew Wiles, and was invited to return to Princeton two years later as a full professor. His dissertation extended, in a dramatic and surprising way, some classical results of Gauss © Mariana Cook 2007, www.marianacook.com on quadratic forms, and he has since gone on to do groundbreaking work in of when Manjul was scheduled to speak. ence center, and at the conclusion of algebraic number theory and combina- The Earle Raymond Hedrick Lecture Manjul’s third lecture, his now-expanded torics. In 2005, he won a Clay Research Series has been a fixture of the MAA’s audience of several hundred erupted Award, and in 2008 he was awarded summer meeting. The list of speakers with applause. For the moment, we were the Cole Prize. over the decades reads like a Who’s all experts in elliptic curves—or at least Just last year, he announced a proof Who in contemporary mathematics— Manjul had made us feel that way. [See of a special case of the Birch and Persi Diaconis, William Thurston, John inset article, next page.] Swinnerton-Dyer conjecture, one of Conway, Timothy Gowers. The invited In the audience, several thoughts the seven Millennium Prize Problems. speaker gives a series of three lectures went through my mind. The first was (These are deemed so important and during the meeting. The first lecture simply being impressed that someone so challenging that a complete proof is always quite popular, but the typical who spends the bulk of his time at the would earn its author $1 million from pattern is for attendance to wane as the frontiers of the hardest problems in the Clay Mathematics Institute.) week goes on and as the complexity of mathematics could so gracefully adapt This dazzling list of accolades formed the ideas goes up. Not so this year. his thinking to the diverse viewpoints a curious contrast with the soft-spoken Manjul’s finely tuned teaching instincts of this crowd of nonspecialists. But and humble friend I had just made on were on display from the beginning, and I was also struck by the warmth and the shuttle bus. There was no hint of the positive buzz from the first lecture generosity of our speaker. Our ques- arrogance, no aloofness one might un- quickly went viral. By day two, napkins tions mattered, our understanding derstandably expect from someone who with scribbled diagrams of rational points mattered, we mattered. All week Manjul does mathematics at such an elite level. on algebraic graphs started appearing in had been ubiquitous at the gamut of I was intrigued and made a careful note the various restaurants near the confer- MathFest events—he was even spotted

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This content downloaded from 65.206.22.38 on Tue, 12 Aug 2014 15:56:23 PM All use subject to JSTOR Terms and Conditions A Quick Tour of Manjul Bhargava’s 2011 Hedrick Lectures on Elliptic Curves Nathan Carter and Eleanor Farrington oesn’t it seem strange that Fermat’s Last Theorem, one of the most famous ques- tions of number theory, was proven using elliptic curves? Why is a theorem about discrete things, like integers, answered Dusing smooth things, like curves? The answer is that the geometry of a curve says a lot about the rational (or integral) points on that curve. For example, in the story “Harvey Plotter and the Circle of Ir- rationality” [see page 10], the characters use the unit circle (a curve) to generate Pythagorean triples (of integers). The technique they use applies just as well to any second- degree polynomial equation in two variables, which is called a conic. Beginning with one rational point on the conic with rational coefficients, consider each line of rational slope through it, and find the other point of Figure 1. How to add two rational points on an elliptic curve. intersection between the line and the conic. This process gives us all the rational points on the conic. Thus, any (Try it!) Degree-two algebraic curves were just dis- conic containing at least one rational point contains in- cussed. For higher degrees, our degree-two method does finitely many. That means there are two possibilities for not apply because a line may intersect such a curve at conics: zero rational points or infinitely many. more than two points. We need a new approach. Algebraic curves with rational coefficients are more Graphing the algebraic curve over (rather than general; they are arbitrary-degree polynomial equations ) gives an object in four dimensions, called a Riemann in two variables. Consider first only degree-one algebra- surface. Faltings’s theorem is the surprising fact that the ic curves; i.e., straight lines. You can easily find topology of this surface (specifically its genus) in tells a method yielding all rational points on such a curve. us something about rational solutions in . It says that

