The book "Love and Mathematics. The heart of the hidden reality "
Hello! We stocked the second edition of the book Love and Mathematics. The heart of the hidden reality ", which was published in conjunction with the fund " Dynasty " Review of the book was already on Habré . Here we will publish a chapter from the book Conquest of the Peak.
“My goal is not to teach you something. I want to give you the opportunity to feel that there is a whole world that is diligently hidden from us - the world of mathematics. This is a portal to an uncharted reality, the key to understanding the deep secrets of the Universe and ourselves. Mathematics is not the only portal, there are others. But in a sense he is the most obvious. And that is why he is so camouflaged, as if a board with the inscription: “You don’t need to come here.” But actually it is necessary. And when we enter it, we remember who we are: not the small cogs of a big machine, not the lonely souls who live in the suburbs of the Universe. We are the Creators of this world, capable of giving each other beauty and love. ” - Edward Frenkel.
')
By the summer I felt ready to share my findings with Fuchs. I knew that Wakimoto’s article would interest him as much as I did. By tradition, I went to the cottage to Fuchs. However, when I arrived there, Fuchs informed me that a small overlap had occurred: he made an appointment with me and his colleague and former pupil Boris Feigin for the same time - according to him, quite by accident, which I, of course, did not believed (and much later, Fuchs confirmed that, indeed, he did it intentionally).
Fuchs introduced me to Fegin a few months earlier. This happened before one of the Gelfand seminars, shortly after I finished my article on the braid groups and began reading the paper of Feigin and Fuchs. On Fuchs advice, I asked Feigin to recommend additional literature, which I should have read. Boris L'vovich — I turned to him then — at that time was thirty-three years old, but he was already considered one of the brightest stars of the Moscow mathematical community. Dressed in jeans and a pair of well-worn sneakers, he looked like a modest, even shy person. Thick glasses were sitting on his nose, and most of our conversation, he looked at the floor, avoiding eye contact. Of course, I, too, was not too confident in myself - just an aspiring student, who had the honor to talk with a famous mathematician. In general, it was not a very exciting conversation. And yet from time to time Feigin looked up and threw a quick glance at me, accompanying him with a broad disarming smile. It melted the ice, and I had no doubt that in reality he was a sincere and benevolent person.
However, Feigin's first recommendation baffled me: he advised me to read the book of Landau and Lifshits called “Statistical Physics”. This perspective really alarmed me - partly because of the similarity (in size and weight) between this thick volume and the textbooks on the history of the Communist Party, which we had to study at the institute.
In defense of Feigin, I will say that the advice turned out to be good. This is an important and useful book; moreover, my own research turned just in that direction (although to my shame I must admit that I still did not finish the book). However, at that moment, the idea of ​​familiarizing myself with this monumental work did not inspire me at all, and I think this is partly why our first conversation quickly faded. In fact, I no longer talked with Feigin until that day at Fuchs' dacha, except for the traditional “hello” when meeting at Gelfand seminars.
Shortly after my arrival at Fuchs' dacha, I saw Feigin in the window - he arrived by bicycle because he lived in a country house nearby. We greeted each other, talked a little about this and that, and sat down at a round table in the kitchen. Fuchs asked me:
- So what's new?
“Well ... I found one interesting article by the Japanese mathematician Wakimoto.”
“Hmmm ...” Fuchs turned to Feigin: “Did you hear about that?”
Feigin shook his head, and Fuchs turned to me again:
- He always knows everything ... It's good that he didn’t come across this article - it means that he will also be interested to listen to you.
I began to describe everything that I learned from the work of Wakimoto. As I thought, it was very interesting for both of them. Then for the first time, Feygin and I had a chance to talk about mathematical concepts, and I immediately felt that we were tuned in to one wave. He listened carefully and asked the right questions. It was quite obvious that he understood the importance of this information, and despite the fact that he seemed calm and relaxed in appearance, he was obviously disturbed by what he heard. Fuchs for the most part just watched us and, I suspect, silently rubbed his hands with joy: his secret plan to get us closer to Feigin worked so wonderfully. It was a truly inspiring conversation. I had no doubt that I was only one step away from some extremely important discovery.
