Boris Feigin Borya Feigin, famous mathematician, stout pillar of the Independent University, MMJ board member, a great personality and a dear friend is fifty. This issue and the next one are our small tribute to him. First a few words on his research achievements. Borya's first research subject was cohomology of infinite-dimensional Lie algebras of ``geometric'' origin (such as Lie algebras of vector fields) pioneered by Gelfand and Fuchs back in the 1960's. In his PhD, he considered the cohomology of the Lie algebra of functions on the circle with values in a finite-dimensional Lie algebra and the continuous cohomology of the corresponding infinite-dimensional group (with trivial coefficients). His further articles in this direction are coauthored with Tsygan. Another very important direction of ``non-commutative geometry'' (initiated also in a joint work with Tsygan) is a new approach to the local Riemann—Roch theorem. Next came the representation theory of Kac—Moody and Virasoro algebras. The first major work here, joint with Fuchs, is a detailed study on the structure of Verma modules over the Virasoro algebras and their ``twisted'' versions (Fock modules). Even more important was the technique introduced in this article (the Feigin—Fuchs integrals), which was crucial for the development of conformal field theory. A series of works with E. Frenkel on the representation theory of Kac—Moody algebras began with a construction of Wakimoto modules, which are ``twisted'' versions of Verma modules, and a (geometric) construction of the corresponding ``twisted'' BGG resolutions. Earlier Borya gave the first mathematical construction of the BRST reduction of math physicists (his term was ``semi-infinite homology''). Using it, he, jointly with E. Frenkel, was able to give a general construction (the quantum Drinfeld—Sokolov reduction) of W-algebras and proved their mysterious duality property. A ``classical'' limit of this result, which is the identification of the center of the enveloping Kac—Moody algebra of critical level and the Gelfand—Dickey algebra for the Langlands dual Lie algebra, was conjectured earlier by Drinfeld; it plays an important role in geometric Langlands theory. Several general finiteness theorems for integrable representations of Kac—Moody algebras and similar representations of Virasoro algebras were obtained in his joint work with Fuchs. They were very important for the development of the global conformal field theory on curves. Some other results of his are mentioned below. There is a quite particular feature of Feigin's gift. His published papers and even his research talks hardly comprise a half of his contribution to the domains he is engaged in. Somehow, Borya's presence alone causes them to develop. This (together with the fact that, in I. M. Gelfand's vein, Borya sired only a few non-coauthored papers) makes the number of Borya's colleagues, who consider him to be their teacher, enormous. He is at the Independent University from its beginning. The very image of it would have been different without Borya. His seminar at the Independent University has been for many years, and is, the very center of involvement of the young. Perhaps it is now the closest approximation to the Gelfand seminar of yore. Borya helps many people in need, either in private or by initiating the wanted support. Both his kindness and his independent thought are quite fabulous. We wish Borya many happy returns of the day.
A. Beilinson, A. Belavin, V. Drinfeld, M. Finkelberg,
* * * Borya's company often presaged a Great Adventure, both in mathematics (this was a highlight of his charisma as a teacher) and in daily routine. For example: • Some thirty summers back, in the attic of a dacha, we found a History of the Popes by a certain Léo Taxil.^{1} An auto-da-fé was organized shortly. The book resisted the burning, and the smoldering remnants were drowned in an outhouse. Remarkably, the kindred materials immediately caught fire issuing billows of black smoke. • One winter afternoon we went to see the Novyi Jerusalem monastery near Moscow, which should have been very close to the railroad station of the same name. The elusive monastery, however, kept hiding behind every other grove; after several hours of wading knee deep in snow, we made a whole circle. • Once in Kyoto, we were to launch a rocket from the balcony of a hotel. For some reason, it headed into the room, hitting the walls haphazardly for several memorable seconds prior to choosing a window. My wife was impressed. Borya is not a subject of delusions; he is not inclined to honor anything administrative. Were I to add an illustration to the Respectful Song, it would be Cesar van Everdingen's ``Socrates''.^{2} Borya's mother Marina Borisovna acquainted me, back in 1970s, with many books that have been my joy ever since. When in Moscow (this mirage over the congealed blackness of cruelty of present day Russia), I always return to her kitchen. Borya's wife Inna was among a pack of volunteers introduced into a Moscow hospital by father Alexander Men' (shortly before his martyrdom in September 1990). Many people owe their lives to Inna's grace and persistence. Here in America, the time of my youth is closing around, with another Brezhnev sending troops to Afghanistan and elsewhere. What is still as bright, did change; Borya is far away. From that distance not diminished by the use of technical devices—my salutations and a bow of gratitude.