at the business meeting. Watching him Math Horizons: Congratulations on try different things, and see what converse with the posse of students an extraordinary series of lectures works. gathered round him in the wake of his this week—the response has been MH: When did this passion start final talk, it became clear to me that this overwhelming. for you? Did you teach in graduate wasn’t a dignitary paying a cordial visit Manjul Bhargava: Thank you. school? to the MAA community, but a member MH: Tell me about your interest in MB: I never taught in graduate of the community intent on engaging it teaching. Teaching is usually consid- school, but I taught a lot as an un- at every level. ered a gift—something you have or dergraduate. I approached him after the lecture. “If you don’t have. MH: Really? you have time later, I would love to do MB: I think teaching is also some- MB: Yeah, that happens a lot. At an informal interview with you for Math thing you can learn. Harvard, the undergraduates are al- Horizons,” I asked, already feeling guilty MH: But your enjoyment of it seems lowed to teach. about adding to Manjul’s busy schedule. very sincere, very much a natural MH: What sorts of things? Despite having a host of valid excuses part of who you are. MB: I was a TA for first-year calculus for politely declining, Manjul agreed and MB: Sure, to do anything well, and the algebra sequence—things was waiting for me in the hotel lobby you have to enjoy it. You need a like that. That’s when I realized I that afternoon. passion to want to get better—to loved teaching. It was my favorite

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This content downloaded from 65.206.22.38 on Tue, 12 Aug 2014 15:56:23 PM All use subject to JSTOR Terms and Conditions any generic algebraic curve (with rational coefficients) braic structure on elliptic curves. We define addition of of degree greater than three contains only finitely many two points on an elliptic curve as follows: Draw the line rational points! But it does not say how to find them. between them as before, find the third point of intersec- So, cubic (degree-three) algebraic curves are special; tion with the curve, and take its x-axis reflection. (See only they exhibit all three possibilities—zero rational figure 1.) This operation satisfies all the properties of an points, finitely many, or infinitely many. Therefore, cubics Abelian group, with the zero (or identity) being a “point are heavily studied, particularly ones with no self-intersec- at infinity,” touched by all vertical lines. The points not tions or spikes; these are called elliptic curves and can be counted in the rank are those with finite order in this written in the canonical form . Despite Abelian group (called torsion points). (For an exercise, this attention, there is no known method for determin- try figuring out what the inverse of a point is.) ing which of the above three possibilities a given cubic A weaker question than the Birch and Swinnerton- falls into. The Birch and Swinnerton-Dyer conjecture is Dyer conjecture asks how many elliptic curves have a famous open problem that proposes such a method. In infinitely many rational points. Goldfeld’s conjecture 2000, the Clay Mathematics Institute listed it as one of states that the average rank of an elliptic curve is ½ and, seven Millennium Prize Problems; a proof is awarded a furthermore, that an elliptic curve chosen at random has million dollars! probability ½ of having rank 0, probability ½ of having We can still use a variant of the degree-two method for rank 1, and a negligible probability of having rank 2 or elliptic curves. Pick at least two rational points on the higher. And that’s where things really get interesting. To curve, and connect them with a line. It will have rational test Goldfeld’s conjecture, people have gathered a lot of slope and thus intersect the curve in exactly one other data on the ranks of elliptic curves and always found an rational point. Reflecting this point about the x-axis gives average rank larger than ½! In fact, no one was able to another new rational point (note the in the canonical say for sure that the average rank was even finite! Until form). Applying this technique again and again, con- last year. necting old points to new ones, generates more and more Manjul Bhargava and his Ph.D. student Arul Shankar rational points. Mordell’s theorem states that for every showed not only that the average rank of elliptic curves elliptic curve, there exists a finite initial set of points is finite, but also that it’s less than one. What’s more, from which this technique will find all rational points on they proved that at least 10 percent of all elliptic curves the curve! The size of that initial set is called the rank of have no rational points (except for the identity point the curve (discounting a few points we’ll explain below). at infinity). That means at least 10 percent of elliptic Many unsolved questions about rank are being actively curves satisfy the Birch and Swinnerton-Dyer conjecture. studied today, including finding an algorithm that gives Did this incredible accomplishment earn Manjul and that initial finite set. Arul 10 percent of the million-dollar prize? Sadly no, but This process gives us a way to reveal an important alge- perhaps they are on their way. Q extracurricular activity. remember I’ve loved math. something. My mother was pretty MH: It was the same for me, actu- MH: What was it that you loved so busy; she was not usually teaching ally, although I was just a tutor. much? me directly, but she was always there MB: Of course, my mom is a teacher, MB: I just