Love and Math

by Frenkel, Edward

  Alexander Alexandrovich Kirillov (or San Sanych, as he was affectionately called) is a charismatic lecturer and a great human being, whom I got to know quite well years later. I think he was teaching a standard undergraduate course on representation theory along the lines of his well-known book. He also had a seminar for graduate students, which we attended as well.

  We got away with this thanks to Kirillov’s good heart. His son Shurik (now professor at the Stony Brook University) studied at the special math school No. 179 together with my classmates Dima Kleinbock and Syoma Hawkin. Needless to say, San Sanych knew about the situation with admissions at MGU. He told me many years later that there was nothing he could do about this – they wouldn’t let him anywhere near the admissions committee, which was largely staffed with the Communist Party apparatchiks. So all he could do was let us sneak into his classes.

  Kirillov did all he could to make Kerosinka students coming to his lectures feel welcome. One of the best memories of my first college year was coming to his lively lectures and seminars. I also attended a seminar given by Alexander Rudakov, which was also a great experience.

  In the meantime, I was learning whatever math I could learn at Kerosinka. I was living in a dorm but coming home for the weekends, and I was still meeting Evgeny Evgenievich every couple of weeks. He advised me on what books to read, and I reported to him on my progress. But I was quickly reaching the point where to maintain my momentum, as well as the motivation for it, I would need an advisor with whom I would meet more regularly and not only learn from, but also get a problem to work on. Because I was not at Mekh-Mat, I could not take advantage of the vast resources that it had to offer. And I was too shy to come up to someone like A.A. Kirillov and ask him to work with me individually, or give me a problem to work on. I felt like an outsider. By the spring semester of 1986 (my second year at Kerosinka), complacency and stagnation were beginning to set in. With all the odds stacked against me, I started to doubt that I could fulfill my dream of becoming a mathematician.

  * At the time, it was known as the Gubkin Institute of Petrochemical and Gas Industry (it was named after the long-time head of the Ministry of Oil and Gas in the USSR, I.M. Gubkin). After I became a student there, it was renamed Gubkin Institute of Oil and Gas, and later, Gubkin University of Oil and Gas.

  Chapter 5

  Threads of the Solution

  I was beginning to despair when, one day, during a break in the lecture at Kerosinka, one of our most respected math professors, Alexander Nikolaevich Varchenko, approached me in the corridor. Varchenko is a former student of Vladimir Arnold, one of the leading Soviet mathematicians, and he is a world-class mathematician himself.

  “Would you be interested in working on a math problem?” he asked.

  “Yes, of course,” I said, “What kind of problem?” as if I would not have been happy to do just anything.

  “There is this question that came up in my research, which I think is a good problem to give to a bright student like you. The expert on this matter is Dmitry Borisovich Fuchs.” That was the name of a famous mathematician, which I had heard before. “I have already spoken to him, and he has agreed to supervise a student’s research on this topic. Here is his phone number. Give him a call, and he’ll tell you what to do.”

  It is quite common for experienced mathematicians like Varchenko to encounter all kinds of unsolved mathematical problems in their research. If Varchenko’s problem had been closely tied to his own research program, he might have tried to solve it himself. But no mathematician does everything alone, so mathematicians often delegate some of such unsolved problems (typically, the ones they consider to be simpler) to their students. Sometimes a problem might be outside of the professor’s immediate interests, but he or she might nonetheless be curious about it, as was the case with my problem. That’s why Varchenko enlisted Fuchs, an expert in this area, to supervise me. All in all, this was for the most part a typical “transaction” in the social workings of the mathematical world.

  What was actually unusual was that Fuchs was not formally affiliated with teaching at any university. But for many years Fuchs had been, along with a number of other top mathematicians, trying to alleviate the effect of the discrimination against Jewish students by privately teaching young talented kids who were denied entry to MGU.

  As part of those efforts, Fuchs was involved in what became known as “Jewish People’s University,” an unofficial evening school, where he and his colleagues gave courses of lectures to students. Some of those lectures had even been held at Kerosinka, although this was before my time.

  The school had been organized by a courageous woman, Bella Muchnik Subbotovskaya, who was its heart and soul. Unfortunately, the KGB got on the case, alarmed that there were unauthorized gatherings of Jewish people. She was eventually called to the KGB and interrogated. Soon after that interview, she was killed by a truck under suspicious circumstances, which led many people to suspect that this was in fact a cold-blooded murder.1 Without her at the helm, the school collapsed.

  I came to Kerosinka two years after this tragic chain of events. Though the evening school did not exist anymore, there was still a small network of professional mathematicians who helped misfortunate outcasts like myself on an individual basis. They sought out promising students and gave them advice, encouragement, and in some cases, full-fledged mentoring and advising. This was the reason that Varchenko gave that problem to me, a student at Kerosinka, rather than a student at Mekh-Mat, where, through his connections, he could have easily found a student willing to take it up. This was also why Fuchs was willing to invest his personal time to supervise me.