Fuchs clearly shared my confidence. When I left, he said to me:
- Well done! It is a pity that you did not write this article. But I think you are ready to take this work to a new level.
Upon returning home, I continued to think about the issues raised in the Wakimoto article. Since Wakimoto did not give any explanations to the formulas given in his work, I had to play the role of a peculiar forensic expert in order to track the connection between these formulas and the overall picture. A few days later, the picture began to emerge. I measured my dorm room thoughtfully when suddenly it hit me: Wakimoto's formulas come from geometry! It was an amazing and unexpected discovery, because Wakimoto's approach was completely algebraic - not a hint of geometry.
In order to explain my geometric interpretation, we need to again visit the Lie group SO (3) symmetries of the sphere and its loop group. As I explained in the previous chapter, each element of the SO (3) loop group is a set of SO (3) elements, one SO (3) element for each loop point. Each of the elements of SO (3) acts on a sphere as a definite rotation. This implies that each element of the loop group SO (3) generates symmetry of the loop space of the sphere.
I realized that this information can be used to search for a Kac-Moody algebra representation related to SO (3). But it still did not give us the Wakimoto formulas. In order to obtain them, it was necessary to modify the formulas in a certain very radical way - to perform an operation with them similar to turning the coat inside out. We can do this with any coat, but in most cases, wearing it after that will not work - at least in public. However, some coats are specially sewn so that they can be worn either side up. As it turned out, this was also true for the Wakimoto formulas.
Armed with this idea, I immediately tried to extend the Wakimoto formulas to other, more complex Kac-Moody algebras. The first, geometric step, I succeeded without any difficulties - everything was the same as in the case of SO (3). However, the next stage - turning the formulas "inside out" - gave some sort of nonsense. The result simply did not make sense. I began to turn the formula and so, and then, but no tricks did not help solve the problem. I began to seriously consider the possibility that this construction works only for SO (3), but not for Kac-Moody algebras of a more general form. I had no opportunity to find out whether this problem is being solved in principle, and if so, can it be reached using the tools at our disposal. I could only work hard and hope for the best.
A week passed, and it was time for our next meeting with Fuchs. I planned to show him my calculations and ask for advice. When I arrived at the dacha, Fuchs said that his wife had to go to Moscow for some urgent business, and he should look after their two young daughters.
“But you know what,” Fuchs continued, “Feigin was here yesterday, and he was delighted with what you told us last week.” Why don't you go to him? Before his giving only fifteen minutes. I warned him that I will send you to him, so he is waiting for you.
He explained the way to me, and I went to Feygin’s country house. Feigin was really expected of me. Boris Lvovich warmly greeted me and introduced me to his charming wife Inna and three children: two lively boys, Roma and Zhenya, eight and ten years old, and a lovely two-year-old daughter Liza. At that time I did not know that the warmest friendship would bind me with this wonderful family for many years.
Feigin's wife offered me a cup of tea and a piece of cake, and we settled on the veranda. It was a beautiful summer evening, the rays of the sun were breaking through the dense foliage of the trees, the birds were chirping - a real idyll. However, of course, our conversation quickly turned to the construction of Wakimoto.
Feigin admitted that he also devoted a lot of time to thinking about them, and the course of his thoughts was similar to mine. From the very beginning of the conversation, we continually ended sentences for each other. It was amazing: he completely understood me, and I understood him.
I began to talk about the failure that I suffered when I tried to extend the construction to other Kac – Moody algebras. Feigin listened to me with great attention, paused a little while thinking about the problem, and then drew my attention to one important thing that I missed. One of the steps of the generalization of the Wakimoto construction is the search for a suitable generalization of a sphere — a manifold on which SO (3) acts by symmetries. In the case of SO (3), the choice is, in fact, unequivocal. However, there may be many more options for other groups. In my calculations, I proceeded from the fact that the so-called projective spaces are natural generalizations of the sphere. I took it for granted. But it is quite possible that I was wrong; It is possible that I did not succeed because of the wrong choice of spaces.
