Thus, braids with n threads may be viewed as paths in the space of collections of n distinct points on the plane.10
The problem that Varchenko gave me, and on which I was about to start working with Fuchs, concerned a part of the braid group called the “commutator subgroup.” Remember that for braids with two threads, we have defined the overlap number. A similar number can be assigned to a braid with any number of threads.11 We use this to define the commutator subgroup B′n of the braid group with n threads. It consists of all braids whose total overlap number is zero.12
The problem I had to solve was to compute the so-called “Betti numbers” of the group B′n. These numbers reflect deep properties of this group, which are important in applications. As an analogy, think of a physical object, like a house. It has various characteristics: some more obvious, like the numbers of floors, rooms, doors, windows, etc., and some less so, like the proportions of the materials from which it is built. Likewise, a group also has various characteristics, and these are the Betti numbers.13 Fuchs had earlier computed the Betti numbers of the braid group Bn itself. He gave me his paper so that I could learn the basics of the subject.
Within a week, I was able to read the entire Fuchs paper on my own, occasionally looking up previously unknown-to-me concepts and definitions in my by-then fairly large library of math books. I called Fuchs.
“Oh, it’s you,” he said. “I was wondering why you hadn’t called. Have you started reading the article?”
“Yes, Dmitry Borisovich. Actually, I have finished it.”
“Finished it?” Fuchs sounded surprised. “Well, then we should meet. I want to hear what you’ve learned.”
Fuchs suggested we meet on the next day at MGU, after a seminar he was going to attend. As I was preparing for the meeting, I kept re-reading the article and practicing my answers to the kinds of questions that I thought Fuchs was likely to ask. A world-class mathematician like Fuchs wouldn’t just take up a new student out of pity. The bar was set high. I understood that my first conversation with Fuchs would be something of an audition, and that’s why I was so eager to make a good impression on him.
We met at the appointed hour and walked the corridors of Mekh-Mat to find a bench where we wouldn’t be bothered. After we sat down, I started telling Fuchs what I learned from his article. He listened attentively, occasionally asking me questions. I think he was pleased by what he was hearing. He was curious where I learned all this stuff, and I told him about my studies with Evgeny Evgenievich, reading books, and attending lectures at Mekh-Mat. We even talked about my exam at the MGU (this was of course nothing new to Fuchs).
Luckily, our meeting went well. Fuchs seemed impressed with my knowledge. He told me that I was ready to tackle Varchenko’s problem and that he would help me with it.
I was elated when I was leaving MGU that evening. I was about to start working on my first math problem, guided by one of the best mathematicians in the world. Less than two years had passed since my entrance exam at Mekh-Mat. I was back in the game.
Chapter 6
Apprentice Mathematician
Solving a mathematical problem is like doing a jigsaw puzzle, except you don’t know in advance what the final picture will look like. It could be hard, it could be easy, or it could be impossible to solve. You never know until you actually do it (or realize that it’s impossible to do). This uncertainty is perhaps the most difficult aspect of being a mathematician. In other disciplines, you can improvise, come up with different solutions, even change the rules of the game. Even the very notion of what constitutes a solution is not clearly defined. For example, if we are tasked with improving productivity in a company, what metrics do we use to measure success? Will an improvement by 20 percent count as a solution of the problem? How about 10 percent? In math, the problem is always well defined, and there is no ambiguity about what solving it means. You either solve it or you don’t.
For Fuchs’ problem, I had to compute the Betti numbers of the groups B′n. There was no ambiguity in what this meant. It means the same thing today to everyone familiar with the language of math as it did in 1986 when I first learned about this problem, and will mean the same thing a hundred years from now.
I knew that Fuchs had solved a similar problem, and I knew how he’d done it. I prepared for my own task by working on similar problems for which solutions had already been known. This gave me intuition, skills, and a toolkit of methods. But I could not know a priori which of these methods would work or which way I should approach the problem – or even whether I could solve it without creating an essentially new technique or an entirely different method.
This quandary besets all mathematicians. Let’s look at one of the most famous problems in mathematics, Fermat’s Last Theorem, to see how one can go about doing math when the problem is easy to state but the solution is far from obvious. Fix a natural number n, that is, 1, 2, 3,..., and consider the equation
on the natural numbers x, y, and z.
If n = 1, we get the equation
which surely has many solutions among natural numbers: just take any x and y and set z = x + y. Note that here we use the operation of addition of natural numbers that we discussed in the previous chapter.
If n = 2, we get the equation
This equation also has many solutions in natural numbers; for instance,
All of this has been known since antiquity. What was unknown was whether the equation had any solutions for n greater than 2. Sounds pretty simple, right? How hard could it be to answer a question like this?
Well, as it turned out, pretty hard. In 1637, a French mathematician, Pierre Fermat, left a note on the margin of an old book saying that if n is greater than 2, then the equation had no solutions x, y, z that are natural numbers. In other words, we cannot find three natural numbers x, y, z such that
cannot find natural numbers x, y, z such that
and so on.
