Love and Math
Page 12
Think of different areas of modern math as languages. We have sentences from these languages that we think mean the same thing. We put them next to each other, and little by little we start developing a dictionary that allows us to translate between different areas of mathematics. André Weil gave us a suitable framework for understanding connections between number theory and geometry, a kind of “Rosetta stone” of modern math.
On one side, we have objects of number theory: rational numbers and other number fields that we discussed in the previous chapter, such as the one obtained by adjoining , and their Galois groups.
On the other side, we have the so-called Riemann surfaces. The simplest example is the sphere.2
The next example is the torus, the surface in the shape of a donut. I want to emphasize that we are considering here the surface of the donut, not its interior.
The next example is the surface of a Danish pastry, shown on the next picture (or you can think of it as the surface of a pretzel).
The torus has one “hole,” and the Danish has two “holes.” There are also surfaces having n holes for any n = 3, 4, 5,... Mathematicians call the number of holes the genus of the Riemann surface.* They are named after the German mathematician Bernhard Riemann, who lived in the nineteenth century. His work opened up several important directions in mathematics. Riemann’s theory of curved spaces, which we now call Riemannian geometry, is the cornerstone of Einstein’s general relativity theory. Einstein’s equations describe the force of gravity in terms of the so-called Riemann tensor expressing the curvature of space-time.
At first glance, number theory has nothing in common with the Riemann surfaces. However, it turns out that there are many analogies between them. The key point is that there is another class of objects between these two.
To see this, we have to realize that a Riemann surface may be described by an algebraic equation. For example, consider again a cubic equation such as
As we noted earlier, when we talk about solutions of such an equation, it is important to specify to what numerical system they belong. There are many choices, and different choices give rise to different mathematical theories.
In the previous chapter, we discussed solutions modulo prime numbers, and that’s one theory. But we can also look for solutions in complex numbers. That’s another theory, which yields Riemann surfaces.
People often ascribe almost mystical qualities to complex numbers, as if these are some incredibly complicated objects. The truth is that they are no more complicated than the numbers we discussed in the previous chapter when trying to make sense of the square root of 2.
Let me explain. In the previous chapter, we adjoined to the rational numbers two solutions of the equation x2 = 2, which we denoted by and . Now, instead of looking at the equation x2 = 2, we look at the equation x2 = –1. Does it look much more complicated than the previous one? No. It has no solutions among rational numbers, but we are not afraid of this. Let’s adjoin the two solutions of this equation to the rational numbers. Denote them by and . They solve the equation x2 = –1, that is,
There is only a minor difference with the previous case. The number is not rational, but it is a real number, so by adjoining it to the rational numbers, we don’t leave the realm of real numbers.
We can think of real numbers geometrically as follows. Draw a line and mark two points on it, which will represent numbers 0 and 1. Then mark the point to the right of 1 whose distance to 1 is equal to the distance between 0 and 1. This point will represent number 2. We represent all other integers in a similar fashion. Next, we mark rational numbers by subdividing the intervals between the points representing the integers. For example, the number is exactly halfway between 0 and 1; the number is one-third of the way from 2 to 3, and so on. Now, the real numbers are, intuitively, in one-to-one correspondence with all points of this line.3
Recall that we encounter number as the length of the hypotenuse of the right triangle with legs of length 1. So we mark on the line of real numbers by finding a point to the right of 0 whose distance to 0 is equal to the length of this hypotenuse. Likewise, we can mark4 on this line the number π, which is the circumference of a circle of diameter 1.
On the other hand, the equation x2 = –1 has no solutions among rational numbers, and it also has no solutions among real numbers. Indeed, the square of any real number must be positive or 0, so it cannot be equal to −1. So unlike and , the numbers and are not real numbers. But so what? We follow the same procedure and introduce them in exactly the same way as we introduced the numbers and . And we use the same rules to do arithmetic with these new numbers.
Let’s recall how we argued before: we noticed that the equation x2 = 2 had no solutions among the rational numbers. So we created two solutions of this equation, called them and , and adjoined them to the rational numbers, creating a new numerical system (which we then called a number field). Likewise, now we take the equation x2 = –1 and notice that it also has no solutions among rational numbers. So we create two solutions of this equation, denote them by and , and adjoin them to the rational numbers. It’s exactly the same procedure! Why should we think of this new numerical system as anything more complicated than our old numerical system, the one with ?
The reason is purely psychological: whereas we can represent as the length of a side of a right triangle, we don’t have such an obvious geometric representation of . But we can manipulate algebraically as effectively as .
Elements of the new numerical system we obtain by adjoining to the rational numbers are called complex numbers. Each of them may be written as follows:
where r and s are rational numbers. Compare this to the formula above expressing general elements of the numerical system obtained by adjoining . We can add any two numbers of this form by adding separately their r-parts and s-parts. We can also multiply any two such numbers by opening the brackets and using the fact that . In a similar way, we can also subtract and divide these numbers.
