I was terrified that in three years they would say this about me, so I constantly felt under pressure to produce and to succeed.
Meanwhile, the economic situation in the Soviet Union was rapidly deteriorating, and the outlook was very uncertain. Observing all of this from the inside and convinced that I had no future in the Soviet Union, my parents took to calling me at regular intervals urging me not to come back. In those days, it was very difficult (and expensive) to call the U.S. from the Soviet Union. My parents were afraid that their home phone was tapped, and because of that, they would travel to the Moscow central post office and place a call from there. Such a trip would take them almost an entire day. But they were determined, even though they missed me terribly, to do everything in their power to convince me to stay in America. They were absolutely sure that this was in my best interests.
Borya also had my best interests in mind, but he took his stance in part on moral grounds. He was going against the grain, and I admired him for this. But I also had to admit that he could afford to do so because of his relatively comfortable situation in Moscow (though that was soon going to change, and he would be forced to spend a few months a year abroad – mostly, in Japan – to provide for his family). My situation was entirely different: I had no place to stay in Moscow, and only a temporary propiska, the right to live there. Though Yakov Isaevich had secured a temporary job as an assistant for me at Kerosinka, it provided a meager salary that would barely be enough to rent a room in Moscow. Because of anti-Semitism, getting enrolled in a graduate school would be an uphill battle, and my future employment prospects looked even bleaker.
At the end of November, Arthur Jaffe called me to his office and offered to extend my stay at Harvard till the end of May. I had to make a decision quickly, but I was torn. I enjoyed my lifestyle in Boston. I felt this was my place to be. With Harvard and MIT, Cambridge was one of the premier centers of mathematics. Some of the world’s most brilliant minds were here, and I could just knock on their doors, ask them questions, learn from them. There were also plenty of seminars where pretty much all exciting discoveries were reported soon after they were made. I was surrounded by the brightest students. This was the most stimulating environment for a young aspiring mathematician one could imagine. Moscow used to be such a place, but not any more.
But this was the first time I was away from home for such a long time. I missed my family and friends. And Borya, my teacher, who was the closest person to me in Cambridge, was adamant that I should go back in December, as planned.
Every morning I woke up terrified, thinking “What should I do?” In retrospect, the decision seems like a no-brainer. But with so many different forces colliding, all at the same time, making that decision wasn’t easy. Finally, after some anguished deliberations, I decided to follow the advice of my parents and stay, and I told Jaffe about it. My friends Reshetikhin and Tsygan did the same.
Borya was unhappy about this, and I felt that I had let him down. It was a moment of sadness and great uncertainty when I saw him off at the Logan Airport, going back to Moscow, in mid-December. We didn’t know what the future held for either of us; we didn’t even know whether we would be able to meet again any time soon. I had ignored Borya’s advice. But I was still afraid that I might fulfill his fears.
*Vera Serganova, the fourth recipient of the Harvard Prize Fellowship, came in the spring.
Chapter 14
Tying the Sheaves of Wisdom
The spring semester brought more visitors to Harvard, one of whom, Vladimir Drinfeld, changed the direction of my research, and in many ways, my mathematical career. And it all happened because of the Langlands Program.
I had heard about Drinfeld before. He was only thirty-six at the time, but already a legend. Six months after we met, he was awarded the Fields Medal, one of the most prestigious prizes in mathematics, considered by many as an equivalent of the Nobel Prize.
Drinfeld published his first math paper at the age of seventeen, and by age twenty he was already breaking new ground in the Langlands Program. Originally from Kharkov, Ukraine, where his father was a well-known math professor, Drinfeld studied at Moscow University in the early 1970s. (At that time, Jews also had trouble gaining admission to MGU, but a certain percentage of Jewish students was admitted.) By the time he received his college degree from the MGU, he was already world-renowned for his work, and he was accepted to the graduate school, which was extraordinary for a Jewish student. His advisor was Yuri Ivanovich Manin, one of the world’s most original and influential mathematicians.