As I explained above, ultimately, I needed to turn the formulas “inside out”. All of my construction was based on the expectation that in some miraculous way the formulas that would result from inversion would remain valid. For Wakimoto, for the simplest group SO (3), everything turned out well. My calculations showed that this is not the case for projective spaces, and yet this did not mean that there is no other, better construction. Feigin suggested that I try to consider the so-called flag varieties.
The flag variety for the SO (3) group is a sphere that has long been familiar to us, so for other groups, such spaces can be considered as natural substitutes for the sphere. At the same time, flag varieties of richer and more diversified projective spaces, so that one could hope that the analogue of the construction of Wakimoto would really work for them.
It was getting dark, and I had to hurry home. We agreed to meet again in a week, I said goodbye to the Feigin family and went to the train station.
On the way home in an empty carriage, through the open windows of which warm summer air penetrated, I continued to reflect on the task. I had to try to solve it - here and now. I pulled out a pen and notebook and began writing formulas for the simplest flag variety. The old train carriage was noisily bouncing at the rail junctions and swaying from side to side. I could not keep the pen straight, and the formulas spread all over the page — I myself could hardly make out my notes. However, in the midst of all this chaos, a system was clearly born. Unlike projective spaces, which I had unsuccessfully tried to tame all week before, with flag varieties, I really began to take shape.
A few more lines of calculations, and ... Eureka! Happened! The “inside-out” formulas worked as clearly and coherently as in the Wakimoto article. The construction was summarized in the most elegant way. I was overwhelmed with joy: I did it! I solved the problem, I found new implementations of Kac – Moody algebras in the free field!
The next morning, I carefully checked my calculations. It all came together. Feigin did not have a telephone at the dacha, so I could not call him and immediately tell me about my findings. To begin with, I put everything in the form of a letter, and told Feygin about the new results next week, when we met again. So began our many years of collaboration. Feigin became my teacher, mentor, leader, friend. At first I addressed him by name and patronymic, Boris Lvovich. But later he insisted that I use a more informal appeal - Boris.
In my life I was incredibly lucky with the teachers. Evgeny Evgenievich revealed to me the beauty of mathematics and helped me fall in love with it. He taught me the basics. Fuchs saved me after a catastrophe at the entrance exam at Moscow State University and ensured the rapid start of my unsure until that point mathematical career. He supervised me throughout my first serious mathematical project, thanks to which I believed in myself, and led me to a fascinating field of research at the interface of mathematics and physics. I was finally ready for the big game. At this stage of my journey, I could not have found a better scientific leader than Boris. My mathematical career began to gain momentum, as if driven by the power of a jet engine.
Without a doubt, Boria Feigin is one of the most original and outstanding mathematicians of his generation all over the world, a seer with an innate instinct for mathematics. He became my guide to the wonderful world of modern mathematics, full of magical beauty and magnificent harmony.
Now that I have got my own students, I will even more appreciate all that Boria has done for me (as well as Evgeny Evgenievich and Fuchs before him). Teaching others is not easy at all! Perhaps in many ways this is similar to parenting experience. You have to sacrifice a lot without asking for anything in return. Of course, the reward can be huge! However, how to decide which direction to tell students when to give them a helping hand, and when it is wiser to push them from the boat into the lake to learn how to swim independently? This is real art. No one can teach this.
Boria really cared about me and my development as a mathematician. He never told me what to do, but communicating with him and learning from him, I was always sure which direction to go on. Somehow, he did so that I always knew what to do. And, feeling that he was always there and always supported me, I had no doubt that I had chosen the right path. I was terribly lucky that my teacher was such an amazing person.
The fall semester began in 1987; it was my fourth year in kerosene. I was nineteen, and my life was never more interesting and exciting. I still lived in a hostel, met with friends, fell in love ... But I did not forget about my studies. By that time, I missed most of the classes, and I was preparing for exams on my own (it also happened that I first took up the textbook just a couple of days before the exam). In all subjects I had only fives. The only exception was the Quartet on Marxist political economy (shame on me).