Alexander Beilinson ^{1} An anti-Catholic writer and fraudster of the end of XIX c. His works inspired, among others, the author(s) of ``Protocols of the Elders of Zion'' and Yemelyan Yaroslavski, and provided the base of the Soviet school curriculum on the history of the Western Church. ^{2} Socrates, His Two Wives, and Alcibiades. Strasbourg, Musée des beaux-arts.
* * * In about 1982, I happened to be at a mathematical conference together with V. Drinfeld, with whom at the moment we were studying the Yang—Baxter equation. From my experience with Instantons and Triangle equations I knew how useful it could be to involve Volodya in a discussion of a mathematical structure arising in the investigation of a physical problem.Therefore I was pestering him with questions on two-dimensional conformal symmetry, since at the same time I was also studying the quantization of Liouville theory together with A. Polyakov and A. Zamolodchikov. Drinfeld advised me to communicate with Borya Feigin, who had just finished a joint paper with D. Fuchs on representations of the Virasoro algebra and who (a mark of destiny!) worked, as I did myself, in Chernogolovka, though not in the Landau Institute, but in the Institute for Condensed Matter Physics. On coming back I found Borya, and we had many talks together. Of what he told us, we could understand almost nothing, yet, in a mysterious way, it was of prime importance for the creation of Conformal Field Theory (A. Belavin, A. Polyakov, A. Zamolodchikov, 1984). Twenty years, filled with a lot of greatly interesting discussions, followed. Borya`s influence on Mathematical Physics of the last 2 decades is difficult to overestimate...
Alexander Belavin
* * * What comes to my mind is not serious. Borya's beloved picture representing a Verma module as an interior angle, oriented—if I remember it well—vertex at the top. At a meeting (was it in Odessa?) Borya explaining his vision of semi-infinite cohomology and BRST, and Sasha Zamolodchikov saying after the talk: now I understand that BRST stands for ``Borya relates string theory'' ^{3}. Here is a funny story rather characteristic for him. In his master year at the university he, just as everyone not allowed to stay for graduate studies, was given some official papers in a stamped envelope porting a severe warning not to be opened by the person concerned but to be given to the authorities at his future place of employment. By pure absentmindedness Borya opened it on the spot, and, fortunately for everyone, since the papers inside were not his but those of another Feigin. I like a lot the photo ^{4} of Borya's class at Moscow high school number 2: Borya is the 3rd from the right in the 2nd row, Andrei Zelevinsky is next to him (2nd from the right).
Vladimir Drinfeld ^{3} The original expression in Russian was ``Боря рассказал старую струну.'' ^{4} See http://www.school2.ru/old-photos/1969-A.htm.
