Love and Math

by Frenkel, Edward

  Let’s go back to Grothendieck’s idea. We start in the middle column of Weil’s Rosetta stone. Then we study curves over finite fields and more general manifolds over finite fields. These manifolds are defined by systems of polynomial equations such as

  which we talked about in Chapter 9.

  Suppose that we have a sheaf on such a manifold. It is a rule assigning to each point of the manifold a vector space, but there is actually more structure. The notion of a sheaf is defined in such a way that any symmetry of the numerical system over which our manifold is defined – which is in this case a finite field – gives rise to a symmetry of this vector space. In particular, the Frobenius, which is an element of the Galois group of the finite field, necessarily gives rise to a symmetry (such as a rotation or a dilation) of this vector space.

  Now, if we have a symmetry of a vector space, we can produce a number out of it. There is a standard technique for doing this. For example, if our vector space is a line, then the symmetry of this space that we obtain from the Frobenius will be a dilation: each element z will be transformed to Az for some number A. Then the number we assign to this symmetry is just A. And for the vector spaces of dimension greater than one, we take what’s called the trace of the symmetry.9 By taking the trace of the Frobenius on the space , we assign a number to the point s.

  The simplest case is that the Frobenius acts as the identity symmetry on the vector space. Then its trace is equal to the dimension of the vector space. So in this case, by taking the trace of the Frobenius, we assign to a vector space its dimension. But if the Frobenius is not the identity, this construction assigns to a vector space a more general number, which is not necessarily a natural number.

  The upshot is that if we have a manifold S over a finite field {0,1,2,..., p−1} (which is what happens if we are in the middle column of Weil’s Rosetta stone) and we have a sheaf on S, then to each point s of S we can assign a number. This gives us a function on S. Therefore we see that in the middle column of Weil’s Rosetta stone we have a way to go from sheaves to functions.

  Grothendieck called this a “sheaves–to–functions dictionary.” It is a curious sort of dictionary, however. Relying on the procedure described above, we obtain a passage from sheaves to functions. Furthermore, natural operations on sheaves are parallel to natural operations on functions. For example, the operation of taking the direct sum of two sheaves, defined similarly to the direct sum of two vector spaces, is parallel to the operation of taking the sum of two functions.

  But there is no natural way to go back from functions to sheaves.10 It turns out we can do this only for some functions, not for all. But if we can do it, then this sheaf will carry a lot of additional information that the function did not have. This information can then be used to get to the heart of that function. A remarkable fact is that most of the functions that appear in the Langlands Program (in the second column of Weil’s Rosetta stone) do come from sheaves.

  Mathematicians studied functions, one of the central notions in all of mathematics, for centuries. This is a concept that we can grasp intuitively by thinking of the temperature or barometric pressure. But what people didn’t recognize before Grothendieck is that if we are in the context of manifolds over finite fields (such as curves over a finite field), we can go beyond functions and work with sheaves instead.

  Functions were, if you will, the concepts of archaic math, and sheaves are the concepts of modern math. Grothendieck showed that in many ways sheaves are more fundamental; the good ol’ functions are their mere shadows.

  This discovery greatly stimulated progress in mathematics in the second half of the twentieth century. The reason is that sheaves are much more vital and versatile objects, with a lot more structure. For example, a sheaf can have symmetries. If we elevate a function to a sheaf, we can exploit these symmetries, and this way we can learn a lot more than we could ever learn using functions.

  What is especially important for us is that sheaves make sense both in the middle column and in the right column of Weil’s Rosetta stone. This opens a path to moving the Langlands Program from the middle to the right column.

  In the right column of the Rosetta stone, we consider manifolds that are defined over the complex numbers. For example, we consider Riemann surfaces such as the sphere or the surface of a donut. In this setting, the automorphic functions that appear in the left and middle columns of Weil’s Rosetta stone do not make much sense. But sheaves do make sense. So once we replace functions by sheaves in the middle column (which we can do because we have Grothendieck’s dictionary), we regain the analogy between the middle and the right columns of Weil’s Rosetta stone.

  Let’s summarize: when we pass from the middle column of Weil’s Rosetta stone to the right column, we have to make some adjustments to both sides of the relation envisioned by the Langlands Program. That’s because the notions of Galois group and automorphic functions don’t have immediate counterparts in the geometry of Riemann surfaces. First, the Galois group finds its analogue in the fundamental group of a Riemann surface, as explained in Chapter 9. Second, we use the Grothendieck dictionary and instead of automorphic functions, we consider sheaves that satisfy properties analogous to the properties of the automorphic functions. We call them automorphic sheaves.

  This is illustrated by the following diagram, in which we have three columns of the Rosetta stone, and the two rows in each column contain the names of the objects on the two sides of the Langlands relation specific to that column.

  The question is then how to construct these automorphic sheaves. This proved to be a very difficult problem. In the early 1980s, Drinfeld proposed the first such construction in the simplest case (building on an earlier unpublished work of Pierre Deligne). Drinfeld’s ideas were further developed by Gérard Laumon a few years later.

