Love and Math

by Frenkel, Edward

  Those who stayed in Moscow were employed at places like the Institute for Seismic Studies or Institute for Signal Processing. Their day jobs consisted of some tedious calculations related to a particular industry to which their institute was attached (though some actually managed to break new ground in those areas, multi-talented as they were). They had to do the kind of mathematical research that was their true passion on the side, in their spare time.

  Gelfand himself was forced out of his teaching job at Mekh-Mat in 1968 after he signed the famous letter of ninety-nine mathematicians demanding the release of the mathematician and human rights activist Alexander Esenin-Volpin (the son of the poet Sergei Esenin) from a politically motivated detention in a psychiatric hospital. That letter was so skillfully written that after it was broadcast on the BBC radio, the worldwide outrage embarrassed the Soviet leadership to release Esenin-Volpin almost immediately.4 But this also greatly angered the authorities. They subsequently found ways to punish everyone who signed it. In particular, many of the signatories were fired from their teaching jobs.5

  So Gelfand was no longer professor of mathematics at MGU, though he was able to preserve his seminar that was still being held at the main MGU building. His official job was at a biological lab of MGU that he had founded to conduct research in biology, which was also his passion.* Fuchs was employed at the same lab.

  Fuchs had earlier urged me to start attending Gelfand’s seminar, and I did come to a couple of meetings at the very end of the spring semester. Those meetings made a great impression on me. Gelfand ran his seminar in the most authoritarian way. He decided its every aspect, and though to an untrained eye his seminars could appear chaotic and disorganized, he actually devoted an enormous amount of time and energy to the preparation and choreographing of the weekly meetings.

  Three years later, when Gelfand asked me to speak about my work, I had the opportunity to see the inner workings of the seminar up close. For now, I was observing it from the vantage point of a seventeen-year-old student just starting his mathematical career.

  The seminar was in many ways the theater of a single actor. Officially, there would be a designated speaker reporting on a designated topic, but typically only part of the seminar would be devoted to it. Gelfand would usually bring up other topics and call other mathematicians, who had not been asked to prepare in advance, to the blackboard to explain them. But he was always at the center of it all. He and only he controlled the flow of the seminar and had the absolute power to interrupt the speaker at any moment with questions, suggestions, and remarks. I can still hear him say “Dayte opredelenie” – “Give the definition” – his frequent admonition to a speaker.

  He also had the habit of launching into long tirades on various topics (sometimes unrelated to the material discussed), telling jokes, anecdotes, and stories of all kinds, many of them quite entertaining. This was where I heard the parable that I mentioned in the Preface: a drunkard may not know which number is larger, 2/3 or 3/5, but he knows that 2 bottles of vodka for 3 people is better than 3 bottles of vodka for 5 people. One of Gelfand’s skills was his ability to “rephrase” questions asked by others in such a way that the answer became obvious.

  Another joke he liked to tell involved the wireless telegraph: “At the beginning of the twentieth century, someone asks a physicist at a party: can you explain how it works? The physicist replies that it’s very simple. First, you have to understand how the ordinary, wired, telegraph works: imagine a dog with its head in London and its tail in Paris. You pull the tail in Paris, and the dog barks in London. The wireless telegraph, says the physicist, is the same thing, but without a dog.”

  After recounting the joke and waiting for the laughter to subside (even from those people in the audience who had heard it a thousand times), Gelfand would pivot to whatever math problem was being discussed. If he thought that the solution of the problem required a radically new approach, he would comment, “What I’m trying to say is we need to do it without a dog.”

  A frequently used device at the seminar was to appoint a kontrol’nyj slushatel’, a test listener, usually a junior member of the audience, who was supposed to repeat at regular intervals what the speaker was saying. If it was deemed that the “test listener” was following the lecture well, this meant that the speaker was doing a good job. Otherwise, the speaker had to slow down and explain better. Occasionally, Gelfand would even discharge a particularly incomprehensible speaker in disgrace and replace him or her with another member of the audience. (Of course, Gelfand would poke fun at the test listener as well.) All of this made the seminar very entertaining.