The fact that I actually had a “second life” —the mathematical work with Boreas, which occupied most of my time and demanded the greatest efforts, I hid from most people in my environment.
As a rule, we met with Boreas twice a week. Officially, he held a position at the Institute of Solid State Physics, but he didn’t have much to do there, so it was enough for him to appear at the institute once a week. On other days, he worked in his mother's apartment, to which it was ten minutes from his house. To Kerosene and my hostel was also close. This was our traditional venue. I came late in the morning or just after noon, and we worked on our projects, sometimes lingering until the very night. Mama Bory came home from work in the evening and fed us dinner, and very often Borey and I left together for about nine or ten hours.
First of all, we made a short article about our results and sent it to the journal Uspekhi Matematicheskikh Nauk. It was published in about a year - rather quickly by the standards of mathematical journals.3 Having dealt with this issue, we focused on the further development of our project. We used the results to better understand the representations of Kac-Moody algebras. Our work also allowed us to detect the implementation of two-dimensional quantum models in a free field. Because of this, we were able to perform calculations in these models that were previously completely inaccessible. As a result, physicists soon began to show interest in our research.
It was a terrific time. In those days when I did not meet with Borey, I worked independently - during the week in Moscow, and on the way out at home. I continued to visit the Scientific Library and swallowed more and more books and articles. Math was my life, my food and water. It was as if I moved into a beautiful parallel Universe, and I wanted to stay there forever, plunging deeper and deeper into this dream. With each new discovery, with each new idea, this magical world became closer and dearer to me.
However, by the autumn of 1988, when the fifth, last year of my studies at the institute began, I had to return to a cruel reality. I had to think about the future. Despite the fact that I was one of the best in the course, the prospects were awaiting me quite bleak. Anti-Semitism reigned not only in graduate school, but also in all institutions where a graduate could get a good job. Complicating matters is also the fact that I did not have a Moscow residence permit. The day of reckoning was near.
More information about the book can be found on the publisher's website.
All books from the New Science series are here.
Table of contents
Excerpt
For readers of this benefit, a 25% discount on the coupon - Frenkel
“My goal is not to teach you something. I want to give you the opportunity to feel that there is a whole world that is diligently hidden from us - the world of mathematics. This is a portal to an uncharted reality, the key to understanding the deep secrets of the Universe and ourselves. Mathematics is not the only portal, there are others. But in a sense he is the most obvious. And that is why he is so camouflaged, as if a board with the inscription: “You don’t need to come here.” But actually it is necessary. And when we enter it, we remember who we are: not the small cogs of a big machine, not the lonely souls who live in the suburbs of the Universe. We are the Creators of this world, capable of giving each other beauty and love. ” - Edward Frenkel.')
Chapter 11. The Conquest of the Peak
By the summer I felt ready to share my findings with Fuchs. I knew that Wakimoto’s article would interest him as much as I did. By tradition, I went to the cottage to Fuchs. However, when I arrived there, Fuchs informed me that a small overlap had occurred: he made an appointment with me and his colleague and former pupil Boris Feigin for the same time - according to him, quite by accident, which I, of course, did not believed (and much later, Fuchs confirmed that, indeed, he did it intentionally).
Fuchs introduced me to Fegin a few months earlier. This happened before one of the Gelfand seminars, shortly after I finished my article on the braid groups and began reading the paper of Feigin and Fuchs. On Fuchs advice, I asked Feigin to recommend additional literature, which I should have read. Boris L'vovich — I turned to him then — at that time was thirty-three years old, but he was already considered one of the brightest stars of the Moscow mathematical community. Dressed in jeans and a pair of well-worn sneakers, he looked like a modest, even shy person. Thick glasses were sitting on his nose, and most of our conversation, he looked at the floor, avoiding eye contact. Of course, I, too, was not too confident in myself - just an aspiring student, who had the honor to talk with a famous mathematician. In general, it was not a very exciting conversation. And yet from time to time Feigin looked up and threw a quick glance at me, accompanying him with a broad disarming smile. It melted the ice, and I had no doubt that in reality he was a sincere and benevolent person.