  When I met Drinfeld, he told me that he had come up with a radically new method to construct automorphic sheaves. But the new construction he envisioned was conditional on a certain conjecture which he thought I could derive from my work with Feigin on Kac–Moody algebras. I couldn’t believe it: my work could be useful for the Langlands Program?

  The chance that I could do something related to the Langlands Program made me eager to learn everything that was known about it. That spring, I went to Drinfeld’s office at Harvard almost every day, and I pestered him with questions about the Langlands Program, which he patiently answered. He would also ask me about my work with Feigin, the details of which were critical for what he was trying to do. The rest of the day I would devour anything that I could find on the Langlands Program at the Harvard library. The subject was so alluring, I tried to fall asleep every night as quickly as I could, so that morning would come sooner and I would immerse myself ever deeper into the Langlands Program. I knew I was embarking on one of the most important projects of my life.

  Something else happened near the end of the spring semester that threw me right back to the Kafkaesque experience of my entrance exams at Moscow University.

  One day, Victor Kac called me at home in Cambridge and told me that someone invited Anatoly Logunov, the President (or Rector, as he was called) of Moscow University to give a lecture at MIT’s physics department. Kac and many of his colleagues were outraged that MIT would give forum to the man directly responsible for the discrimination against Jewish students at the entrance exams to MGU. Kac and the others felt that his actions amounted to a crime, and hence the invitation was scandalous.

  Logunov was a very powerful man: he was not only the President of MGU but also director of the Institute of High Energy Physics, member of the Central Committee of the Communist Party of the USSR, and more. But why would someone at MIT invite him? In any case, Kac and several of his colleagues protested and asked for the visit and the lecture to be canceled. After some negotiations, a compromise was found: Logunov would come and give his lecture, but after the lecture there would be a public discussion of the situation at the MGU, and people would have a chance to c
onfront him about the discrimination. It would be something like a town-hall meeting.

  Naturally, Kac asked me to come to that meeting to present my story as firsthand evidence of what was happening at MGU under Logonov’s leadership. I was a bit reluctant to do this. I was sure that Logunov would be accompanied by “assistants,” who would be recording everything. Remember, this was May 1990, more than a year before the failed putsch of August 1991 that started the collapse of the Soviet Union. And I was about to go back home for the summer. If I were to say something even mildly embarrassing for such a high-profile Soviet official as Logunov, I could easily get in trouble. At the very least, they could prevent me from leaving the Soviet Union and returning to Harvard. Still, I couldn’t deny Kac’s request. I knew how important my testimony could be at this meeting, so I told Victor that I would come and, if needed, tell my story. Kac tried to reassure me.

  “Don’t worry, Edik,” he said, “if they put you in jail for this, I’ll do everything I can to get you out.”

  The word about the upcoming event quickly spread, and the lecture hall was packed for Logunov’s lecture. People did not come to learn anything from his talk. Everybody knew that Logunov was a weak physicist who made his career trying to disprove Einstein’s relativity theory (I wonder why). As expected, the talk – on his “new” theory of gravity – had very little substance. But it was quite unusual in many respects. First of all, Logunov did not speak English and delivered his lecture in Russian, which was simultaneously translated by a tall man in a black suit and tie who spoke perfect English. He might as well have had “KGB” written on his forehead in big block letters. His clone (as in the movie The Matrix) was sitting in the audience looking around.

  Before the talk, one of Logunov’s MIT hosts introduced him in a very peculiar fashion. He projected a slide of the first page of a paper in English, co-authored by Logunov and a few other people and published a decade earlier. I guess the point was to show that Logunov was not a total idiot, but actually he had to his credit some publications in refereed journals. I’d never seen anyone being introduced in this way. It was clear that Logunov was not invited to MIT for his scientific brilliance.

  There were no protests at the lecture, though Kac had distributed among the audience members photocopies of some damning documents. One of them was the transcript of a fellow with a Jewish last name from a decade earlier. He had A’s on all subjects and yet during his last year at MGU he was expelled for “academic failure.” A short note added to the transcript informed the reader that this student was spotted by specially dispatched agents at the Moscow synagogue.

  After the lecture, people went to another room and sat around a large rectangular table. Logunov was sitting on one side, close to an end, flanked by the two plain-clothed “assistants,” who were translating, and Kac and other accusers were sitting directly across the table from them. I sat quietly with a few friends at the opposite end of the table, to Logunov’s side, so he wasn’t paying any attention to us.

  At first, Kac and others spoke and said that they had heard many stories about Jewish students not being admitted to MGU. They asked Logunov if he, as the Rector of Moscow University, would comment on this. Of course, he flatly denied everything, no matter what they said to him. At some point, one of the plain-clothed guys said, in English, “You know, Professor Logunov is a very modest person, so he would never tell you this. But I will. He has actually helped many Jewish people with their careers.”

  The other plain-clothed guy then said to Kac and others, “You should either put up or shut up. If you have any concrete cases that you want to talk about, bring them up. Otherwise, Professor Logunov is very busy and he has other things to attend to.”

  At this point, of course, Kac said, “Actually, we do have a concrete case that we want to talk to you about,” and he gestured toward me.