  Most seminars proceed at a steady pace, with people in the audience listening politely (and some perhaps dozing off) – too complacent, too polite, or simply afraid to ask the speaker any questions, and perhaps learning little. There is no doubt that the uneven pace and the generally subversive character of the Gelfand seminar not only kept people awake (not an easy task given that the seminar often lasted till midnight), but stimulated them in ways that other seminars simply couldn’t. Gelfand demanded a lot of his speakers. They worked hard, and so did he. Whatever one can say about Gelfand’s style, people never left the seminar empty-handed.

  However, it seems to me that a seminar like this could only exist in a totalitarian society, like the Soviet Union. People were accustomed to the kind of dictatorial powers and behavior that Gelfand displayed. He could be cruel, at times insulting, to people. I don’t think many would tolerate this kind of treatment in the West. But in the Soviet Union, this was not considered to be out of the ordinary, and no one protested. (Another famous example like this was Lev Landau’s seminar on theoretical physics.)

  When I first started coming to the seminar, Gelfand had a young physicist, Vladimir Kazakov, present a series of talks about his work on so-called matrix models. Kazakov used methods of quantum physics in a novel way to obtain deep mathematical results that mathematicians could not obtain by more conventional methods. Gelfand had always been interested in quantum physics, and this topic had traditionally played a big role at his seminar. He was particularly impressed with Kazakov’s work and was actively promoting it among mathematicians. Like many of his foresights, this proved to be golden: a few years later this work became famous and fashionable, and it led to many important advances in both physics and math.

  In his lectures at the seminar, Kazakov was making an admirable effort to explain his ideas to mathematicians. Gelfand was more deferential to him than usual, allowing him to speak without interruptions longer than other speakers.

  While these lectures were going on, a new paper arrived, by John Harer and Don Zagier, in which they gave a beautiful solution to a very difficult combinatorial problem.6 Zagier has a reputation for solving seemingly intractable problems; he is also very quick. The word was that the solution of this problem took him six months, and he was very proud of that. At the next seminar, as Kazakov was continuing his presentation, Gelfand asked him to solve the Harer–Zagier problem using his work on the matrix models. Gelfand had sensed that Kazakov’s methods could be useful for solving this kind of problem, and he was right. Kazakov was unaware of the Harer–Zagier paper, and this was the first time he heard this question. Standing at the blackboard, he thought about it for a couple of minutes and immediately wrote down the Lagrangian of a quantum field theory that would lead to the answer using his methods.

  Everyone in the audience was stunned. But not Gelfand. He asked Kazakov innocently, “Volodya, how many years have you been working on this topic?”

  “I am not sure, Israel Moiseevich, perhaps six years or so.”

  “So it took you six years plus two minutes, and it took Don Zagier six months. Hmmm... You see how much better he is?”

  And this was a mild “joke,” compared with some others. You had to have thick skin in order to survive in this environment. Unfortunately, some speakers took these kinds of public put-downs personally, and this caused them a lot of torme
nt. But I have to add that Gelfand always had a sharper tongue for the older, more established mathematicians, and he was much more gentle to young mathematicians, especially to students.

  He used to say that at the seminar he welcomed all undergraduates, talented graduate students, and only brilliant professors. He understood that in order to keep the subject moving, it was very important to prepare new generations of mathematicians, and he always surrounded himself with young talent. They kept him young as well (he was actively doing cutting-edge research till he was in his late eighties). Often, he would even invite high school students to the seminar and make them sit in the front row to make sure that they were following what was going on. (Of course, these were no ordinary high school students. Many of them went on to become world-renowned mathematicians.)

  By all accounts, Gelfand was very generous with his students, spending hours talking to them on a regular basis. Very few professors do this. It wasn’t easy to be his student; he gave them a kind of tough love, and they had to cope with his various quirks and dictatorial habits. But my impression from talking to many of them is that they were all loyal to him and felt they owed him a tremendous debt.

