February 10, 2016 at 14:20The book “Love and Mathematics. The heart of hidden reality
Hello! In our warehouse, the second edition of the book Love and Math. Heart of the hidden reality "which was published together with the Dynasty fund The book review was already on Habré . Here we will publish a chapter from the book “Conquering the summit”
“My goal is not to teach you anything. I want to give you the opportunity to feel that there is a whole world that is carefully hiding from us - the world of mathematics. This is a portal to uncharted reality, the key to understanding the deepest secrets of the Universe and ourselves. Mathematics is not the only portal, there are others. But in a sense, it is the most obvious. And that is why it is so camouflaged, as if a board with the inscription: “You do not need to be here” is nailed on it. But actually it is necessary. And when we enter it, we remember who we are: not the small cogs of a large machine, not the lonely souls who dwell on the outskirts 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 that I was ready to share my findings with Fuchs. I knew that an article by Wakimoto would interest him no less than I did. By tradition, I went to the cottage to Fuchs. However, when I arrived there, Fuchs informed me that there was a slight overlap: he made an appointment with me and his colleague and former student Boris Feigin at the same time - he said, completely by accident, which I, of course, didn’t believed (and much later Fuchs confirmed that he had indeed done it on purpose).
Fuchs introduced me to Feigin a few months earlier. This happened before one of Gelfand's seminars, shortly after I finished my article on braid groups and began reading an article by Feigin and Fuchs. On the advice of Fuchs, I asked Feigin to recommend additional literature that I should read. Boris Lvovich — as I then addressed him — at that time was thirty-three years old, but he was already considered one of the brightest stars in the Moscow mathematical community. Dressed in jeans and a pair of well-worn sneakers, he looked a modest, even shy man. Thick glasses sat 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 a novice student who had the honor of talking with the famous mathematician. In general, it was not a very interesting conversation. Nevertheless, from time to time Feigin raised his eyes and cast a quick glance at me, accompanying him with a wide disarming smile. This melted the ice, and I had no doubt that in reality he was a sincere and friendly person.
However, Feigin’s first recommendation baffled me: he advised me to read Landau and Lifshitz’s book entitled Statistical Physics. This prospect 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 that we had to study at the institute.
In defense of Feigin, I will say that the advice turned out to be sensible. 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 have not read 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 died out. In fact, I didn’t speak with Feigin until that day at Fuchs’s cottage, except for the traditional “hello” when I met at Gelfand’s seminars.
Soon after my arrival at the Fuchs country house, I saw Feigin out of the window - he came by bicycle, because he lived in a nearby country house. We greeted each other, talked a little about this and sat at a round table in the kitchen. Fuchs asked me:
“So, what's new?”
“Well ... I found one interesting article by Japanese mathematician Wakimoto.”
- Hmmm ... - Fuchs turned to Feigin: - Have you heard about this?
Feigin shook his head in the negative, and Fuchs again turned to me:
“He always knows everything ... It’s good that he didn’t come across this article, which 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, this interested them both very much. Then for the first time, Feigin and I had a chance to talk about mathematical concepts, and I immediately felt that we were tuned to the same wavelength. 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, what he heard was clearly excited. Fuchs, for the most part, was just watching us and, I suspect, was rubbing his hands in joy: his secret plan to introduce Feigin and me closer worked so wonderfully. It was a truly inspiring conversation. I had no doubt that I was 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're ready to take this work to the next 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 the work, I had to act as a kind of forensic expert in order to track the connection between these formulas and the big picture. A few days later, the picture began to emerge. In thoughtfulness, I paced my dorm room, when suddenly it dawned on me: Wakimoto's formulas come from geometry! It was an amazing and unexpected discovery, because the approach of Wakimoto 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) of the symmetries of the sphere and its group of loops. As I explained in the previous chapter, each element of the SO (3) loop group is a collection of SO (3) elements, one SO (3) element for each loop point. Each of the elements of SO (3) acts on the sphere as a specific rotation. This implies that each element of the loop group SO (3) generates the symmetry of the loop space of the sphere.
I realized that this information can be used to search for a representation of the Katz - Moody algebra associated with SO (3). But that still didn't give us Wakimoto's formula. In order to get 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 with either side up. As it turned out, this was also true for Wakimoto's formulas.