However, Feigin's first recommendation baffled me: he advised me to read the book of Landau and Lifshits called “Statistical Physics”. This perspective really alarmed me - partly because of the similarity (in size and weight) between this thick volume and the textbooks on the history of the Communist Party, which we had to study at the institute.
In defense of Feigin, I will say that the advice turned out to be good. This is an important and useful book; moreover, my own research turned just in that direction (although to my shame I must admit that I still did not finish the book). However, at that moment, the idea of ​​familiarizing myself with this monumental work did not inspire me at all, and I think this is partly why our first conversation quickly faded. In fact, I no longer talked with Feigin until that day at Fuchs' dacha, except for the traditional “hello” when meeting at Gelfand seminars.
Shortly after my arrival at Fuchs' dacha, I saw Feigin in the window - he arrived by bicycle because he lived in a country house nearby. We greeted each other, talked a little about this and that, and sat down at a round table in the kitchen. Fuchs asked me:
- So what's new?
“Well ... I found one interesting article by the Japanese mathematician Wakimoto.”
“Hmmm ...” Fuchs turned to Feigin: “Did you hear about that?”
Feigin shook his head, and Fuchs turned to me again:
- He always knows everything ... It's good that he didn’t come across this article - it means that he will also be interested to listen to you.
I began to describe everything that I learned from the work of Wakimoto. As I thought, it was very interesting for both of them. Then for the first time, Feygin and I had a chance to talk about mathematical concepts, and I immediately felt that we were tuned in to one wave. He listened carefully and asked the right questions. It was quite obvious that he understood the importance of this information, and despite the fact that he seemed calm and relaxed in appearance, he was obviously disturbed by what he heard. Fuchs for the most part just watched us and, I suspect, silently rubbed his hands with joy: his secret plan to get us closer to Feigin worked so wonderfully. It was a truly inspiring conversation. I had no doubt that I was only one step away from some extremely important discovery.
Fuchs clearly shared my confidence. When I left, he said to me:
- Well done! It is a pity that you did not write this article. But I think you are ready to take this work to a new level.
Upon returning home, I continued to think about the issues raised in the Wakimoto article. Since Wakimoto did not give any explanations to the formulas given in his work, I had to play the role of a peculiar forensic expert in order to track the connection between these formulas and the overall picture. A few days later, the picture began to emerge. I measured my dorm room thoughtfully when suddenly it hit me: Wakimoto's formulas come from geometry! It was an amazing and unexpected discovery, because Wakimoto's approach was completely algebraic - not a hint of geometry.
In order to explain my geometric interpretation, we need to again visit the Lie group SO (3) symmetries of the sphere and its loop group. As I explained in the previous chapter, each element of the SO (3) loop group is a set of SO (3) elements, one SO (3) element for each loop point. Each of the elements of SO (3) acts on a sphere as a definite rotation. This implies that each element of the loop group SO (3) generates symmetry of the loop space of the sphere.
I realized that this information can be used to search for a Kac-Moody algebra representation related to SO (3). But it still did not give us the Wakimoto formulas. In order to obtain them, it was necessary to modify the formulas in a certain very radical way - to perform an operation with them similar to turning the coat inside out. We can do this with any coat, but in most cases, wearing it after that will not work - at least in public. However, some coats are specially sewn so that they can be worn either side up. As it turned out, this was also true for the Wakimoto formulas.
Armed with this idea, I immediately tried to extend the Wakimoto formulas to other, more complex Kac-Moody algebras. The first, geometric step, I succeeded without any difficulties - everything was the same as in the case of SO (3). However, the next stage - turning the formulas "inside out" - gave some sort of nonsense. The result simply did not make sense. I began to turn the formula and so, and then, but no tricks did not help solve the problem. I began to seriously consider the possibility that this construction works only for SO (3), but not for Kac-Moody algebras of a more general form. I had no opportunity to find out whether this problem is being solved in principle, and if so, can it be reached using the tools at our disposal. I could only work hard and hope for the best.