liked playing around; try- as a resource. And she let me skip and I used to go and attend her ing to understand why something my school to go to hers. classes. was happening. One of my earliest MH: Number theory, with its very MH: She’s a mathematics professor memories was stacking oranges in a primitive questions, is a common at Hofstra [University], right? She let pyramid and trying to figure out that introduction to mathematics—but you do this? if you had n oranges on a side, how you have stayed with it. MB: All the time when I was younger many did you need to make the whole MB: Hmm. This I would say, you know, I don’t want pyramid? That was one of the first thing is a number theory problem—I to go to school today, can I just problems I solved: n times (n +1) just liked these kinds of questions come with you. She was very cool times (n +2) over 6. [laughs] It took from the beginning. about that. me many months to figure it out. MH: That’s never gone away? MH: What part did this play in your MH: Do you remember how old you MB: [shrugs] That’s never gone away. decision to become a mathematician? were? MH: Do you remember where you MB: Actually, as far back as I can MB: I must’ve been around eight or were when Andrew Wiles announced

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This content downloaded from 65.206.22.38 on Tue, 12 Aug 2014 15:56:23 PM All use subject to JSTOR Terms and Conditions his proof of Fermat’s Last Theorem? MB: That was late in college for me. The math students were very excited—emails were flying around. MH: And then you ended up work- ing with Wiles. That must’ve been exciting. MB: It was wonderful. He has such a great sense of the global picture of number theory. Even though I worked in an area that was different from what he does, he stayed inter- ested in my work and was able to say when an idea represented a promis- ing direction. MH: Were you star struck at all when you first met him? Laura McHugh (MAA) MB: I think I met him at the Joint Manjul Bhargava, center, talks with members of the audience after one of his lectures Meetings. I had joined Princeton at MathFest 2011. as a graduate student, but I hadn’t UMP: Keep up the good work. certain passion and interest coming met him yet. [dawning on him] He MB: Thank you. What’s your name? from the teacher about the particu- was getting the Cole Prize and I was UMP: My name is Joe. I’ve got a lar subject being taught. It’s not so getting the Morgan Prize, and so we master’s in statistics, and I teach much which mathematics is included were on stage together. [laughs] high school mathematics. I like to as much as what can the teacher be MH: And you leaned over and said, poke around at the meetings, I saw really enthusiastic about, what topics “Hey, I’m Manjul . . . ”? your [talk], and I really like the way have they thought about from every MB: Hey, I’m at Princeton too! The you broke it down. Keep doing that. possible angle. next time I met him—or the next MB: Okay, thanks. MH: So what do you like to do? time we really talked—was my oral UMP: The best mathematicians can MB: I start with music. generals exam. break it down for the lowest common MH: Really? MH: Was that scary? denominator, and I like to say that MB: The first third of the course is MB: Oh yeah. Three professors— I’m the lowest common denominator mathematics and music—the math- in the room. Bye-bye now. Conway was the chair, Wiles, and ematics of rhythm, the mathematics [He leaves.] [Charles] Fefferman—all sitting in of pitch. I’m a tabla player . . . MH: So, tell me about your liberal the audience while I am at the chalk- MH: I’ve heard—word is that you are arts math course that Joe Gallian board answering questions. very accomplished. mentioned in his introduction. Is this MH: Ouch. That’s what you get for MB: A lot of the examples I use come something you currently teach at going to Princeton. from ancient India—mathematics Princeton? that was written by percussionists [At this moment, an unknown Math- MB: Yes, the first time I had about Fest participant (UMP) walks up to 100 students—the next year demand and poets of ancient times. Much of our table and addresses Manjul, who nearly tripled. this is not really known by mathema- is unfazed and perfectly cordial.] MH: And how is the course struc- ticians; in fact, many of the people UMP: Can I interrupt? To the tured? Do you use a particular text? who sit in on my class are mathema- youngest full professor at Princeton, MB: I don’t use a text. The inspi- ticians and math majors who aren’t I just want to say I found your talk ration came from a similar course actually allowed to take it. enjoyable and understandable. So developed by Dick Gross and Joe MH: You make a really good point. many times these Ph.D.s just want to Harris at Harvard, and they wrote A lot of these courses for nonmajors talk up here [gestures above his head a text that I considered using, but cover fascinating topics . . . with his hand ], but you put it right the thing that makes these courses MB: Right, that our regular students here—and I really appreciated it. is that they have to be individual never get a chance to learn! There is MB: Well, I appreciate it. to the instructor. There has to be a also a lot of history of mathematics