  I was not Gelfand’s student – I was his “grand-student,” as both of my teachers, Fuchs and Feigin (who was not yet in my life), were at least partially Gelfand’s students. Hence I always considered myself as being part of the “Gelfand mathematical school.” Much later, when he and I were in the United States, Gelfand asked me directly about this, and by the look of satisfaction on his face when I said yes, I could tell how important the issue of his school and recognition of who was part of it was to him.

  This school, of which the seminar was the focal point, its window to the world, had an enormous impact not only on mathematics in Moscow, but around the world. Foreign mathematicians came to Moscow just to meet Gelfand and attend his seminar, and many considered it an honor to lecture there.

  Gelfand’s fascinating and larger-than-life personality played a big role in the seminar’s reputation. A few years later, he became interested in my work and asked me to speak at his seminar. I spent many hours talking to him, not just about mathematics, but about a lot of other things. He was very interested in the history of mathematics and his own legacy in particular. I remember vividly how, when I first came to visit him at his Moscow apartment (I had just turned twenty-one), he informed me that he considered himself the Mozart of mathematics.

  “Most composers are remembered for particular pieces they wrote,” he said. “But in the case of Mozart, that’s not so. It’s the totality of his work that makes him a genius.” He paused and continued: “The same goes for my mathematical work.”

  Putting aside some interesting questions raised by such self-assessment, I think it’s actually an apt comparison. Though Gelfand did not prove any famous long-standing conjectures (such as Fermat’s Last Theorem), the cumulative effect of his ideas on mathematics was staggering. Perhaps more importantly, Gelfand possessed an excellent taste for beautiful mathematics as well as an astute intuition about which areas of mathematics were the most interesting and promising. He was like an oracle who had the power to predict in which directions mathematics would move.

  In the subject that was becoming increasingly fractured and specialized, he was one of the last remaining Renaissance men able to bridge different areas. He epitomized the unity of mathematics. Unlike most seminars, which focus on one area of math, if you came to Gelfand’s seminar, you could see how all these different parts fit together. That’s why all of us gathered every Monday night on the fourteenth floor of the main MGU building and eagerly awaited the word of the master.

  And it was to this awe-inspiring man that Fuchs suggested I submit my first math paper. Gelfand’s journal, Functional Analysis and Applications, was published in four slim issues a year, about a hundred pages each (a pitiful amount for a journal like this, but the publisher refused to give more, so one had to cope), and it was held in extremely high regard around the world. It was translated into English, and many science libraries around the world subscribed to it.

  It was very difficult to get a paper published in this journal, partly because of the severe page limitations. There were in fact two types of papers that were published: research articles, each typically ten–fifteen pages long, containing detailed proofs, and short announcements, in which only the results were stated, without proofs. The announcement could not be longer than two pages. In theory, such a short paper was supposed to be followed eventually by a detailed article containing all proofs, but in reality quite often that did not happen because publishing a longer article was extremely difficult. Indeed, it was nearly impossible for a mathematician in the USSR to publish abroad (one needed to get all sorts of security clearances, which could easily take more than a year and a lot of effort). On the other hand, the number of math publications in the Soviet Union, considering the number of mathematicians there, was very small. Unfortunately, many of them were controlled by various groups, which would not allow outsiders to publish, and anti-Semitism was also prevalent in some of them.

  Because of all this, a certain subculture of math papers emerged in the USSR, which came to be referred to as the “Russian tradition” of math papers: extremely terse writing, with few details provided. What many mathematicians outside of the Soviet Union did not realize was that this was largely done by necessity, not by choice.

  It was this kind of short announcement that Fuchs was aiming at for my first article.

  Each article submitted to Functional Analysis and Applications, including short announcements, had to be screened and approved by Gelfand. If he liked it, he would then let the article go through the standard refereeing process. This meant that for my article to be considered, I had to meet Israel Moiseevich in person. So before one of the first seminars of the fall semester of 1986, Fuchs introduced me to him.

