Love and Math

by Frenkel, Edward

A few days later, the picture started to emerge. In a flash of insight, as I was pacing around my dorm room, I realized that Wakimoto’s formulas came from geometry. This was a startling discovery because Wakimoto’s approach was entirely algebraic – there were no hints of geometry.

  To explain my geometric interpretation, let’s revisit the Lie group SO(3) of symmetries of the sphere and its loop group. As explained in the previous chapter, an element of the loop group of SO(3) is a collection of elements of SO(3), one element of SO(3) for every point of the loop. Each of those elements of SO(3) acts on the sphere by a particular rotation. This implies that each element of the loop group of SO(3) gives rise to a symmetry of the loop space of the sphere.1

  I realized that I could use this information to obtain a representation of the Kac–Moody algebra associated to SO(3). This does not yet give us Wakimoto’s formulas. To get them, we have to modify the formulas in a certain radical way. Think about it as turning a coat inside out. We can do this to any coat, but in most cases, the garment then becomes unusable – we can’t wear it in public. However, there are coats that can be worn on either side. And the same was true of Wakimoto’s formulas.

  Armed with this new understanding, I immediately tried to generalize Wakimoto’s formulas to other, more complicated Kac–Moody algebras. The first, geometric, step worked fine, just like in the case of SO(3). But when I tried to turn the formulas “inside out,” I got nonsense. The resulting math simply didn’t add up. I tried to fiddle with the formulas but could not find my way around the problem. I had to consider the very real possibility that this construction only worked for SO(3) and not for more general Kac–Moody algebras. There was no way of knowing for sure whether the problem had a solution, and if so, whether a solution could be obtained using the available means. I just had to work as hard as I could and hope for the best.

  A week passed, and it was time to meet Fuchs again. I was planning to tell him about my calculations and ask for advice. When I arrived at the dacha, Fuchs told me that his wife had to go to Moscow to run some errands, and he had to take care of his two young daughters.

  “But you know what,” he said, “Feigin was here yesterday, and he was all excited about the stuff you told us last week. Why don’t you go visit him – his dacha is just fifteen minutes away. I told him that I would send you his way today, so he is expecting you.”

  He explained to me the directions, and I went to Feigin’s dacha.

  Feigin was indeed expecting me. He greeted me warmly and introduced me to his charming wife, Inna, and his three kids: two energetic boys Roma and Zhenya, ages eight and ten, and an adorable two-year old daughter Lisa. I could not know at that moment that I would be very close to this wonderful family for many years to come.

  Feigin’s wife offered us some tea and pie, and we sat down on the terrace. It was a beautiful summer afternoon, rays of sun protruding between the leafy trees, birds chirping – idyllic countryside. But of course, the conversation quickly turned to the Wakimoto construction.

  It turned out that Feigin was also thinking about it, and along similar lines. At the beginning of our conversation, in fact, we were basically completing each other’s sentences. It was a special feeling: he understood me completely, and I understood him.

  I started telling him about my failure to generalize the construction to other Kac–Moody algebras. Feigin listened intently, and after sitting quietly for a while, thinking about this, he drew my attention to an important point that I had missed. In trying to generalize Wakimoto’s construction, we need to find a proper generalization of the sphere – the manifold on which SO(3) acts by symmetries. In the case of SO(3), this choice is practically unique. But for other groups there are many choices. In my calculations, I had taken it for granted that the natural generalizations of the sphere were the so-called projective spaces. But that was not necessarily the case; the fact that I wasn’t getting anywhere with this could just be because my choice of spaces was poor.

  As I explained above, at the end of the day, I needed to turn the formulas “inside out.” The whole construction hinged on the expectation that, miraculously, the resulting formulas would still be sound. This is what happened in Wakimoto’s case, for the simplest group SO(3). My calculations indicated that for the projective spaces, this was not the case, but this didn’t mean that a better construction couldn’t be found. Feigin suggested I try instead the so-called flag manifolds.2

  The flag manifold for the group SO(3) is the familiar sphere, so for other groups these spaces may be viewed as natural substitutes for the sphere. But flag manifolds are richer and more versatile than the projective spaces, so there was a chance that an analogue of the Wakimoto construction would work for them.

  It was already getting dark, time to go home. We agreed to meet again the following week, and then I waved good-bye to Feigin’s family and went back to the train station.

  On the train ride home, in an empty train car, with its open windows letting in the warm summer air, I couldn’t stop thinking about the problem. I had to try to do it, right there and then. I pulled out a pen and a pad and started writing the formulas for the simplest flag manifold. The old train car, making a staccato noise, was shaking back and forth, and I couldn’t hold my pen steady, so the formulas I was writing were all over the place. I could hardly read what I was writing. But in the midst of this chaos, there was a pattern emerging. Things definitely worked better for the flag manifolds than for the projective spaces that I had tried, unsuccessfully, to tame the previous week.

  A few more lines of computations, and... Eureka! It was working. The “inside out” formulas worked as nicely as in Wakimoto’s work. The construction generalized beautifully. I was overwhelmed with joy: this was the real deal. I did it, I found new free-field realizations of Kac–Moody algebras!