Even Drinfeld, however, was not able to escape anti-Semitism entirely. After getting his Ph.D., he was unable to get a job in Moscow and had to spend three years at a provincial university in Ufa, an industrial city in the Ural Mountains. Drinfeld was reluctant to go to Ufa, not least because there were no mathematicians there working in the areas he cared about. But as the result of his stay in Ufa, Drinfeld wrote an important work in the theory of integrable systems, a subject that was quite far from his interests, together with a local mathematician Vladimir Sokolov. The integrable systems they created are now known as the Drinfeld–Sokolov systems. After three years in Ufa, Drinfeld was finally able to secure a job in his hometown, at the Kharkov Institute for Low Temperature Physics. This was a relatively comfortable job, and he could stay close to his family, but being in Kharkov, Drinfeld was isolated from the Soviet mathematical community, which was concentrated in Moscow and, to a lesser extent, St. Petersburg.
Despite all this, working essentially alone, Drinfeld kept producing marvelous results in diverse areas of math and physics. In addition to proving important conjectures within the Langlands Program and opening a new chapter in the theory of integrable systems with Sokolov, he also developed the general theory of quantum groups (originally discovered by Kolya Reshetikhin and his co-authors), and many other things. The breadth of his contributions was staggering.
Attempts were made to hire Drinfeld in Moscow. I’ve been told that physicist Alexander Belavin, for example, tried to bring Drinfeld to the Landau Institute for Theoretical Physics near Moscow. To raise the chances of success, Belavin and Drinfeld solved together an important problem of classification of solutions to the “classical Yang–Baxter equation,” which many physicists were interested in at the time. Their paper was published to much acclaim in Gelfand’s journal Functional Analysis and Applications (I believe that this was the longest article ever published by Gelfand, which is saying a lot about its importance). It was that work that led Drinfeld to the theory of quantum groups, which revolutionized many areas of mathematics. Alas, none of these hiring plans worked. Anti-Semitism and Drinfeld’s lack of propiska in Moscow were a deadly combination. Drinfeld remained in Kharkov, visiting Moscow only rarely.
Drinfeld was invited to visit Harvard in the spring of 1990, and this proved to be serendipitous for me. He arrived in late January. Having heard all the legends about him, I was a bit intimidated at first, but he turned out to be extremely nice and generous. Soft-spoken, carefully weighing his words, Drinfeld was also the very model of clarity when he talked about mathematics. When he explained things to you, he did not try to do it in a self-aggrandizing way, as if he was unveiling a big mystery which you would never be able to fully understand on your own (which unfortunately is the case for some of our colleagues, who shall remain nameless). On the contrary, he was always able to put things in the simplest and clearest possible way, so after he explained something to you, you felt like you’d known it all along.
More importantly, Drinfeld told me right away that he was very interested in my work with Feigin, which he wanted to use for his new project related to the Langlands Program.
Let’s recall from Chapter 9 the three columns of André Weil’s Rosetta stone:
The Langlands Program was originally developed within the left and the middle columns: number theory and curves over finite fields. The idea of the Langlands Program is to set up a relation between the repr
esentations of a Galois group and automorphic functions. The concept of the Galois group makes perfect sense in the left and the middle columns of the Rosetta stone, and there are suitable automorphic functions that can be found in another area of mathematics called harmonic analysis.
Prior to Drinfeld’s work, it was not clear whether there was an analogue of the Langlands Program for the right column, the theory of Riemann surfaces. The means to include Riemann surfaces started to emerge in the early 1980s in Drinfeld’s work, which was followed by the French mathematician Gérard Laumon. They realized that it was possible to make a geometric reformulation of the Langlands Program that makes sense for both the middle column and the right column of André Weil’s Rosetta stone.