Finally, we extend the definition of complex numbers by allowing r and s in the above formula to be arbitrary real numbers (not just the rational numbers). Then we obtain the most general complex numbers. Note that it is customary to denote by i (for “imaginary”), but I chose not to do this to emphasize the algebraic meaning of this number: it really is just a square root of −1, nothing more and nothing less. It is just as concrete as the square root of 2. There is nothing mysterious about it.
We can get a feel for how concrete these numbers are by representing them geometrically. Just as the real numbers may be represented geometrically as points of a line, complex numbers may be represented as points of a plane. Namely, we represent the complex number as a point on the plane with coordinates r and s:5
Let’s go back to our cubic equation
and let’s look for solutions x and y that are complex numbers.
A remarkable fact is that the set of all such solutions turns out to be exactly the set of points of a torus depicted earlier. In other words, each point of the torus can be assigned to one and only one pair of complex numbers x, y solving the above cubic equation, and vice versa.6
If you’ve never thought about complex numbers before, your head might be starting to hurt just about now. This is completely natural. Wrapping one’s mind around a single complex number is challenging enough, let alone pairs of complex numbers solving some equation. It’s not obvious at all that these pairs are in one-to-one correspondence with points on the surface of a donut, so don’t be alarmed if you don’t see why this is so. In fact, many professional mathematicians would be hard-pressed to prove this surprising and non-trivial result.7
In order to convince ourselves that solutions of algebraic equations give rise to geometric shapes, let’s look at a simpler situation: solutions over the real numbers instead of complex numbers. For example, consider the equation
and let’s mark its solutions as points on the plane with coordinates x and y. The set of all such solutions is a circle of radiu
s one, centered at the origin. Likewise, solutions of other algebraic equations in two variables x and y form curves on this plane.8
Now, complex numbers are in some sense doubles of real numbers (indeed, each complex number is determined by a pair of real numbers), so it’s not surprising that solutions of such algebraic equations in complex numbers form Riemann surfaces.
In addition to real and complex solutions, we may also look for solutions x, y of these equations that take values in a finite field {0,1,2,..., p−2, p−1}, where p is a prime number. This means that when we substitute x, y in the above cubic equation, say, the left- and right-hand sides become integers that are equal to each other up to an integer multiple of p. This gives us an object that mathematicians call a “curve over a finite field.” Of course, these are not really curves. The terminology is due to the fact that when we look for solutions in real numbers, we obtain curves on the plane.9
A deep insight of Weil was that the most fundamental object here is an algebraic equation, like the cubic one above. Depending on the choice of the domain where we look for solutions, the same equation gives rise to a surface, a curve, or a bunch of points. But those are nothing but avatars of an ineffable being, which is the equation itself, the way Vishnu has ten avatars, or incarnations, in Hinduism. Somewhat serendipitously, in the letter to his sister, André Weil invoked the Bhagavad-Gita,10 a sacred text of Hinduism, in which the doctrine of avatars of Vishnu is believed to appear for the first time.11 Weil wrote poetically about what happens when the inkling of an analogy between two theories is turned into concrete knowledge:12
Gone are the two theories, gone their troubles and delicious reflections in one another, their furtive caresses, their inexplicable quarrels; alas, we have but one theory, whose majestic beauty can no longer excite us. Nothing is more fertile than these illicit liaisons; nothing gives more pleasure to the connoisseur.... The pleasure comes from the illusion and the kindling of the senses; once the illusion disappears and knowledge is acquired, we attain indifference; in the Gita there are some lucid verses to that effect. But let’s go back to algebraic functions.
The connection between Riemann surfaces and curves over finite fields should now be clear: both come from the same kind of equations, but we look for solutions in different domains, either finite fields or complex numbers. On the other hand, “any argument or result in number theory can be translated, word for word,” to curves over finite fields, as Weil put it in his letter.13 Weil’s idea was therefore that curves over finite fields are the objects that intermediate between number theory and Riemann surfaces.
Thus, we find a bridge, or a “turntable” – as Weil called it – between number theory and Riemann surfaces, and that is the theory of algebraic curves over finite fields. In other words, we have three parallel tracks, or columns:
Weil wanted to exploit this in the following way: take a statement in one of the three columns and translate it into statements in the other columns. He wrote to his sister:14
My work consists in deciphering a trilingual text; of each of the three columns I have only disparate fragments; I have some ideas about each of the three languages: but I know as well there are great differences in meaning from one column to another, for which nothing has prepared me in advance. In the several years I have worked at it, I have found little pieces of the dictionary.
Weil went on to find one of the most spectacular applications of his Rosetta stone: what we now call the Weil conjectures. The proof of these conjectures15 greatly stimulated the development of mathematics in the second half of the twentieth century.