Fermat wrote that he had found a simple proof of this statement, for all n greater than 2, but “this margin is too small to contain it.” Many people, professional mathematicians as well as amateurs, took Fermat’s note as a challenge and tried to reproduce his “proof,” making this the most famous mathematical problem of all time. Prizes were announced. Hundreds of proofs were written and published, only to be crashed later on. The problem remained unsolved 350 years later.
In 1993, a Princeton mathematician, Andrew Wiles, announced his own proof of Fermat’s Last Theorem. But his proof, at first glance, had nothing to do with the original problem. Instead of proving Fermat’s Last Theorem, Wiles tackled the so-called Shimura–Taniyama–Weil conjecture, which is about something entirely different and is a lot more complicated to state. But a few years earlier, a Berkeley mathematician named Ken Ribet had proved that the statement of this conjecture implies Fermat’s Last Theorem. That’s why a proof of the conjecture would also prove Fermat’s Last Theorem. We will talk about all this in detail in Chapter 8; the point I want to make now is that what looks like a simple problem may not necessarily have an elementary solution. It is clear to us now that Fermat could not have possibly proved the statement attributed to him. Entire fields of mathematics had to be created in order to do this, a development that took a lot of hard work by many generations of mathematicians.1
But is it possible to predict all that, given this innocent-looking equation?
Not at all!
With any math problem, you never know what the solution will involve. You hope and pray that you will be able to find a nice and elegant solution, and perhaps discover something interesting along the way. And you certainly hope that you will actually be able to do it in a reasonable period of time, that you won’t have to wait for 350 years to reach the conclusion. But you can never be sure.
In the case of my problem, I was lucky; there was in fact an elegant solution that I was able to find in a relatively short period of time, about two months. But it didn’t come easily to me. It never does. I tried many d
ifferent methods. As each of them failed, I felt increasingly frustrated and anxious. This was my first problem, and inevitably I questioned whether I could be a mathematician. This problem was my first test of whether I had what it takes.
Working on this problem didn’t excuse me from taking classes and passing exams at Kerosinka, but my highest priority was the problem, and I spent endless hours with it, nights and weekends. I was putting way too much pressure on myself. I was starting to have trouble sleeping, the first time this ever happened to me. The insomnia I acquired while working on this problem was the first “side effect” of my mathematical research. It haunted me for many months afterward, and from that point on I never allowed myself to get lost so completely in a math problem.
I met with Fuchs every week or so at the Mekh-Mat building, where I told him about my progress, or lack of it (by then he was able to get me an ID, so I did not have to scale the fence anymore). Fuchs was always supportive and encouraging, and each time we met he would tell me about a new trick or suggest a new insight, which I would try to apply to my problem.
And then, suddenly, I had it. I found the solution, or perhaps more accurately, the solution presented itself, in all of its splendor.
I was trying to use one of the standard methods for computing Betti numbers, which Fuchs had taught me, called “spectral sequence.” I was able to apply it in a certain way, which allowed me in principle to compute the Betti numbers of the group B′n from the knowledge of the Betti numbers of all the groups B′m with m < n. The caveat was, of course, that I did not know what those other Betti numbers were either.
But this gave me a way to attack the problem: if I could guess the right answer, I would then have a path to proving it by following this method.
That’s easy to say, but coming up with such a guess required many sample computations, which only became more and more complicated. For a long time, no pattern seemed to emerge.
Suddenly, as if in a stroke of black magic, it all became clear to me. The jigsaw puzzle was complete, and the final image was revealed to me, full of elegance and beauty, in a moment that I will always remember and cherish. It was an incredible feeling of high that made all those sleepless nights worthwhile.
For the first time in my life, I had in my possession something that no one else in the world had. I was able to say something new about the universe. It wasn’t a cure for cancer, but it was a worthy piece of knowledge, and no one could ever take it away from me.
If you experience this feeling once, you will want to go back and do it again. This was the first time it happened to me, and like the first kiss, it was very special. I knew then that I could call myself mathematician.
The answer was actually quite unexpected, and much more interesting than what Fuchs or I could imagine. I found that for each divisor of the natural number n (the number of threads in the braids we are considering), there is a Betti number of the group B′n that is equal to the celebrated “Euler function” of that divisor.2
The Euler function assigns to any natural number d another natural number, called φ(d). This is the number of integers between 1 and d that are relatively prime with d; that is, have no common divisors with d (apart from 1, of course).
For example, take d = 6. Then 1 is relatively prime with 6, 2 is not (it is a divisor of 6), 3 is not (it is also a divisor of 6), 4 is not (4 and 6 share a common divisor; namely, 2), 5 is relatively prime with 6, and 6 is not. So there are two natural numbers between 1 and 6 that are relatively prime with 6: namely, 1 and 5. Hence the Euler function of 6 is equal to 2. We write this as φ(6) = 2.
The Euler function has many applications. For example, it is employed in the so-called RSA algorithm used to encrypt credit card numbers in online transactions (this is explained in endnote 7 to Chapter 14). It is named in honor of the eighteenth-century Swiss mathematician Leonhard Euler.