A week passed, and it was time for our next meeting with Fuchs. I planned to show him my calculations and ask for advice. When I arrived at the dacha, Fuchs said that his wife had to go to Moscow for some urgent business, and he should look after their two young daughters.
“But you know what,” Fuchs continued, “Feigin was here yesterday, and he was delighted with what you told us last week.” Why don't you go to him? Before his giving only fifteen minutes. I warned him that I will send you to him, so he is waiting for you.
He explained the way to me, and I went to Feygin’s country house. Feigin was really expected of me. Boris Lvovich warmly greeted me and introduced me to his charming wife Inna and three children: two lively boys, Roma and Zhenya, eight and ten years old, and a lovely two-year-old daughter Liza. At that time I did not know that the warmest friendship would bind me with this wonderful family for many years.
Feigin's wife offered me a cup of tea and a piece of cake, and we settled on the veranda. It was a beautiful summer evening, the rays of the sun were breaking through the dense foliage of the trees, the birds were chirping - a real idyll. However, of course, our conversation quickly turned to the construction of Wakimoto.
Feigin admitted that he also devoted a lot of time to thinking about them, and the course of his thoughts was similar to mine. From the very beginning of the conversation, we continually ended sentences for each other. It was amazing: he completely understood me, and I understood him.
I began to talk about the failure that I suffered when I tried to extend the construction to other Kac – Moody algebras. Feigin listened to me with great attention, paused a little while thinking about the problem, and then drew my attention to one important thing that I missed. One of the steps of the generalization of the Wakimoto construction is the search for a suitable generalization of a sphere — a manifold on which SO (3) acts by symmetries. In the case of SO (3), the choice is, in fact, unequivocal. However, there may be many more options for other groups. In my calculations, I proceeded from the fact that the so-called projective spaces are natural generalizations of the sphere. I took it for granted. But it is quite possible that I was wrong; It is possible that I did not succeed because of the wrong choice of spaces.
As I explained above, ultimately, I needed to turn the formulas “inside out”. All of my construction was based on the expectation that in some miraculous way the formulas that would result from inversion would remain valid. For Wakimoto, for the simplest group SO (3), everything turned out well. My calculations showed that this is not the case for projective spaces, and yet this did not mean that there is no other, better construction. Feigin suggested that I try to consider the so-called flag varieties.
The flag variety for the SO (3) group is a sphere that has long been familiar to us, so for other groups, such spaces can be considered as natural substitutes for the sphere. At the same time, flag varieties of richer and more diversified projective spaces, so that one could hope that the analogue of the construction of Wakimoto would really work for them.
It was getting dark, and I had to hurry home. We agreed to meet again in a week, I said goodbye to the Feigin family and went to the train station.
On the way home in an empty carriage, through the open windows of which warm summer air penetrated, I continued to reflect on the task. I had to try to solve it - here and now. I pulled out a pen and notebook and began writing formulas for the simplest flag variety. The old train carriage was noisily bouncing at the rail junctions and swaying from side to side. I could not keep the pen straight, and the formulas spread all over the page — I myself could hardly make out my notes. However, in the midst of all this chaos, a system was clearly born. Unlike projective spaces, which I had unsuccessfully tried to tame all week before, with flag varieties, I really began to take shape.
A few more lines of calculations, and ... Eureka! Happened! The “inside-out” formulas worked as clearly and coherently as in the Wakimoto article. The construction was summarized in the most elegant way. I was overwhelmed with joy: I did it! I solved the problem, I found new implementations of Kac – Moody algebras in the free field!
The next morning, I carefully checked my calculations. It all came together. Feigin did not have a telephone at the dacha, so I could not call him and immediately tell me about my findings. To begin with, I put everything in the form of a letter, and told Feygin about the new results next week, when we met again. So began our many years of collaboration. Feigin became my teacher, mentor, leader, friend. At first I addressed him by name and patronymic, Boris Lvovich. But later he insisted that I use a more informal appeal - Boris.