My first meeting with Borya It was a warm sunny day, and Summer was filling the air as I was walking from the train station, aptly named Otdyh (rest), to the dacha of Dmitry Borisovich Fuchs. It was our usual weekly appointment day. During the year we would normally meet at Fuchs' apartment in the southwest end of Moscow. But during the Summer months Fuchs and his family rented a house in Otdyh, a woodsy vacation neighborhood about an hour train ride away from the city, and I would come to visit him there. I had met Fuchs a year earlier, in 1986, when I was a second year student at the Kerosinka (the Moscow University for Oil and Gas which had become home for many students who, like myself, were denied admission to Moscow State University because of their Jewish ethnic background). That year I wrote, under Fuchs' direction, my first paper on the cohomology of braid groups, and after that, following his advice and with his generous help, I started studying representations of infinite-dimensional Lie algebras. Our weekly meetings were the focal point of my studies, and they played a crucial role in my learning of the subject. I felt very excited that day, because I was going to tell Fuchs about a very interesting paper that I had come across. Its author, M. Wakimoto, gave an explicit construction of representations of the affine Lie algebra \widehat{sl}_{2} on a bosonic Fock space. This was fascinating to me, because a few years earlier Fuchs, together with his former student and collaborator Boris Feigin, had given a similar construction of representations of the Virasoro algebra, and I had spent a good part of the preceding year studying their work. Wakimoto's result was both elegant and unexpected, and the similarity between the two constructions was striking. I knew that Fuchs would share my enthusiasm about this paper. When I arrived at the dacha, Fuchs told me that there was a small problem: he had made appointments with me and with Feigin on the same day—inadvertently, he said. I knew that Fuchs had regular meetings with Feigin, who was his collaborator, but until then our appointments had never overlapped (I have always suspected that this ``double booking'' did not happen by accident). I had met Feigin only once before, in the lobby in front of the Gelfand seminar auditorium a few months earlier. Our conversation had a strong impression on me. I had just finished my first paper on braid groups and was thinking about what to do next. Feigin suggested that I study the book ``Statistical mechanics'' by Landau and Lifshitz, a prospect that I found absolutely terrifying at the time, partly because of the resemblance, in size and weight, between that book and the textbook on the history of the Communist Party that we all had to study at school. After Feigin had arrived (by bicycle, from a dacha nearby where he was staying with his family) and we exchanged pleasantries, I set out to describe Wakimoto's work to Feigin and Fuchs. As expected, they were both very interested and asked many questions. This was the first time that I had a chance to talk to Feigin about math, and I had a feeling that we clicked instantly. All in all, this was a very productive discussion. I was so inspired that on the train ride back home I wrote down the formulas generalizing Wakimoto's for \widehat{sl}_{3}, and shortly afterwards for \widehat{sl}_{n}. I started meeting with Feigin on a regular basis, and by the end of the summer we understood the meaning of Wakimoto's construction and generalized it to an arbitrary affine Kac—Moody algebra. Thus Wakimoto modules were born, and my collaboration with Feigin began. I addressed him as Boris Lvovich, in the Russian old-fashioned way, including the patronymic name. It was later, when we were both in the US, that I switched, at his insistence, to the more informal Borya. Borya was the best advisor that I could possibly dream of. He was extremely generous, both with his time and with sharing his ideas and insights. He opened the world of mathematics for me, full of beauty, hidden connections and surprises. Borya is a visionary who has an incredible taste for beautiful mathematics. Talking with him and learning from him always gave me a sense of direction. He never formulated concrete problems for me to work on, but he was somehow able to make sure that I always knew what I wanted to do next. I can appreciate this even more now that I have my own students. A birthday is a good occasion for reflection and recollections. We have had many memorable and funny moments over the years: like greeting the New Year with a bottle of champagne in the Champs Elysées and then missing the last train to Bures-sur-Yvette where we stayed and spending the night wandering around Paris, without a penny in our pockets. Or surviving the 1991 coup in the Soviet Union while visiting Kyoto; we felt so happy when the coup was finally defeated—as evidenced by a photograph ^{5} of both of us, smiling and pumping fists in the air, that was published the next day in Yomiuri... But more importantly, I realize that no one else has made a greater impact on how I do and think about mathematics.
Edward Frenkel ^{5} See http://math.berkeley.edu/~frenkel/Feigin.jpg.