In the left and the middle columns of the Rosetta stone, Langlands Program relates the Galois groups and the automorphic functions. The question is then to find the right analogues of the Galois groups and the automorphic functions in the geometric theory of Riemann surfaces. We have already seen in Chapter 9 that in the geometric theory the role of the Galois group is played by the fundamental group of a Riemann surface. But we left the geometric analogues of the automorphic functions unexplored.
It turns out that the right geometric analogues are not functions, but what mathematicians call sheaves.
To explain what they are, let’s talk about numbers. We have natural numbers: 1,2,3,..., and of course they have many uses. One is that they measure dimensions. As we discussed in Chapter 10, a line is one-dimensional, a plane is two-dimensional, and for any natural number n we have an n-dimensional flat space, also known as a vector space.1 Now imagine a world in which natural numbers are replaced by vector spaces; that is, instead of number 1 we have a line, instead of number 2 we have a plane, and so on.
Addition of numbers is replaced in this new world by what mathematicians call the direct sum of vector spaces. Given two vectors spaces, each with its own coordinate system, we create a new one, which combines the coordinates of the two vector spaces, so its dimension is the sum of two dimensions. For example, a line has one coordinate, and a plane has two. Combining them, we obtain a vector space with three coordinates. This is our three-dimensional space.
Multiplication of natural numbers is replaced by another operation on vector spaces: given two vectors spaces, we produce a third one, called their tensor product. I will not give a precise definition of the tensor product here; the important point is that if the two vector spaces we start with have dimensions m and n, then their tensor product has dimension m · n.
Thus, we have operations on vectors spaces that are analogous to the operations of addition and multiplication of natural numbers. But this parallel world of vector spaces is so much richer than the world of natural numbers! A given number has no inner structure. Number 3, for example, taken by itself, has no symmetries. But a three-dimensional space does. In fact, we have seen that any element of the Lie group SO(3) gives rise to a rotation of the three-dimensional space. The number 3 is a mere shadow of the 3-dimensional space, reflecting only one attribute of this space, its dimensionality. But this number cannot do justice to other aspects of the vector spaces, such as its symmetries.
In modern math, we create a new world in which numbers come alive as vector spaces. Each of them has a rich and fulfilling personal life, and they also have more meaningful relations with each other, which cannot be reduced to mere addition and multiplication. Indeed, we can subtract 1 from 2 in only one way. But we can embed a line in a plane in many different ways.
Unlike natural numbers, which form a set, vector spaces form a more sophisticated structure, which mathematicians call a category. A given category has “objects,” such as vector spaces, but in addition, there are “morphisms” from any object to any other object.2 For example, morphisms from an object to itself in a given category are essentially the symmetries of that object that are allowed within this category. The language of categories therefore enables us to focus not on what the objects consist of, but on how they interact with each other. Because of that, the mathematical theory of categories turns out to be particularly well-adapted to computer science.3 The development of functional programming languages, such as Haskell, is just one example of a myriad of recent applications.4 It seems inevitable that the next generations of computers will be based more on category theory than on set theory, and categories will enter our daily lives, whether we realize it or not.
The paradigm shift from sets to categories is also one of the driving forces of modern math. It is referred to as categorification. We are, in essence, creating a new world, in which the familiar concepts are elevated to a higher level. For example, numbers get replaced by vector spaces. The next question is: what should become of functions in this new world?
To answer this question, let’s revisit the notion of a function. Suppose we have a geometric shape, like a sphere or a circle, or the surface of a donut. Call it S. As we already discussed before, mathematicians refer to such shapes as manifolds. A function f on a manifold S is a rule that assigns to each point s in S a number, called the value of the function f at the point s. We denote it by f(s).
An example of a function is temperature, with our manifold S simply the three-dimensional space we live in. At each point s we can measure the temperature, which is a number. This gives us a rule assigning to each point a number, so we get a function. Likewise, barometric pressure also gives us a function.