  I stood up. Everyone turned to me, including Logunov and his “assistants,” their faces betraying some anxiety. I was now facing Logunov directly.

  “Very interesting,” said Logunov in Russian – this was to be translated into English for everybody – and then added quietly to his assistants, but I could hear: “Don’t forget to take down his name.”

  I have to say, it was a bit scary, but I had reached the point of no return. I introduced myself and said, “I was failed at the entrance exams to the Mekh-Mat six years ago.”

  And then I briefly described what had happened at the exams. The room went quiet. This was a “concrete” eyewitness account from one of the victims of Logunov’s policy, and there was no way he could deny that this had happened. The two assistants rushed to limit the damage.

  “So you were failed at the MGU. And where did you apply after this?” one of them asked.

  “I went to the Institute of Oil and Gas,” I said.

  “He went to Kerosinka,” translated the assistant to Logunov, who then vigorously nodded yes – of course, he knew that this was one of the few places in Moscow where students like me could be accepted.

  “Well,” continued the assistant, “Perhaps, the competition at the Oil and Gas Institute was not as stiff as at MGU. Maybe that’s why you got into one, but not the other?”

  This was false: I knew for a fact there was very little competition at Mekh-Mat among those who were not discriminated against. I’d been told that getting one B and three C’s at the four exams was sufficient to get in. Entrance exams to Kerosinka were, on the contrary, very competitive. At this point Kac interjected: “While a student, Edward did some groundbreaking mathematical work and was invited as a Visiting Professor to Harvard at the age of twenty-one, less than five years after he was failed at MGU. Are you going to suggest that the competition for the Harvard position was also lower than at the entrance exam to MGU?”

  Long silence. Then, suddenly, Logunov became very animated.

  “I am outraged by this!” he yelled. “I will investigate and punish those responsible for failing this young man. I will not allow this kind of things to happen at MGU!”

  And he went on like this for a few minutes.

  What could one possibly say to this? No one at the table believed that Logunov’s outburst was genuine and that he would really do anything. Logunov was very clever. By expressing his feigned outrage about one case, he deflected a much bigger issue: that of thousands of other students who were ruthlessly failed as the result of a carefully orchestrated discrimination policy that was clearly approved by the top brass at MGU, including the Rector himself.

  We couldn’t possibly bring up those cases at that meeting and prove that there was a concerted policy of anti-Semitism at the entrance exams to Mekh-Mat. And while there was a certain measure of satisfaction that I was able to face my tormentor directly and force him to admit that I was indeed wronged by his subordinates, we all knew that the bigger question remained unanswered.

  Logunov’s hosts, who were clearly embarrassed by all the negative publicity surrounding his visit, wanted to get this over with as quickly as possible. They adjourned the meeting and whisked him away. He was never invited back.

  Chapter 15

  A Delicate Dance

  In the fall of 1990, I became a Ph.D. student at Harvard, which I had to do to move from Visiting Professor to something more permanent. Joseph Bernstein agreed to be my official advisor. By then, I already had more than enough material for a Ph.D. thesis, and Arthur Jaffe got the Dean to waive the usual two-year enrollment requirement for me so that I could get my Ph.D. in one year. Because of that, my “demotion” from professor to a graduate student didn’t last very long.

  In fact, I wrote my Ph.D. thesis about a new project, which I completed during that year. It all started from my discussions with Drinfeld on the Langlands Program in the spring. Here’s one of them, in the form of a screenplay.

  FADE IN:

  INT. DRINFELD’S OFFICE AT HARVARD

  DRINFELD is pacing in front of the blackboard. EDWARD, sitting in a chair, is taking notes. A tea
cup is on the desk by his side.

  DRINFELD

  So, the Shimura-Taniyama-Weil conjecture gives us a link between cubic equations and modular forms, but Langlands went much farther than this. He envisioned a more general relation, in which the role of modular forms is played by the automorphic representations of a Lie group.

  EDWARD

  What’s an automorphic representation?

  DRINFELD

  (after a long pause)

  The precise definition is not important now. And anyway, you can read it in a book. What’s important is that it is a representation of a Lie group G - for example, the group SO(3) of rotations of a sphere.

  EDWARD

  OK. And what are these automorphic representations related to?

  DRINFELD

  Well, that’s the most interesting part: Langlands predicted that they should be related to representations of the Galois group in another Lie group.1

  EDWARD

  I see. You mean this Lie group is not the same group G?

  DRINFELD

  No! It’s another Lie group, which is called the Langlands dual group of G.

  DRINFELD writes the symbol LG on the blackboard.

  EDWARD

  Is the L for Langlands?

  DRINFELD

  (hint of a smile)

  Well, Langlands’ original motivation was to understand something called L-functions, so he called this group an L-group...

  EDWARD

  Let me see if I understand this. For every Lie group G, there is another Lie group called LG, correct?

  DRINFELD

  Yes. And it appears in the Langlands relation, which looks schematically as follows.

  DRINFELD draws a diagram on the blackboard:2

  EDWARD

  I don’t understand it... at least, not yet. But let me ask you a simpler question: what’s the Langlands dual group of SO(3), say?