Armed with this idea, I immediately tried to extend Wakimoto's formulas to other, more complex Katz-Moody algebras. The first, geometric step was a success for me 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 nonsense. The result simply did not make sense. I began to twirl the formulas, and so, and so, but no tricks helped to solve the problem. I began to seriously consider the possibility that this construction works only for SO (3), but not for Katz - Moody algebras of a more general form. I had no way of knowing if this problem is being solved in principle, and if so, is it possible to come to a solution using the tools at our disposal. I could only work hard and hope for the best.
A week passed, and the time came for our next meeting with Fuchs. I planned to show him my calculations and ask for advice. When I arrived at the cottage, Fuchs said that his wife had to go to Moscow on some urgent matters, and he should watch over their two little daughters.
“But you know what,” Fuchs continued, “Feigin was here yesterday, and he is completely delighted with what you told us last week.” Why don't you go visit him? Only 15 minutes to his summer residence. I warned him that I would send you to him, so he was waiting for you.
He explained the way to me, and I went to the cottage to Feigin. Feigin really was waiting for me. Boris Lvovich warmly greeted me and introduced me to his charming wife Inna and three children: two brisk boys Roma and Zhenya eight and ten years old and a lovely two-year-old daughter Lisa. Then I did not know that with this wonderful family I would be connected by the warmest friendship for many years.
Feigin's wife offered me a cup of tea and a piece of cake, and we sat on the porch. It was a beautiful summer evening, the rays of the sun breaking through the dense foliage of trees, birds chirped - 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 his train of thought was similar to mine. From the very beginning of the conversation we continually ended sentences for each other. It was amazing: he fully understood me, and I understood him.
I began to talk about the failure that I suffered while trying to extend the construction to other Katz-Moody algebras. Feigin listened with great attention to me, paused a little pondering the problem, and then turned my attention to one important thing that I missed. One of the steps in the process of generalizing the construction of Wakimoto is the search for a suitable generalization of the sphere - the manifold on which SO (3) acts by symmetries. In the case of SO (3), the choice is essentially unambiguous. However, for other groups, there may be many more options. 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 entirely possible that I was mistaken; it is possible that I did not succeed precisely because of the wrong choice of spaces.
As I explained above, ultimately, I had to turn the inside out formulas. My whole construction was based on the expectation that, in some miraculous way, the formulas that result from inversion will remain valid. For Wakimoto, for the simplest group SO (3), everything turned out that way. 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 group SO (3) 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 are richer and more versatile of projective spaces, so one could hope that the analogue of the Wakimoto construction would really work on 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 railway 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 to write down formulas for the simplest flag variety. The old train car bounced noisefully at the joints of the rails and swayed 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, some kind of system was clearly emerging. Unlike projective spaces, which I tried unsuccessfully to tame all the week before, with flag varieties I really started to emerge.
A few more lines of calculations, and ... Eureka! Happened! The “turned inside out” formulas worked as clearly and harmoniously as in Wakimoto’s 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 Katz - Moody algebras in a free field!
The next morning, I carefully checked my calculations. It all came together. Feigin didn’t have a phone in the country, so I could not call him and immediately tell about my findings. To begin with, I set out everything in the form of a letter, and told Feigin about the new results next week, when we met again. Thus began our many years of joint work. Feigin became my teacher, mentor, leader, friend. At first I turned to him by name and patronymic, Boris Lvovich. But later he insisted that I use a more informal appeal - Borya.
In my life, I was incredibly lucky with teachers. Evgeny Evgenievich discovered the beauty of mathematics and helped me fall in love with it. He taught me the basics. Fuchs saved me after the catastrophe at the entrance exam at Moscow State University and ensured a swift start to my mathematical career, which was still uncertain. He supervised me during my first serious mathematical project, thanks to which I believed in myself, and led me to an amazing field of research at the intersection of mathematics and physics. Finally, I was ready for the big game. At this stage of my journey, I could not find a better scientific adviser than Borya. My mathematical career began to gain momentum, as if driven by the power of a jet engine.
Without a doubt, Borya Feigin is one of the most original and outstanding mathematicians of her generation in the whole world, a visionary 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 my students, I will still appreciate all that Borya 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 parental 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 students should take when they should lend a helping hand, and when it would be wiser to push them from a boat into the lake so that they learn to swim on their own? This is real art. No one can teach this.
Borya 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 in which direction to move on. Somehow, he made sure that I always knew what to do. And, feeling that he was always there and always support me, I had no doubt that I had chosen the right path. I was very lucky that my teacher was such a terrific person.