  And I am glad he did. Looking back, it is clear to me that without Fuchs’ kindness and generosity, I would have never become a mathematician. I was studying math at Kerosinka and sitting in at the lectures at MGU, but by itself that was not enough. In fact, it is virtually impossible for students to do their own research without someone guiding their work. Having an advisor is absolutely essential.

  At the time, though, all I knew was that I had in my hand the phone number of Fuchs, a renowned mathematician, and I was about to embark on a project supervised by him. This was unbelievable! I didn’t know where this would end up, but I knew right away that something big had happened.

  That evening, having mustered all my courage, I called Fuchs from a pay phone and explained who I was.

  “Yes, I know,” said Fuchs, “I have to give you a paper to read.”

  We met the next day. Fuchs had the physical appearance of a giant, not at all how I imagined him. He was very business-like.

  “Here,” he said, handing me an offprint of an article, “try to read this, and as soon as you see a word that you don’t understand, call me.”

  I felt like he had just handed me the Holy Grail.

  This was an article, a dozen pages long, which he had written some years earlier, on the subject of “braid groups.” That evening I started reading it.

  The preceding three years of studying with Evgeny Evgenievich and on my own were not spent in vain. Not only did I understand all the words in the title, I could make sense of the content as well. I decided to try to read the whole thing on my own. It was a matter of pride. I was already imagining how impressed Fuchs would be when I told him that I understood everything on my own.

  I had heard of the “braid groups” before. These are excellent examples of groups, the concept we discussed in Chapter 2. Evgeny Evgenievich had introduced this concept in the context of symmetries, and so elements of the groups that we considered were symmetries of some object. For example, the circle group consisted of the symmetries of a round table (or any other round object), and the group of four rotations was the group of symmetries of a square table (or any other square object). Once we have the notion “group,” we can look for other examples. It turns out there are many examples of groups that have nothing to do with symmetries, which was our motivation to introduce the concept of a group in
the first place. This is actually a typical story. The creation of a mathematical concept may be motivated by problems and phenomena in one area of math (or physics, engineering, and so forth), but later it may well turn out to be useful and well adapted to other areas.

  It turns out that many groups do not come from symmetries. And the braid groups are such groups.

  I did not know yet about the real-world applications of braid groups to such areas as cryptography, quantum computing, and biology, which we will talk about later. But I was mesmerized by the innate beauty of these mathematical abstractions.

  There is one braid group for each natural number n = 1, 2, 3,... We can use those numbers to get a name for each braid group. In general, we call them Bn, and so for n = 1 we have a group called B1, for n = 2 we have a group called B2, and so on.

  To describe the group Bn, we have to describe first its elements, as we did with the rotational symmetries of the round and square tables. The elements of the group Bn are the so-called braids with n threads, such as the one shown on the picture below, with n = 5. Imagine two solid, transparent plates with five nails in each, with one thread connecting each nail in one plate to one nail in the other. Since the plates are transparent, we can see each of the threads in its entirety. Each thread is allowed to weave around any other thread any way we like but is not allowed to get entangled with itself. Each nail must connect to exactly one thread.

  This whole thing – two plates and however many threads – constitutes a single braid, just as a car has four wheels, one transmission, four doors, and so forth. We are not considering those parts separately; we are focusing on the braid as a whole.

  Those are the braids with n threads. Now we need to show that all braids with n threads form a group. This means that we need to describe how to make the composition of two such braids. In other words, for each pair of braids with n threads, we have to produce another braid with n threads, just as applying two rotations one after another gave us a third rotation. And then we will have to check that this composition satisfies the properties listed in Chapter 2.

  So suppose we have two braids. In order to produce a new braid out of them, we put one of them on top of the other, aligning the nails, as shown on the picture. And then we remove the middle plates while connecting the upper threads to the lower ones attached to the matching nails.

  The resulting braid will be twice as tall, but this is not a problem. We’ll just shorten the threads so that the resulting braid will have the same height as the original ones, while preserving the way the threads go around each other. Voilà! We started out with two braids and produced a new one. This is the rule of composition of two braids in our braid group.

  Since a braid group does not come from symmetries, it is sometimes better to think of this operation not as “composition” (which was natural in the case of groups of symmetries), but as “addition” or “multiplication,” similar to the operations that we perform on numbers. From this point of view, braids are like numbers – these are some “hairy numbers,” if you will.

  Given two whole numbers, we can add them to each other and produce a new number. Likewise, given two braids, we produce a new one by the rule described above. So we can think of this as the “addition” of two braids.

  Now we need to check that this addition of braids satisfies all properties (or axioms) of a group. First, we need the identity element. (In the circle group, this was the point corresponding to the rotation by 0 degrees.) This will be the braid with all threads going straight down without any weaving as shown on the next picture. It is a kind of “trivial” braid, in which no braiding actually occurs, the same way rotation by 0 degrees makes no rotation at all.2

  Next, we need to find the inverse braid of a given braid b (in the case of the circle group, this was the rotation by the same angle but in the opposite direction). It should be such that if we add this braid to the braid b, according to the rule described above, we will get the identity braid.