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This content downloaded from 65.206.22.38 on Tue, 12 Aug 2014 15:56:23 PM All use subject to JSTOR Terms and Conditions that is not known to mathemati- Grothendieck was trying to do. cians, because, for instance, it was “When I was learning MH: Tell me more. written in Sanskrit. Sanskrit poetry, I MB: A lot of the work of [Alexander] MH: Which you also know something Grothendieck was taking all of the about. had no idea I was geometric intuition that people had MB: My grandfather was a scholar of encountering collected over the years, putting it on Sanskrit. One way I got interested in mainstream a very solid algebraic foundation, and mathematics when I was young was making an equivalence between cer- through what he taught me about mathematics.” tain geometric language and algebraic the rhythms of Sanskrit poetry. language—making that connection so MB: Well, I just meant in that par- MH: Does that end up being number strong that in the future people could ticular area. theory too? just think geometrically and know MH: Of course, but I wanted to be MB: Number theory and combinator- that that geometry had an algebraic provocative and quote you out of ics. So here was my grandfather, a translation. Nowadays, algebraic ge- context so that I could ask whether Sanskrit scholar, teaching me all this ometers can just proceed geometrically you think geometry—in modern amazing combinatorics, and then and be totally confident that what mathematics—is being shortchanged they are visualizing really is proved. when I went to study mathematics I in some way. thought, gee, I’ve already seen these That’s one way of thinking about MB: What do you mean? ideas, from Sanskrit. That connec- Grothendieck’s contributions—making MH: The human brain, it seems to tion was amazing to me, and I like geometric intuition broadly usable, in me, thinks geometrically much more a rigorous way. to talk about those experiences from naturally than it thinks algebraically. my life in the courses I teach. MH: So that it becomes a properly In your lecture today, when you were justified way of doing mathematics. MH: Do you play the tabla in class? explaining the group structure on the MB: Exactly. MB: Sure, and I often bring guest rational points along elliptic curves, I know we are talking about musicians so we can show some of MH: you explained it by drawing chords research mathematics here, but the ways these rhythmic techniques along the graphs. watching my kids’ journey through are actually implemented in practice. MB: Yes, it’s a geometric construction. secondary school, I would say that MH: That sounds fantastic. Do you But then you have to ask what kind of geometry is buried amid a lot of play a lot outside of class? structure do you get, and that’s where symbolic manipulation techniques. MB: I try to—it gets harder every you have to do the algebra. I think that makes mathematics year. The mathematics keeps me MH: Your point is well taken, but harder than it needs to be. busier and busier. there are some subjects like, say, MB: You’re right; the human mind is MH: Do you consciously seek complex analysis that have divorced very visual. In number theory, when out connections between music themselves from their geometric you first state the problems, you and mathematics, or Sanskrit and roots and are frequently taught as don’t see geometry right away. But mathematics? largely algebraic subjects, as though somehow, the way people go about MB: Yeah, but I love it when I’m casting something in purely algebraic solving these problems often requires not looking for them. When I was terms makes it more true. some way of visualizing the question. learning Sanskrit poetry, I had no MB: Well, it doesn’t make it more At this moment we are joined by idea I was encountering mainstream true. Sometimes geometry allows another guest in the hotel lobby—a mathematics—rediscovering those you to visualize and get a feeling for dignified older gentleman whom Man- ideas later in a math class, under what should be true, and those kinds jul knows well. It is Richard Guy, the a totally different name, and in a of discoveries couldn’t have been distinguished British number theorist, totally different guise—wow! That’s obtained simply by manipulating and it becomes apparent that he and what makes me so excited, the un- formulas—you would just get a mess, Manjul have dinner plans. Well into expected unity of two very different you wouldn’t know where to go. So his 90s, Professor Guy is still sharp subjects. the geometry is very critical. as a tack and has come equipped with MH: You made a comment in your MH: But sometimes I would like the pen, paper, and several mathematical first lecture this week that I wanted geometry to be more than that—for queries for his young friend. Manjul to discuss with you. You said that the visualization to be enough on its Bhargava, to no one’s surprise, is “geometry was for intuition and alge- own. happy to oblige. Q bra was for proofs.” MB: That was, in a large part, what http://dx.doi.org/10.4169/mathhorizons.19.2.5

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