The fact that the Betti numbers I found were given by the Euler function suggested the existence of some hidden connections between braid groups and number theory. Therefore, the problem I had solved could potentially have implications far beyond its original scope.
Of course, I was eager to tell Fuchs about my results. It was already June 1986, almost three months after he and I first met. By then, Fuchs had left Moscow with his wife and two young daughters to spend the summer at his dacha near Moscow. Luckily for me, it was situated along the same train line as my hometown, about halfway, and so it was easy for me to visit him there on my way home.
After offering me a customary cup of tea, Fuchs asked me about my progress.
“I solved the problem!”
I couldn’t contain my excitement, and I guess the account of the proof that I gave was pretty rambling. But no worries – Fuchs understood everything quickly. He looked pleased.
“This is great,” he said, “Well done! Now you have to start writing a paper about this.”
It was the first time I wrote a math paper, and it turned out to be no less frustrating than my mathematical work, but much less fun. Searching for new patterns on the edge of knowledge was captivating and exciting. Sitting at my desk, trying to organize my thoughts and put them on paper, was an entirely different process. As someone told me later, writing papers was the punishment we had to endure for the thrill of discovering new mathematics. This was the first time I was so punished.
I came back to Fuchs with different drafts, and he read them carefully, pointing out deficiencies and suggesting improvements. As always, he was extremely generous with his help. From the beginning, I put Fuchs’ name as one of the coauthors, but he flatly refused. “This is your paper,” he said. Finally, Fuchs declared that the article was ready, and he told me that I should submit it to Functional Analysis and Applications, the math journal run by Israel Moiseevich Gelfand, the patriarch of the Soviet mathematical school.
A compact charismatic man, then in his early seventies, Gelfand was a legend in the Moscow mathematical community. He presided over a weekly seminar held at a grand auditorium on the fourteenth floor of the main MGU building. This was an important mathematical and social event, which had been running for more than fifty years and was renowned all over the world. Fuchs was a former collaborator of Gelfand (their work on what became known as “Gelfand–Fuchs cohomology” was widely known and appreciated) and one of the most senior members of Gelfand’s seminar. (The others included A.A. Kirillov, who was Gelfand’s former student, and M.I. Graev, Gelfand’s longtime collaborator).
The seminar was unlike any other seminars I have ever attended. Usually, a seminar has fixed hours – in the U.S. one hour or an hour and a half – and there is a speaker who prepares a talk on a particular topic chosen in advance. Occasionally, the audience members ask questions. It was not at all like this at the Gelfand seminar. It met every Monday evening, and the official starting time was 7:00 pm. However, the seminar rarely started before 7:30, and it usually began around 7:45 to 8:00. During the hour or so before the start, the members of the seminar, including Gelfand himself (who usually arrived around 7:15–7:30), would wander around and talk to each other inside the auditorium and in the large foyer outside. Clearly, this was what Gelfand had intended. This was as much a social event as a math seminar.
Most of the mathematicians coming to the Gelfand seminar worked at various places that were not affiliated with MGU. Gelfand’s seminar was the only place where they could meet their peers, find out what was happening in the world of mathematics, share their ideas, and forge collaborations. Since Gelfand was himself Jewish, his seminar was considered as one of “safe havens” for Jews and even hailed as “the only game in town” (or one of very few) in which Jewish mathematicians could participate (though, in fairness, many other seminars at MGU were open to the public and were run by people who were not prejudiced against any ethnicities). No doubt, Gelfand gladly took advantage of this.
The anti-Semitism that I had experienced at my entrance exam to MGU spread to all levels of academia in the Soviet Union
. Earlier, in the 1960s and early 1970s, even though there were restrictions, or “quotas,” for students of Jewish background, they could still get in as undergraduates at the Mekh-Mat (the situation gradually worsened throughout the 1970s and early 1980s, to the point where in 1984, when I was applying to Mekh-Mat, almost no Jewish students were accepted).3 But even in those years, it was nearly impossible for these students to enter graduate school. The only way Jewish students could do this was to go to work somewhere for three years after getting the bachelor’s degree, and then one could be sent to graduate school by their employer (often, located somewhere in a faraway province). And even if they managed to overcome this hurdle and get a Ph.D., it was impossible for them to find an academic job in mathematics in Moscow (at MGU, for example). Either they had to settle for a job somewhere in the province or join one of many research institutes in Moscow that had little or nothing to do with mathematics. The situation was even more difficult for those who were not originally from Moscow, because they did not have propiska, a Moscow residency stamp in their interior passport, which was required for employment in the capital.
Even the most exceptional students got such treatment. Vladimir Drinfeld, a brilliant mathematician and future Fields Medal winner about whom we will talk more later, was allowed to become a graduate student at Mekh-Mat right after obtaining his bachelor’s degree (though from what I’ve heard it was very difficult to arrange), but being a native of Kharkov, Ukraine, he could not be employed in Moscow. He had to settle for a teaching job at a provincial university in Ufa, an industrial city in the Ural Mountains. Eventually, he got a job as a researcher at the Institute for Low Temperature Physics in Kharkov.
Love and Math Page 7