  This inverse braid will be the reflection of b with respect to the bottom plate. If we compose it with the original braid according to our rule, we will be able to rearrange all threads so that the result will be the identity braid.

  Here I need to make an important point, which up to now I have kind of swept under the rug: we will not distinguish the braids that can be obtained from one another by pulling the threads, stretching and shrinking them any way we like so long as we do not cut or resew the threads. In other words, the threads should be attached to the same nails, and we do not allow the threads to go through each other, but otherwise we can tweak them any way we like. Think of this as grooming our braid. When we do that, it will still be the same braid (only prettier!). It is in this sense that the addition of a braid and its mirror image is “the same” as the identity braid; it is not literally the same but becomes one after we tweak the threads.3

  So we see now that the axioms of a group – composition (or addition), identity, and inverse – are satisfied. We have proved that braids with n threads form a group.4

  To see what the braid groups are more concretely, let’s look closely at the simplest one: the group B2 of braids with two threads. (The group with B1 with one thread has only one element, so there is nothing to discuss.5) We will assign to each such braid an integer N. By an integer, here I mean a natural number: 1, 2, 3,...; or 0; or a negative of a natural number: −1, −2, −3,...

  First of all, to the identity thread we will assign the number 0. Second, if the thread starting at the left nail on the top plate goes underneath the other thread, then we assign to it 1. If it goes around it, then we assign to it 2, and so on, as shown on the pictures.

  If this thread goes on top of the other thread, then we assign to the braid the negative number −1, if it goes around it as shown on the picture below, we assign to it −2, and so on.

  Let’s call the number assigned to a braid in this way the “number of overlaps.” If we have two braids with the same number of overlaps, we can transform one into another by “tweaking” the threads. In other words, the braid is completely determined by the number of overlaps. So we have a one-to-one correspondence between braids with two threads and integers.

  Here it is useful to note something we always take for granted: the set of all integers is itself a group! Namely, we have the operation of addition, the “identity element” is number 0, and for any integer N, its “inverse” is −N. Then all properties of a group listed in Chapter 2 are satisfied. Indeed, we have N + 0 = N and N + (−N) = 0.

  What we have just found is that the group of braids with two threads has the same structure as the group of integers.6

  Now, in the group of integers the sum of two integers a and b is the same in two different orders:

  This is also so in the braid group B2. Groups satisfying this property are called “commutative” or “abelian” (in honor of the Norwegian mathematician Niels Henrik Abel).

  In a braid with 3 threads or more, the threads can be entangled among themselves in a much more complicated fashion than in a braid with only 2 threads. The knotting pattern can no longer be described merely by the numbers of overlaps (look at the above picture of a braid with 5 threads). The pattern in which the overlaps occur is also important. Furthermore, it turns out that the addition of two braids with 3 or more threads does depend on the order in which it is taken (that is to say, which of the two braids is on top in the picture above, describing the addition of braids). In other words, in the group Bn with n = 3,4,5,... we have in general

  Such groups are called “non-commutative” or “non-abelian.”

  Braid groups have many important practical applications. For example, they are used to construct efficient and robust public key encryption algorithms.7

  Another promising direction is designing quantum computers based on creating complex braids of quantum particles known as axions. Their trajectories weave around each other, and their overlaps are used to build “logic gates” of the quantum comp
uter.8

  There are also applications in biology. Given a braid with n threads, we can number the nails on the two plates from 1 to n from left to right. Then, connect the ends of the threads attached to the nails with the same number on the two plates. This will create what mathematicians call a “link”: a union of loops weaving around each other.

  In the example shown on this picture, there is only one loop. Mathematicians’ name for it is “knot.” In general, there will be several closed threads.

  The mathematical theory of links and knots is used in biology: for example, to study bindings of DNA and enzymes.9 We view a DNA molecule as one thread, and the enzyme molecule as another thread. It turns out that when they bind together, highly non-trivial knotting between them may occur, which may alter the DNA. The way they entangle is therefore of great importance. It turns out that the mathematical study of the resulting links sheds new light on the mechanisms of recombination of DNA.

  In mathematics, braids are also important because of their geometric interpretation. To explain it, consider all possible collections of n points on the plane. We will assume that the points are distinct; that is, for any two points, their positions on the plane must be different. Let’s choose one such collection; namely, n points arranged on a straight line, with the same distance between neighboring points. Think of each point as a little bug. As we turn on the music, these bugs come alive and start moving on the plane. If we view the time as the vertical direction, then the trajectory of each bug will look like a thread. If the positions of the bugs on the plane are distinct at all times – that is, if we assume that the bugs don’t collide – then these threads will never intersect. While the music is playing, they can move around each other, just like the threads of a braid. However, we demand that when we stop the music after a fixed period of time, the bugs must align on a straight line in the same way as at the beginning, but each bug is allowed to end up in a position initially occupied by another bug. Then their collective path will look like a braid with n threads.