  The next morning I checked my calculations carefully. Everything worked out. There was no phone at Feigin’s dacha, so I couldn’t call him and tell him about my new findings. I started writing them down in a form of a letter, and when we met the following week, I told him about the new results.

  This was the beginning of our work together. He became my teacher, mentor, advisor, friend. I addressed him at first as Boris Lvovich, in the old-fashioned Russian way, including the patronymic name. But later he insisted that I switch to the more informal Borya.

  I’ve been incredibly lucky with my teachers. Evgeny Evgenievich showed me the beauty of mathematics and made me fall in love with it. He also helped me learn the basics. Fuchs saved me after the MGU entrance exam catastrophe and jump-started my faltering mathematical career. He led me through my first serious math project, which gave me confidence in my abilities, and steered me to an exciting area of research on the interface of math and physics. Finally, I was ready for the big leagues. Borya proved to be the best advisor that I could possibly dream of at that stage of my journey. It was as though my mathematical career was getting turbocharged.

  Borya Feigin is undoubtedly one of the most original mathematicians of his generation in the entire world, a visionary who has the deepest sense of mathematics. He guided me into the wonderland of modern math, full of magic beauty and sublime harmony.

  Now that I’ve had students of my own, I appreciate even more what Borya has done for me (and what Evgeny Evgenievich and Fuchs did for me earlier). It’s hard work being a teacher! I guess in many ways it’s like having children. You have to sacrifice a lot, not asking for anything in return. Of course, the rewards can also be tremendous. But how do you decide in which direction to point students, when to give them a helping hand and when to throw them in deep waters and let them learn to swim on their own? This is art. No one can teach you how to do this.

  Borya cared deeply for me and my development as a mathematician. He never told me what to do, but talking with him and learning from him always gave me a sense of direction. Somehow, he was able to make sure that I always knew what I wanted to do next. And, with him by my side, I always fe
lt confident that I was on the right track. I was very fortunate to have him as my teacher.

  It was already the beginning of the fall semester of 1987, my fourth year at Kerosinka. I was nineteen, and my life had never been more exciting. I was still living in the dorm, hanging out with friends, falling in love... I was also keeping up with my studies. By then, I was skipping most of my classes and studying for the exams on my own (occasionally, just a few days prior to the exam). I was still getting straight A’s – the only exception being a B in Marxist Political Economy (shame on me).

  I kept secret from most people the fact that I had a “second life” – which took up most of my time and energy – my mathematical work with Borya.

  I would usually meet with Borya twice a week. His official job was at the Institute for Solid State Physics, but he did not have to do much there, and only had to show up once a week. On other days, he would work at his mother’s apartment, which was ten minutes’ walk from his home. It was also close to Kerosinka and to my dorm. This was our usual meeting place. I would come in late morning or early afternoon, and we would work on our projects, sometimes all day. Borya’s mother would come from work in the evening and feed us dinner, and often we would leave together around nine or ten o’clock.

  As our first order of business, Borya and I wrote a short summary of our results and sent it to the journal Russian Mathematical Surveys. It was published within a year, pretty fast by the standards of math journals.3 Having gotten this out of the way, Borya and I focused on developing our project further. Our construction was powerful, and it opened up many new directions of research. We used our results to understand better representations of Kac–Moody algebras. Our work also enabled us to develop a free-field realization of two-dimensional quantum models. This allowed us to make calculations in these models that were not accessible before, which soon made physicists interested in our work.

  Those were exciting times. On the days Borya and I weren’t meeting, I was working on my own – in Moscow during the week, at home on the weekends. I continued going to the Science Library and devouring more and more books and articles on closely related subjects. I was living and eating and drinking this stuff. It was as though I was immersed in this beautiful parallel universe, and I wanted to stay there, getting ever deeper into this dream. With every new discovery, every new idea, this magical world was becoming more and more my home.

  But in the fall of 1988, as I entered the fifth, and last, year of my studies at Kerosinka, I was pulled back to reality. It was time to start thinking about the future. Though I was at the top of my class, my prospects looked bleak. Anti-Semitism ruled out graduate school and the best jobs available to graduates. Not having a propiska, residency in Moscow, complicated things even more. The day of reckoning was coming.

  Chapter 12

  Tree of Knowledge

  Even though I knew that I would never be allowed to pursue a career in academia, I continued doing mathematics. Mark Saul talks about this in his article1 (referring to me by the diminutive form of my first name, Edik):

  What impelled Edik and others to continue, like so many salmon swimming upstream? There was every indication that the discrimination they faced at the university level would continue into their professional lives. Why then should they prepare themselves so intensively and against such odds for a career in mathematics?

  I was not expecting to receive anything in return other than the pure joy and passion of intellectual pursuit. I wanted to dedicate my life to mathematics simply because I loved doing it.