* * * Some people (including Borya himself) call Borya Feigin a student of mine. I proudly accept the title of his teacher, but there are some doubts. His formal advisor in his undergraduate years was I. M. Gelfand; in the graduate school in Yaroslavl he was under A. L. Onishchik. Certainly, besides a ``formal advisor'' some people have an ``informal advisor'' who is their actual teacher (for example, in my own student's years, I had a chain of formal advisors, and only one informal advisor: Albert Schwarz). But at the informal level, I am much more a student of Borya than his teacher. Thank you, Borya, for your lessons. Borya was a third year undergraduate student when we first consciously met. He studied our cohomological papers with Gelfand, we were busy writing new papers, and very soon Borya joined us as a co-author of some of them. But it is not for Borya to follow the ideas of other people. Soon we set up a day for our weekly meetings, and every time Borya appeared with a new idea. It was always very interesting, I waited with impatience for a further development, but next week Borya appeared with a new idea, sometimes even more exciting; last week's project, however, did not interest him any longer. It is this style that makes Borya a unique mathematician: he is always full of ideas, and he willingly shares them with friends. It is good if he is surrounded by people who believe in him and are willing to understand him (some people, not me, find it difficult); otherwise, some precious ideas of him stand the sad chance to be lost. In his young years his style could turn against him. One day I had to insist that he finish some work: he needed a ``diploma work'' to graduate from the University. As a result of my most stubborn pressure, there appeared his article `On characteristic classes of flags of foliations'', and Borya got his degree. The times were hard, ``the period of flourishing of the period of stagnation'', as Sasha Kirillov once said. Borya's Jewish origin, aggravated by a ``bad'' political record (for example, Borya used to cross out the names of the official candidates in the ballot right before the eyes of astonished election committee members at the days of so-called ``elections''), barred him from graduate school. Borya was hired as a computer programmer by a remote firm. His experience there was diverse: on one hand, he was at a constant conflict with his bosses, because of his full inability to come to his workplace on time; on the other hand, years after his leaving the firm, it kept on receiving monetary rewards for programs written by him, and sometimes Borya even received his share (which was important for him, since his family has never been well off). Then there was a turn to the better. Friends helped him to get into the graduate school in Yaroslavl, and he eventually got his PhD. Then he got a position at the Institute of Theoretical Physics, and as a result of this he was able to return to active mathematical research. At the late 70-ies, we started working together. We did some work on cohomology of Lie algebras (this work was nicknamed ``parabolic'', because of the most characteristic picture from it), and then began studying representation of the Virasoro algebra (a ``hyperbolic'' work). Sometimes, other people joined us (we wrote a paper about representations of the Kac—Moody algebras with F. Malikov, and a work on multiplications in the cohomology of Lie algebras of vector fields with V. Retach). Simultaneously, Borya participated in some other projects (he worked with B. Tsygan, A. Odesski, and E. Frenkel; in the last case, I played the important role of a matchmaker—see some details in Edward Frenkel's note). When (in 1990) I said to Borya that I was going to spend a year at Stanford, he said, ``nobody ever returns from America; and you will never return''. I do not remember a single case when Borya's prediction proved to be wrong. Now we meet from time to time. Time is passing by. Borya writes joint papers with his son Eugene (whom I first met when he was 8 days old). He travels between Russia, Japan and America, but still lives in Moscow, in the same apartment as 25 years ago, and his phone number has not changed.