The fall semester of 1987 began; this was my fourth year in Kerosinka. I was nineteen, and my life was never more interesting and exciting. I still lived in a dormitory, met friends, fell in love ... But I did not forget about studying. By that time I was missing most of the classes, and I was preparing for the exams myself (it also happened that for the first time I took up a textbook just a couple of days before the exam). In all subjects I had only five. The only exception was the four on Marxist political economy (shame on me).
The fact that I actually had a “second life” —a mathematical work with Boreas, which occupied most of my time and required the greatest efforts, I hid from most people in my environment.
As a rule, we met with Northwind twice a week. Officially, he held a position at the Institute of Solid State Physics, but there he did not have too many things, so it was enough for him to show up at the institute once a week. The rest of the days he worked in his mother’s apartment, which was ten minutes walk from his house. Kerosinka and my hostel were also nearby. This was our traditional venue. I arrived late in the morning or immediately after noon, and we worked on our projects, sometimes staying up until the night. Mama Bori came home from work in the evening and fed us dinner, and very often Borea and I left together for about nine or ten hours.
First of all, we wrote a short article about our results and sent it to the journal Uspekhi Matematicheskikh Nauk. It was published about a year later - quite 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 in order to better understand the representations of Kac - Moody algebras. Our work also allowed us to discover the implementation of two-dimensional free-field quantum models. Thanks to 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 studies.
It was an amazing time. In those days when I did not meet with Borey, I worked on my own - a week in Moscow, and on weekends at home. I continued to visit the Science Library and swallowed more and more books and articles. Mathematics was my life, my food and water. It was as if I moved to a beautiful parallel Universe, and I wanted to stay there forever, plunging deeper into this dream. With each new discovery, with each new idea, this magical world became closer and dearer to me.
However, by the fall of 1988, when the fifth, last year of my studies at the institute began, I had to return to harsh reality. I had to think about the future. Despite the fact that I was one of the best on the course, my prospects were completely bleak. Anti-Semitism reigned not only in graduate school, but also in all institutions where a graduate could get a good job. Complicating matters was the fact that I did not have a Moscow residence permit. The day of reckoning was drawing near.
More detailed information on the book is available on the website of the publishing house
all the books from the series New Science are here
Contents
Excerpt
To the readers of this benefit a 25% discount coupon - Frankel
“My goal is not to teach you anything. I want to give you the opportunity to feel that there is a whole world that is carefully hiding from us - the world of mathematics. This is a portal to uncharted reality, the key to understanding the deepest secrets of the Universe and ourselves. Mathematics is not the only portal, there are others. But in a sense, it is the most obvious. And that is why it is so camouflaged, as if a board with the inscription: “You do not need to be here” is nailed on it. But actually it is necessary. And when we enter it, we remember who we are: not the small cogs of a large machine, not the lonely souls who dwell on the outskirts of the Universe. We are the Creators of this world, capable of giving each other beauty and love. ” - Edward Frenkel.Chapter 11. Conquering the Peaks
By the summer, I felt that I was ready to share my findings with Fuchs. I knew that an article by Wakimoto would interest him no less than I did. By tradition, I went to the cottage to Fuchs. However, when I arrived there, Fuchs informed me that there was a slight overlap: he made an appointment with me and his colleague and former student Boris Feigin at the same time - he said, completely by accident, which I, of course, didn’t believed (and much later Fuchs confirmed that he had indeed done it on purpose).
Fuchs introduced me to Feigin a few months earlier. This happened before one of Gelfand's seminars, shortly after I finished my article on braid groups and began reading an article by Feigin and Fuchs. On the advice of Fuchs, I asked Feigin to recommend additional literature that I should read. Boris Lvovich — as I then addressed him — at that time was thirty-three years old, but he was already considered one of the brightest stars in the Moscow mathematical community. Dressed in jeans and a pair of well-worn sneakers, he looked a modest, even shy man. Thick glasses sat 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 a novice student who had the honor of talking with the famous mathematician. In general, it was not a very interesting conversation. Nevertheless, from time to time Feigin raised his eyes and cast a quick glance at me, accompanying him with a wide disarming smile. This melted the ice, and I had no doubt that in reality he was a sincere and friendly person.
However, Feigin’s first recommendation baffled me: he advised me to read Landau and Lifshitz’s book entitled Statistical Physics. This prospect 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 that we had to study at the institute.