  Gelfand shook my hand, smiled, and said, “I am pleased to meet you. I’ve heard about you.”

  I was totally star-struck. I could swear that I saw a halo around Gelfand’s head.

  Then Gelfand turned to Fuchs and asked him to show my article, which Fuchs handed to him. Gelfand started to turn the pages. There were five of them, which I neatly typed (slowly, with two fingers) on a typewriter I borrowed at Kerosinka, and then inserted formulas by hand.

  “Interesting,” Gelfand said approvingly, and then turned to Fuchs: “But why is this important?”

  Fuchs started to explain something about the discriminant of polynomials of degree n with distinct roots, and how my result could be used to describe the topology of the fiber of the discriminant, and... Gelfand interrupted him: “Mitya,” he said, using the diminutive form of Fuchs’ first name, “Do you know how many subscribers the journal has?”

  “No, Israel Moiseevich, I don’t.”

  “More than a thousand.” That was a pretty large number given how specialized the journal was. “I cannot send you with every issue so that you would explain to each subscriber what this result is good for, now can I?”

  Fuchs shook his head.

  “This has to be written clearly in the paper, OK?” Gelfand made the point of saying all of this to Fuchs, as though it was all his fault. Then he said to both of us: “Otherwise, the paper looks good to me.”

  With that, he smiled again at me and went to talk to someone else.

  Quite an exchange! Fuchs waited until Gelfand was out of the earshot and said to me, “Don’t worry about this. He just wanted to impress you.” (And he sure did!) “We’ll just have to add a paragraph to this effect at the beginning of the paper, and after that he will probably publish it.”

  That was the best possible outcome. After adding a paragraph required by Gelfand, I officially submitted the article, and eventually it appeared in the journal.7 With that, my first math project was complete. I crossed my first threshold and was at the beginning of a path that would lead me into the magical world of modern math.

 
This is the world I want to share with you.

  *It is also worth noting that Gelfand was not elected as a full member of the Academy of Sciences of the USSR until the mid-1980s because the Mathematical Branch of the Academy was for decades controlled by the director of the Steklov Mathematical Institute in Moscow, Ivan Matveevich Vinogradov, nicknamed the “Anti-Semite-in-Chief of the USSR.” Vinogradov had put in place draconian anti-Semitic policies at the Academy and the Steklov Institute, which was in his grip for almost fifty years.

  Chapter 7

  The Grand Unified Theory

  The solution of the first problem was my initiation into the temple of mathematics. Somewhat serendipitously, the next mathematical project I did with Fuchs brought me into the midst of the Langlands Program, one of the deepest and most exciting mathematical theories to emerge in the past fifty years. I will tell you about my project below, but my goal in this book is to describe much more than my own experience. It is to give you a sense of modern math, to prove that it is really about originality, imagination, groundbreaking insights. And the Langlands Program is a great example. I like to think of it as a Grand Unified Theory of Mathematics because it uncovers and brings into focus mysterious patterns shared by different areas of math and thus points to deep, unexpected connections between them.

  Mathematics consists of many subfields. They often feel like different continents, with mathematicians working in those subfields speaking different languages. That’s why the idea of “unification,” bringing together the theories coming from these diverse fields and realizing that they are all part of a single narrative, is so powerful. It’s as if you suddenly realized that you could understand another language, one you had desperately tried to learn without much success.

  It’s useful to think about mathematics as a whole as a giant jigsaw puzzle, in which no one knows what the final image is going to look like. Solving this puzzle is a collective enterprise of thousands of people. They work in groups: here are the algebraists laboring over their part of the puzzle, here are the number theorists, here are the geometers, and so on. Each group has been able to create a small “island” of the big picture, but through most of the history of mathematics, it has been hard to see how these little islands will ever join up. As a result, most people work on expanding those islands of the puzzle. Every once in a while, however, someone will come who will see how to connect the islands. When this happens, important traits of the big picture emerge, and this gives a new meaning to the individual fields.