# Fields Medals at Madrid

Andrei Okounkov

Okounkov was born in Moscow in 1969 and is now at Princeton University. His work centres on representation theory, combinatorics, and applications to physics. Representation theory studies how a given abstract group can be realised as a group of linear transformations. A triumph of classical mathematics is the complete characterisation of all representations of the symmetric group—the group of all permutations of some finite set, usually taken to be all integers from 1 to n for some chosen n. A permutation is just a way to rearrange the order of the numbers. If n = 9, for instance, then a typical permutation looks like 461782935.

A novel feature of Okounkov’s research is the introduction of an element of randomness. For example, suppose that the integer n is very large, and we choose a random permutation of n. What is the probability that this permutation has some specific feature—for example, that it contains an increasing subsequence of numbers of specified length? (In the permutation 461782935, the increasing subsequences include 46789 and 1235.)

Surprisingly, questions of this general nature crop up in mathematical physics, when thinking about quantum fluctuations. A fundamental concept in quantum theory is that of a Hermitian matrix , which is a square matrix of complex numbers with useful technical properties. In particular, the eigenvalues of a Hermitian matrix are always real. These eigenvalues are related to the spectrum of the quantum system—the possible energy-levels—and these are things that experimentalists can observe.  In particular, the statistical fluctuations in the largest eigenvalue of a randomly chosen Hermitian matrix provide insights in to quantum theory. The probability distribution of these fluctuations—how likely a fluctuation of given size is—has been calculated. It turns out to be identical to the distribution of fluctuations in a combinatorial quantity determined by random permutations. Namely, how does the size of the longest increasing subsequence fluctuate?

This unexpected connection led to a more general conjecture, and Okounkov proved the conjecture using representation theory and his own theory of random surfaces. The link to surfaces enriched the area by introducing ideas from algebraic geometry. This in turn allowed Okounkov to develop new results in algebraic geometry. He also worked out some applications to statistical mechanics—the physics of large numbers of atoms subjected to heat, which causes them to vibrate, effectively at random. His ideas shed light on how a crystal melts as it is heated. The standard example here is to consider a crystal in the form of a cube, assumed to be assembled from a gigantic number of tiny sub-cubes in a regular lattice (common salt and lead sulphide have this type of crystal structure, for example). The process of melting in effect removes random sub-cubes from the surface of the cube. Even though the process is random, one particular pattern emerges with very high probability. The melted part of the cube, viewed from the diagonal direction, always looks like a well-known algebraic curve called a cardioid.