In defense of Feigin, I will say that the advice turned out to be sensible. 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 have not read 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 died out. In fact, I didn’t speak with Feigin until that day at Fuchs’s cottage, except for the traditional “hello” when I met at Gelfand’s seminars.
Soon after my arrival at the Fuchs country house, I saw Feigin out of the window - he came by bicycle, because he lived in a nearby country house. We greeted each other, talked a little about this and sat at a round table in the kitchen. Fuchs asked me:
“So, what's new?”
“Well ... I found one interesting article by Japanese mathematician Wakimoto.”
- Hmmm ... - Fuchs turned to Feigin: - Have you heard about this?
Feigin shook his head in the negative, and Fuchs again turned to me:
“He always knows everything ... It’s good that he didn’t come across this article, which 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, this interested them both very much. Then for the first time, Feigin and I had a chance to talk about mathematical concepts, and I immediately felt that we were tuned to the same wavelength. 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, what he heard was clearly excited. Fuchs, for the most part, was just watching us and, I suspect, was rubbing his hands in joy: his secret plan to introduce Feigin and me closer worked so wonderfully. It was a truly inspiring conversation. I had no doubt that I was 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're ready to take this work to the next 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 the work, I had to act as a kind of forensic expert in order to track the connection between these formulas and the big picture. A few days later, the picture began to emerge. In thoughtfulness, I paced my dorm room, when suddenly it dawned on me: Wakimoto's formulas come from geometry! It was an amazing and unexpected discovery, because the approach of Wakimoto 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) of the symmetries of the sphere and its group of loops. As I explained in the previous chapter, each element of the SO (3) loop group is a collection of SO (3) elements, one SO (3) element for each loop point. Each of the elements of SO (3) acts on the sphere as a specific rotation. This implies that each element of the loop group SO (3) generates the symmetry of the loop space of the sphere.
I realized that this information can be used to search for a representation of the Katz - Moody algebra associated with SO (3). But that still didn't give us Wakimoto's formula. In order to get 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 with either side up. As it turned out, this was also true for Wakimoto's formulas.
Armed with this idea, I immediately tried to extend Wakimoto's formulas to other, more complex Katz-Moody algebras. The first, geometric step was a success for me 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 nonsense. The result simply did not make sense. I began to twirl the formulas, and so, and so, but no tricks helped to solve the problem. I began to seriously consider the possibility that this construction works only for SO (3), but not for Katz - Moody algebras of a more general form. I had no way of knowing if this problem is being solved in principle, and if so, is it possible to come to a solution using the tools at our disposal. I could only work hard and hope for the best.
A week passed, and the time came for our next meeting with Fuchs. I planned to show him my calculations and ask for advice. When I arrived at the cottage, Fuchs said that his wife had to go to Moscow on some urgent matters, and he should watch over their two little daughters.
“But you know what,” Fuchs continued, “Feigin was here yesterday, and he is completely delighted with what you told us last week.” Why don't you go visit him? Only 15 minutes to his summer residence. I warned him that I would send you to him, so he was waiting for you.
He explained the way to me, and I went to the cottage to Feigin. Feigin really was waiting for me. Boris Lvovich warmly greeted me and introduced me to his charming wife Inna and three children: two brisk boys Roma and Zhenya eight and ten years old and a lovely two-year-old daughter Lisa. Then I did not know that with this wonderful family I would be connected by the warmest friendship for many years.
Feigin's wife offered me a cup of tea and a piece of cake, and we sat on the porch. It was a beautiful summer evening, the rays of the sun breaking through the dense foliage of trees, birds chirped - 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 his train of thought was similar to mine. From the very beginning of the conversation we continually ended sentences for each other. It was amazing: he fully understood me, and I understood him.
I began to talk about the failure that I suffered while trying to extend the construction to other Katz-Moody algebras. Feigin listened with great attention to me, paused a little pondering the problem, and then turned my attention to one important thing that I missed. One of the steps in the process of generalizing the construction of Wakimoto is the search for a suitable generalization of the sphere - the manifold on which SO (3) acts by symmetries. In the case of SO (3), the choice is essentially unambiguous. However, for other groups, there may be many more options. 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 entirely possible that I was mistaken; it is possible that I did not succeed precisely because of the wrong choice of spaces.