  In the stagnant life of the Soviet period, talented youth could not apply their energy in business; the economy had no private sector. Instead, it was under tight government control. Likewise, communist ideology controlled intellectual pursuit in the spheres of humanities, economics, and social sciences. Every book or scholarly article in these areas had to start with quotations of Marx, Engels, and Lenin and unequivocally support the Marxist point of view of the subject. The only way to write a paper on foreign philosophy, say, would be to present it as a condemnation of the philosophers’ “reactionary bourgeois views.” Those who did not follow these strict rules were themselves condemned and persecuted. The same applied to art, music, literature, and cinema. Anything that could be even remotely considered as critical of the Soviet society, politics, or lifestyle – or simply deviated from the canons of “socialist realism” – was summarily censored. The writers, composers, and directors who dared to follow their artistic vision were banned, and their work was shelved or destroyed.

  Many areas of science were also dominated by the party line. For example, genetics was banned for many years because its findings were deemed to contradict the teachings of Marxism. Even linguistics was not spared: after Stalin, who considered himself an expert in this subject (as well as many others), wrote his infamous essay On Certain Questions of Linguistics, this whole field was reduced to interpreting that largely meaningless treatise. Those who did not follow it were repressed.

  In this environment, mathematics and theoretical physics were oases of freedom. Though communist apparatchiks wanted to control every aspect of life, these areas were just too abstract and difficult for them to understand. Stalin, for one, never dared to make any pronouncements about math. At the same time, Soviet leaders also realized the importance of these seemingly obscure and esoteric areas for the development of nuclear weapons, and that’s why they did not want to “mess” with these areas. As the result, mathematicians and theoretical physicists who worked on the atomic bomb project (many of them reluctantly, I might add) were tolerated, and some even treated well, by Big Brother.

  Thus, on the one hand, mathematics was abstract and inexpensive, and on the other hand, it was useful in the areas that the Soviet leaders cared deeply about – especially defense, which ensured the regime’s survival. That’s why mathematicians were largely allowed to do their research and were not subjected to the constraints that were imposed on other fields (unless they tried to meddle in politics, as with the “Letter of the 99,” which I mentioned earlier).

  I believe that this was the main reason why so many talented young students chose mathematics as their profession. This was one area in which they could engage in free intellectual pursuit.

  But passion and joy of doing mathematics notwithstanding, I needed a job. Because of this, in parallel with my main mathematical research work, which I did in secret with Borya, I had to do some “official” research at Kerosinka.

  My advisor at Kerosinka was Yakov Isaevich Khurgin, who was a professor at the Department of Applied Mathematics and one of the most charismatic and revered faculty members. A former student of Gelfand, Yakov Isaevich was then in his late sixties, but he was one of the “coolest” professors we had. Because of his engaging teaching style and sense of humor, his classes had the highest student attendance. Even though I was skipping most lectures starting from my third year, I always tried to come to his lectures on probability theory and statistics. I started to work with him in my third year.

  Yakov Isaevich was very kind to me. He made sure that I was treated well and, whenever I needed help, he was always there for me. For example, when I had some issues at my dorm, he used his leverage to intervene. Yakov Isaevich was a smart man who learned well how to “work the system”: even though he was Jewish, he occupied a prestigious position at Kerosinka, as a full professor and the head of a laboratory that did work in areas ranging from oil exploration to medicine.

  He was also a popularizer of mathematics, having written several best-selling books about mathematics for non-specialists. I especially liked one of them, entitled So What? It is about his collaboration with scientists, engineers, and doctors. Through dialogues with them, he explains in accessible and entertaining ways interesting mathematical concepts (mostly, concerning probability and statistics, his main areas of expertise) and their applications. The title of the book is meant to represent the sense of curiosity with which a mathematician a
pproaches real-life problems. These books and his passion for making mathematical ideas accessible to the public have greatly inspired me.

  For many years, Yakov Isaevich worked with medical doctors, mostly urologists. His original motivation was personal. He was enrolled as a student at Mekh-Mat when he was called to the front lines of World War II, where he caught a serious kidney disease in the frigid trenches. This was actually lucky for him because he was taken to the hospital, and this saved his life – most of his classmates who were with him in the army were killed in battle. But from that point on, he had to deal with kidney problems. In the Soviet Union medicine was free, but the quality of medical services was low. In order to get a good treatment, one had to have a personal connection with a doctor or have a bribe to offer. But Yakov Isaevich had something else to offer, which very few people could: his expertise as a mathematician. He used it to befriend the best specialists in urology in Moscow.

  This was a great deal for him because whenever his kidneys were acting out, he would get the best treatment by top urologists at the best Moscow hospital. And this was a great deal for the doctors too because he would help them analyze their data, which often revealed interesting and previously unknown phenomena. Yakov Isaevich used to say that doctors’ thinking was well adapted to analyzing particular patients and making decisions on a case-by-case basis. But this also made it sometimes difficult for them to focus on the big picture and try to find general patterns and principles. This is where mathematicians become useful because our thinking is entirely different: we are trained to look for and analyze these kinds of general patterns. Yakov Isaevich’s doctor friends appreciated this.

  When I became his student, Yakov Isaevich enlisted me in his medical projects. Ultimately, in the about two and a half years that I worked with Yakov Isaevich, we developed three different projects in urology. The results were used by three young urologists for their doctoral theses. (In Russia, there was a further degree in medicine after M.D., at the level of Ph.D., which required writing a thesis containing original medical research.) I became a co-author of publications in medical journals and even co-authored a patent.