Dmitry B. Fuchs
* * * It was March, 1990, when I first met Borya, at the Steklov institute, Moscow. At the entrance I took off my fur coat, which was given to me for a second hand use from my father. Someone who also took off his coat came to me and said, ``I am Boris Feigin''. It was my third visit to Moscow. It was the time of very rapid change in Russia. In my previous visits I stayed in the Academy Hotel. This was the only place where foreign scientists could stay. I was surprised when I knew that my hotel was not the Academy Hotel. One morning that week Borya and Sasha Beilinson came to the hotel and we walked to the subway station by a short cut across somebody's yard covered by snow. On the train to his flat, we talked about Russia, how it would change; something we never discussed several years ago. I had a mission from the organizing committee of ICM 90 in Kyoto, to obtain Borya's agreement on the chairman of his plenary talk. He said he did not care. I had my own purpose to meet Borya, to invite him to the satellite conference in Okayama, and also to the RIMS 91 project on Infinite Analysis. He said he would come. I was very pleased with his answers. However, neither Borya nor I imagined at that moment that his visits would repeat every year. Borya likes Kyoto. I believe he likes the atmosphere of Kyoto in mathematics. I know another thing he finds interesting. It is cycling. If someone, who will visit Kyoto in summer, asks me about availability of bicycles, I write ``Ask Borya''. He owns 7 bicycles for him, his family and his friends. Several years ago, I invited him and other people to my dacha, one and a half hours away from the city by car across mountains. They came by train. Next weekend, he and Loktev showed up again on their bikes after a 6 hour ride! They stayed for lunch and went back to the city by a most complicated route, spending another 8 hours. I was lucky that I was abroad when RIMS was called by the police and asked to pick up a Russian mathematician stopped by the police just before entering a tunnel on the Kyoto—Shiga highway.
Tetsuji Miwa
* * * I met Borya in early 1982. As a Jew, I was not admitted to Moscow State University in 1975, but I was able to get to Kiev University. Then, in 1980, I was not admitted to graduate school, so I had to work in a place that had very little to do with mathematics. I had to be there from eight to five to do some work, but they were understanding when I spent some time on mathematics. The education in Kiev had its limitations, but I did have some first-rate instruction, including excellent courses on algebraic geometry (Yu. A. Drozd) and on Lie algebras and their cohomology (A. K. Tolpygo). My undergraduate advisor Yu. S. Samoilenko introduced me to gauge groups. So it was quite natural for me to get interested in the following topic: how the topology of a manifold (say, a circle) can be algebraically described in terms of the cohomology of the Lie algebra of matrix-valued functions. This led me to the computation of this cohomology in terms of what is now called cyclic cohomology. The latter was introduced by Connes that very year. On the advise of A. M. Vershik, I called Borya and formulated the new theorem to him. He got quite interested. Soon he called back, not just with the proof but with a couple of new results in that direction. Thus started our collaboration which continued for a number of years. We met in person several times in 1982, in Moscow, Kiev, and Leningrad. That summer, I remember, Borya amazed me by saying that he did not know that the soccer world cup was going on. He compensated for that lack of erudition by wonderful displays of knowledge of various subjects of mathematics, as well as history and literature. I especially remember a fascinating two-hour lecture on etale cohomology at the Palace square in Leningrad. (Another, unexpected piece of knowledge puzzled me at that time. In Kiev, he said confidently that Lenin had never been there. Now, I grew up as a young pioneer in Kiev, and we were told about that unfortunate fact in the Communist spirit of openness and honesty. But I could not figure out how Borya knew it. I tried to think about any other person about whom I knew for sure that he or she had never been to Kiev. Napoleon came to mind quickly; I could not name any Russian. In addition to a very original angle at which Borya sees things, he knows a lot of hard facts.) It was Borya who, together with A. M. Vershik, came up with the idea that I should apply to the Moscow University graduate school, with Yu. I. Manin as advisor. That seemed hopeless to me, but it exploited a bug in the discrimination system that was in place at that time. Namely, those who applied directly from undergraduate school needed to clear several hurdles, like to get recommendations from the Party committee, etc. It was at that stage that Jews were excluded. But those few who applied two years or more after college needed no such thing. So I applied and got accepted, and got a wonderful opportunity to spend three years in the wonderful mathematical atmosphere of Moscow. During these years I had a great delight that the contact with Borya can offer. We worked together on various topics of what is now called noncommutative geometry. He was always willing to tell me