Love & Math - Edward Frenkel

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 more 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