Let’s go back to the Langlands Program. Langlands’ original ideas concerned the left column of Weil’s Rosetta stone; that is, number theory. Langlands related representations of the Galois groups of number fields, which are objects studied in number theory, to automorphic functions, which are objects in harmonic analysis – an area of mathematics that is far removed from number theory (and also far away from other columns of the Rosetta stone). Now we can ask whether this kind of relation may also be found if we replace the Galois groups by some objects in the the middle and the right columns of Weil’s Rosetta stone.
It is fairly straightforward to translate Langlands’ relation to the middle column because all the necessary ingredients are readily available. Galois groups of number fields should be replaced here by the Galois groups relevant to curves over finite fields. There also exists a branch of harmonic analysis that studies suitable automorphic functions. Already in his original work, Langlands related representations of the Galois groups and automorphic functions relevant to the middle column.
However, it is not at all clear how to translate this relation to the right column of the Rosetta stone. In order to do this, we have to find geometric analogues of the Galois groups and automorphic functions in the theory of Riemann surfaces. When Langlands first formulated his ideas, the former was known, but the latter was a big mystery. It wasn’t until the 1980s that the appropriate notion was found, starting with the pioneering work by a brilliant Russian mathematician Vladimir Drinfeld. This enabled the translation of the Langlands relation to the third column of the Rosetta stone.
Let’s discuss first the geometric analogue of the Galois group. It is the so-called fundamental group of a Riemann surface.
The fundamental group is one of the most important concepts in the mathematical field of topology, which focuses on the most salient features of geometric shapes (such as the number of “holes” in a Riemann surface).
Consider, for example, a torus. We pick a point on it – call it P – and look at the closed paths starting and ending at this point. Two such paths are shown on the picture.
Likewise, the fundamental group of any given Riemann surface consists of such closed paths on this Riemann surface starting and ending at the same fixed point P.16
Given two paths starting and ending at the point P, we construct another path as follows: we move along the first path and then move along the second path. This way we obtain a new path, which will also start and end at the point P. It turns out that this “addition” of closed paths satisfies all properties of a group listed in Chapter 2. Thus, we find that these paths indeed form a group.17
You may have noticed that the rule of addition of paths in the fundamental group is similar to the rule of addition of braids in the braid groups, as defined in Chapter 5. This is not accidental. As explained in Chapter 5, braids with n threads may be viewed as paths on the space of collections of n distinct points on the plane. In fact, the braid group Bn is precisely the fundamental group of this space.18
It turns out that the two paths on the torus shown on the above picture commute with each other; that is, adding them in two possible orders gives us the same element of the fundamental group.19 The most general element of the fundamental group of the torus is therefore obtained by following the first path M times and then following the second path N times, where M and N are two integers (if M is negative, then we follow the first path −M times in the opposite direction, and similarly for negative N). Since the two basic paths commute with each other, the order in which we follow these paths does not matter; the result will be the same.
For other Riemann surfaces, the structure of the fundamental group is more complicated.20 Different paths do not necessarily commute with each other. This is similar to braids with more than two threads not commuting with each other, as we discussed in Chapter 5.
It has been known for some time that there is a deep analogy between the Galois groups and the fundamental groups.21 This provides the answer to our first question: what is the analogue of the Galois group in the right column of Weil’s Rosetta stone? It is the fundamental group of the Riemann surface.
Our next question is to find suitable analogues of the automorphic functions, the objects that appear on the other side of the Langlands relation. And here we have to make a quantum leap. The good old functions turn out to be inadequate. They need to be replaced by m
ore sophisticated objects of modern mathematics called sheaves, which will be described in Chapter 14.
This was proposed by Vladimir Drinfeld in the 1980s. He gave a new formulation of the Langlands Program that applies to the middle and the right columns, which concern curves over finite fields and Riemann surfaces, respectively. This formulation became known as the geometric Langlands Program. In particular, Drinfeld found the analogues of the automorphic functions suitable for the right column of Weil’s Rosetta stone.
I met Drinfeld at Harvard University in the spring of 1990. Not only did he get me excited about the Langlands Program, he also told me that I had a role to play in its development. That’s because Drinfeld saw a connection between the geometric Langlands Program and the work I did as a student in Moscow. The results of this work were essential in Drinfeld’s new approach, and this in turn shaped my mathematical life: the Langlands Program has played a dominant role in my research ever since.
So let us return to Moscow, and see where I went after finishing my first paper, on braid groups.
*My editor tells me that the pretzels at the German bar near his house are genus–3 (and delicious).
Chapter 10
Being in the Loop
In Moscow in the fall of 1986, I was in the third year of my studies at Kerosinka. With the braid group paper finished and submitted, Fuchs had a question for me: “What do you want to do next?”
I wanted another problem to solve. It turned out that for several years Fuchs had been working with his former student Boris Feigin on representations of “Lie algebras.” Fuchs said it was an active area with many unsolved problems and with close ties to quantum physics.
That sure caught my attention. Even though Evgeny Evgenievich had “converted” me to math, and even though I was enchanted by mathematics, I had never lost my childhood fascination with physics. That the worlds of math and quantum physics might come together was exciting for me.