about other subjects, be it statistical mechanics, strings, representation theory, or number theory. Later, he tried to understand the appearance of the Riemann—Roch theorem in string theory. That led him to a fascinating idea how to use noncommutative geometry and formal geometry to understand the Riemann—Roch. This idea is still a cornerstone of my current research. I have the fondest memories about our human contacts with Borya and all his family, in particular with his wonderful mother Marina Borisovna, to whom I am very grateful both for her warm and generous hospitality and for exciting discussions of music, culture, religion, and many other things. (It was at her place that I first read ``Gulag Archipelago'' and ``1984''. I remember that, when I was reading the latter, Borya entered the room. He asked me what I was reading; before answering, I looked at the ceiling to see where the telescreen was.) On a related topic, that was the time when you needed a special paper to enter the building of the University. It was not easy to get. I remember Borya and me in the courtyard, looking at Sasha Beilinson balancing on the top of the grill with a large bag, negotiating with a policeman who caught him trying to climb over. Borya sighs calmly and says: ``And the bag is full of antisovietchina ^{6}. A dangerous man, god damn it.'' After I left for America in 1989, we met a few times, but it was not before 2000 when I had a long contact with Borya. That was in Kyoto, a city of great beauty which gives you a feeling of peace, ease, and freedom. It was in that surroundings that I understood what, for me, is so appealing about Borya. Being around him gives me a sense of ease and freedom. I am not sure how to describe it, and maybe not everybody will apply these words to conversations full of long silences, unanswered questions, and cryptic remarks. To complement Sasha's story about New Jerusalem: I participated in a traditional Feigin family bike trip to the ancient capital of Nara. Nara and Kyoto are both very beautiful, but the Route 24 that connects them is just a very busy road in commercial and industrial surroundings. It is a few hours one way, and it is very, very hot there in August. (When I opened a bank account in Kyoto, I arrived at the bank after a leisurely 15 minute bike ride. They usually give a small present to a new customer. After looking at me, they gave me a bar of soap.) Whatever could be saved on train tickets was spent on ice-cream and water. But it was a great trip. Let me finish by discussing two of Borya's ideas that show, in my view, the power of his insight. First, he told me in mid-80s that Floer cohomology should be, in some nontrivial way, related to Ext functors between modules over a deformed algebra of functions on a symplectic manifold. Since then, Floer cohomology was related to mirror symmetry through the works of Kontsevich and Fukaya. Noncommutative geometry of deformed algebras now routinely enters the study of dualities, like T-duality; and the Fukaya category is more and more treated as if it were the derived category of modules over a ring; in particular, its Hochschild and cyclic complexes are being studied. Secondly, during our joint work on noncommutative geometry, he always insisted on, and made a lot of effort towards, a systematic study of operations acting on Hochschild and cyclic complexes of algebras. Such a study was later crucial for Tamarkin's approach to Kontsevich formality theorem. In Kyoto, Borya shared with me his really fascinating and original ideas about formality. I believe that they have a future as big and as fruitful as many of his other ideas in various fields. Happy Birthday, Borya!
Boris Tsygan ^{6} Literature considered to be anti-Soviet by the authorities.
* * * Borya Feigin and I have been classmates since the seventh grade and continued studying together all the way through the graduate school at Moscow State University. He joined our class at Moscow School number 2 in the middle of the school year, using this opportunity for skipping a grade when transferring from another school. This was a well-known exam school at the time, and our class was full of bright and ambitious kids with a wide range of interests in all things intellectual: mathematics, literature, history, etc. Despite being a year younger than most of his classmates, Borya was not lost in this crowd. It was immediately clear that there was something very special about him. He knew and thought about a lot of things, and he already possessed his typical cryptic way of expressing himself, and his enigmatic smile. I have always found talking to him enormously stimulating and thought-provoking. Unfortunately, we do not see each other very often these days. The last time he visited me at my present home in Sharon, Massachusetts, I took him to the beach at our lake Massapoag which is somewhat similar to the Kratovo lake where we used to hang out and swim in old days. It was late in the evening on a rather cold April day, but Borya did not hesitate for a moment: he just jumped into the water and would undoubtedly have made it all five miles to the other side of the lake, if we only had more time. Happy birthday, Borya, and may you have many more amazing achievements.
Andrei Zelevinsky |
Moscow Mathematical Journal |