In my life I was incredibly lucky with the teachers. Evgeny Evgenievich revealed to me the beauty of mathematics and helped me fall in love with it. He taught me the basics. Fuchs saved me after a catastrophe at the entrance exam at Moscow State University and ensured the rapid start of my unsure until that point mathematical career. He supervised me throughout my first serious mathematical project, thanks to which I believed in myself, and led me to a fascinating field of research at the interface of mathematics and physics. I was finally ready for the big game. At this stage of my journey, I could not have found a better scientific leader than Boris. My mathematical career began to gain momentum, as if driven by the power of a jet engine.
Without a doubt, Boria Feigin is one of the most original and outstanding mathematicians of his generation all over the world, a seer with an innate instinct for mathematics. He became my guide to the wonderful world of modern mathematics, full of magical beauty and magnificent harmony.
Now that I have got my own students, I will even more appreciate all that Boria has done for me (as well as Evgeny Evgenievich and Fuchs before him). Teaching others is not easy at all! Perhaps in many ways this is similar to parenting experience. You have to sacrifice a lot without asking for anything in return. Of course, the reward can be huge! However, how to decide which direction to tell students when to give them a helping hand, and when it is wiser to push them from the boat into the lake to learn how to swim independently? This is real art. No one can teach this.
Boria really cared about me and my development as a mathematician. He never told me what to do, but communicating with him and learning from him, I was always sure which direction to go on. Somehow, he did so that I always knew what to do. And, feeling that he was always there and always supported me, I had no doubt that I had chosen the right path. I was terribly lucky that my teacher was such an amazing person.
The fall semester began in 1987; it was my fourth year in kerosene. I was nineteen, and my life was never more interesting and exciting. I still lived in a hostel, met with friends, fell in love ... But I did not forget about my studies. By that time, I missed most of the classes, and I was preparing for exams on my own (it also happened that I first took up the textbook just a couple of days before the exam). In all subjects I had only fives. The only exception was the Quartet on Marxist political economy (shame on me).
The fact that I actually had a “second life” —the mathematical work with Boreas, which occupied most of my time and demanded the greatest efforts, I hid from most people in my environment.
As a rule, we met with Boreas twice a week. Officially, he held a position at the Institute of Solid State Physics, but he didn’t have much to do there, so it was enough for him to appear at the institute once a week. On other days, he worked in his mother's apartment, to which it was ten minutes from his house. To Kerosene and my hostel was also close. This was our traditional venue. I came late in the morning or just after noon, and we worked on our projects, sometimes lingering until the very night. Mama Bory came home from work in the evening and fed us dinner, and very often Borey and I left together for about nine or ten hours.
First of all, we made a short article about our results and sent it to the journal Uspekhi Matematicheskikh Nauk. It was published in about a year - rather quickly by the standards of mathematical journals.3 Having dealt with this issue, we focused on the further development of our project. We used the results to better understand the representations of Kac-Moody algebras. Our work also allowed us to detect the implementation of two-dimensional quantum models in a free field. Because of this, we were able to perform calculations in these models that were previously completely inaccessible. As a result, physicists soon began to show interest in our research.
It was a terrific time. In those days when I did not meet with Borey, I worked independently - during the week in Moscow, and on the way out at home. I continued to visit the Scientific Library and swallowed more and more books and articles. Math was my life, my food and water. It was as if I moved into a beautiful parallel Universe, and I wanted to stay there forever, plunging deeper and deeper into this dream. With each new discovery, with each new idea, this magical world became closer and dearer to me.
However, by the autumn of 1988, when the fifth, last year of my studies at the institute began, I had to return to a cruel reality. I had to think about the future. Despite the fact that I was one of the best in the course, the prospects were awaiting me quite bleak. Anti-Semitism reigned not only in graduate school, but also in all institutions where a graduate could get a good job. Complicating matters is also the fact that I did not have a Moscow residence permit. The day of reckoning was near.
More information about the book can be found on the publisher's website.
All books from the New Science series are here.
Table of contents
Excerpt
For readers of this benefit, a 25% discount on the coupon - Frenkel
Source: https://habr.com/ru/post/367913/
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