As I explained above, ultimately, I had to turn the inside out formulas. My whole construction was based on the expectation that, in some miraculous way, the formulas that result from inversion will remain valid. For Wakimoto, for the simplest group SO (3), everything turned out that way. 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 group SO (3) 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 are richer and more versatile of projective spaces, so one could hope that the analogue of the Wakimoto construction would really work on 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 railway 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 to write down formulas for the simplest flag variety. The old train car bounced noisefully at the joints of the rails and swayed 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, some kind of system was clearly emerging. Unlike projective spaces, which I tried unsuccessfully to tame all the week before, with flag varieties I really started to emerge.
A few more lines of calculations, and ... Eureka! Happened! The “turned inside out” formulas worked as clearly and harmoniously as in Wakimoto’s 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 Katz - Moody algebras in a free field!
The next morning, I carefully checked my calculations. It all came together. Feigin didn’t have a phone in the country, so I could not call him and immediately tell about my findings. To begin with, I set out everything in the form of a letter, and told Feigin about the new results next week, when we met again. Thus began our many years of joint work. Feigin became my teacher, mentor, leader, friend. At first I turned to him by name and patronymic, Boris Lvovich. But later he insisted that I use a more informal appeal - Borya.
In my life, I was incredibly lucky with teachers. Evgeny Evgenievich discovered the beauty of mathematics and helped me fall in love with it. He taught me the basics. Fuchs saved me after the catastrophe at the entrance exam at Moscow State University and ensured a swift start to my mathematical career, which was still uncertain. He supervised me during my first serious mathematical project, thanks to which I believed in myself, and led me to an amazing field of research at the intersection of mathematics and physics. Finally, I was ready for the big game. At this stage of my journey, I could not find a better scientific adviser than Borya. My mathematical career began to gain momentum, as if driven by the power of a jet engine.
Without a doubt, Borya Feigin is one of the most original and outstanding mathematicians of her generation in the whole world, a visionary 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 my students, I will still appreciate all that Borya 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 parental 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 students should take when they should lend a helping hand, and when it would be wiser to push them from a boat into the lake so that they learn to swim on their own? This is real art. No one can teach this.
Borya 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 in which direction to move on. Somehow, he made sure that I always knew what to do. And, feeling that he was always there and always support me, I had no doubt that I had chosen the right path. I was very lucky that my teacher was such a terrific person.
The fall semester of 1987 began; this was my fourth year in Kerosinka. I was nineteen, and my life was never more interesting and exciting. I still lived in a dormitory, met friends, fell in love ... But I did not forget about studying. By that time I was missing most of the classes, and I was preparing for the exams myself (it also happened that for the first time I took up a textbook just a couple of days before the exam). In all subjects I had only five. The only exception was the four on Marxist political economy (shame on me).
The fact that I actually had a “second life” —a mathematical work with Boreas, which occupied most of my time and required the greatest efforts, I hid from most people in my environment.
As a rule, we met with Northwind twice a week. Officially, he held a position at the Institute of Solid State Physics, but there he did not have too many things, so it was enough for him to show up at the institute once a week. The rest of the days he worked in his mother’s apartment, which was ten minutes walk from his house. Kerosinka and my hostel were also nearby. This was our traditional venue. I arrived late in the morning or immediately after noon, and we worked on our projects, sometimes staying up until the night. Mama Bori came home from work in the evening and fed us dinner, and very often Borea and I left together for about nine or ten hours.
First of all, we wrote a short article about our results and sent it to the journal Uspekhi Matematicheskikh Nauk. It was published about a year later - quite 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 in order to better understand the representations of Kac - Moody algebras. Our work also allowed us to discover the implementation of two-dimensional free-field quantum models. Thanks to 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 studies.
It was an amazing time. In those days when I did not meet with Borey, I worked on my own - a week in Moscow, and on weekends at home. I continued to visit the Science Library and swallowed more and more books and articles. Mathematics was my life, my food and water. It was as if I moved to a beautiful parallel Universe, and I wanted to stay there forever, plunging deeper into this dream. With each new discovery, with each new idea, this magical world became closer and dearer to me.
However, by the fall of 1988, when the fifth, last year of my studies at the institute began, I had to return to harsh reality. I had to think about the future. Despite the fact that I was one of the best on the course, my prospects were completely bleak. Anti-Semitism reigned not only in graduate school, but also in all institutions where a graduate could get a good job. Complicating matters was the fact that I did not have a Moscow residence permit. The day of reckoning was drawing near.
More detailed information on the book is available on the website of the publishing house
all the books from the series New Science are here
Contents
Excerpt
To the readers of this benefit a 25% discount coupon - Frankel