For a more abstract example, let S be the circle. Each point of the circle is determined by an angle, which, as before, we will call φ. Let f be the sine function. Then the value of this function at the point of the circle corresponding to the angle φ is sin(φ). For example, if φ = 30 degrees (or π/6, if we measure angles in radians rather than degrees), then the value of the sine function is 1/2. If φ = 60 degrees (or π/3), then it is , and so on.
Now let’s replace numbers by vector spaces. So a function will become a rule that assigns to each point s in a manifold S, not a number, but a vector space. Such a rule is called a sheaf. If we denote a sheaf by the symbol , then the vector space assigned to a point s will be denoted by .
Thus, the difference between functions and sheaves is in what we assign to each point of our manifold S: for functions, we assign numbers to points, and for sheaves, we assign vector spaces. For a given sheaf, these vector spaces can be of different dimensions for different points s. For example, on the picture below, most of these vector spaces are planes (that is, two-dimensional vector spaces), but there is one that is a line (that is, a one-dimensional vector space). Sheaves are categorifications of functions, in the same way as vector spaces are categorifications of numbers.
Though this is beyond the scope of this book, a sheaf is actually more than just a disjoint collection of vector spaces assigned to the points of our manifold. The fibers of a given sheaf at different points have to be related to each other by a precise set of rules.5
What matters to us at the moment is that there is a deep analogy between functions and sheaves, discovered by the great French mathematician Alexander Grothendieck.
The influence of Grothendieck on modern mathematics is virtually unparalleled. If you ask who was the most important mathematician of the second half of the twentieth century, many mathematicians will say without hesitation: Grothendieck. Not only did he almost singlehandedly create modern algebraic geometry, he also transformed the way we think about mathematics as a whole. The dictionary between functions and sheaves, which we use in the geometric reformulation of the Langlands Program, is an excellent example of the profound insights that are characteristic of Grothendieck’s work.
To give you a gist of Grothendieck’s idea, I recall the notion of a finite field from Chapter 8. For each prime number p, there is a finite field with p elements: {0,1,2,..., p−1}. As we discussed, these p elements comprise a numerical system with operations of addition, subtraction, multiplication, and division modulo p, which obey the same rules as the corresponding operations on
the rational and real numbers.
But there is also something special about this numerical system. If you take any element of the finite field {0,1,2,..., p−1} and raise it to pth power – in the sense of the arithmetic modulo p that we discussed earlier – you will get back the same number! In other words,
This formula was proved by Pierre Fermat, the mathematician who came up with Fermat’s Last Theorem. Unlike the proof of the latter, though, the proof of the above formula is fairly simple. It could even fit in the margin of a book. I’ve put it in the back of this one.6 To distinguish this result from Fermat’s Last Theorem (sometimes also referred as Fermat’s Great Theorem), it is called Fermat’s little theorem.
For example, set p = 5. Then our finite field is {0,1,2,3,4}. Let’s raise each of them to the 5th power. Surely, 0 to any power is 0, and 1 to any power is 1, so no surprises here. Next, let’s raise 2 to the 5th power: we then get 32. But 32 = 2+5·6, so modulo 5 this is 2 – we get back 2, as promised. Let’s take the 5th power of 3: we get 243, but this is 3+5·48, so it is 3 modulo 5. Again, we get back the number we started with. And finally, let’s try the same with 4: its 5th power is 1024, which is 4 modulo 5. Bingo! I encourage you to check that a3 = a modulo 3, and a7 = a modulo 7 (for larger primes you might need a calculator to verify Fermat’s little theorem).
What’s also remarkable is that a similar equation forms the basis of the RSA encryption algorithm widely used in online banking.7
This formula ap = a is more than a neat discovery – it means that the operation of raising numbers to the pth power, sending a to ap, is an element of the Galois group of the finite field. It is called the Frobenius symmetry, or simply the Frobenius. It turns out that the Galois group of the finite field of p elements is generated by this Frobenius.8
Love and Math Page 18