Pavel Etingof (MIT): Representation theory without vector spaces

A modern view of representation theory is that it is a study not just of individual representations (say, finite dimensional representations of an affine (super)group scheme G over an algebraically closed field k) but also of the category Rep(G) formed by them. The properties of Rep(G) can be summarized by saying that it is a symmetric tensor category (shortly, STC) which uniquely determines G. It is therefore natural to ask: does the study of STC reduce to group representation theory, or is it more general? In other words, do there exist STC other than Rep(G)? If so, this would be interesting, since one can do algebra in any STC, and in categories other than Rep(G) this would be a new kind of algebra. The answer turns out to be “yes”, and beautiful examples in characteristic zero were provided by Deligne-Milne in 1981. These very interesting categories are interpolations of representation categories of classical groups GL(n), O(n), Sp(n) to arbitrary values of n in k. Deligne later generalized them to symmetric groups and also to characteristic p, where, somewhat unexpectedly, one needs to interpolate n to p-adic integer values rather than elements of k. All these categories violate an obvious necessary condition for a STC to have any realization by finite dimensional vector spaces (and in particular to be of the form Rep(G)): for each object X the length of the n-th tensor power of X grows at most exponentially with n. We call this property “moderate growth”. So it is natural to ask if there exist STC of moderate growth other than Rep(G). A remarkable theorem of Deligne (2002) says that in characteristic zero, the answer is “no”: any such category is of the form Rep(G), where G is an affine supergroup scheme; in other words, it can be realized in supervector spaces. In particular, algebra in these categories is just the usual one with equivariance under some supergroup G. However, in characteristic p the situation is much more interesting. Namely, Deligne’s theorem is known to fail in any characteristic p>0. The simplest exotic symmetric tensor category of moderate growth (i.e., not of the form Rep(G)) for p>3 is the semisimplification of the category of representations of Z/p, called the Verlinde category. For example, for p=5, this category has an object X such that X^2=X+1, so X cannot be realized by a vector space (as its dimension would have to be the golden ratio or its conjugate). I will discuss some aspects of algebra in these categories, in particular failure of PBW theorem for Lie algebras (and how to fix it) and Ostrik’s generalization of Deligne’s theorem in characteristic p. I will also discuss new STC in characteristic p constructed in my joint work with Dave Benson and Victor Ostrik. There is a hope that any STC of moderate growth in characteristic p is the representation category of an affine group scheme in one of them.

In an instance of the (metric) traveling salesperson problem (TSP), we are given a list of n cities and their pairwise symmetric distances satisfying the triangle inequality, and we want to find the shortest tour that visits all cities exactly once and returns back to the starting point. I will talk about an algorithm that provably returns a tour whose cost is at most 50-eps percent more than the optimum where eps>0 is a small constant independent of n. This slightly improves classical algorithms of Christofides and Serdyukov from 1970s.

The proof exploits a deep connection to the rapidly evolving area of geometry of polynomials. In this area, we encode discrete phenomena, in our case uniform spanning tree distributions, in coefficients of complex multivariate polynomials, and we understand them via the interplay of the coefficients, zeros, and function values of these polynomials. Over the last fifteen years, this perspective has led to several breakthroughs in computer science and Mathematics such as simpler proofs of the van-der-Waerden conjecture, and resolutions of the Kadison-Singer and the Feder-Mihail conjectures. I will discuss how properties of strongly Rayleigh distributions, a family of probability distributions whose generating polynomial is real stable, can be used to design and analyze algorithms for TSP.

Based on a joint work with Anna Karlin and Nathan Klein

A random planar map is a canonical model for a discrete random surface which is studied in probability, combinatorics, mathematical physics, and geometry. Liouville quantum gravity is a canonical model for a random 2d Riemannian manifold with roots in the physics literature. After introducing these objects, I will present a joint work with Xin Sun where we prove convergence of random planar maps to Liouville quantum gravity under a discrete conformal embedding which we call the Cardy embedding.

In the beginning of 19th century Napoleon set a prize for the best mathematical explanation of Chladni’s resonance experiments. Nodal geometry studies the zeroes of solutions to elliptic differential equations such as the visible curves that appear in these physical experiments. We will discuss geometrical and analytic properties of zero sets of harmonic functions and eigenfunctions of the Laplace operator. For harmonic functions on the plane there is an interesting relation between local length of the zero set and the growth of harmonic functions. The larger the zero set is, the faster the growth of harmonic function should be and vice versa. Zero sets of Laplace eigenfunctions on surfaces are unions of smooth curves with equiangular intersections. Topology of the zero set could be quite complicated, but Yau conjectured that the total length of the zero set is comparable to the square root of the eigenvalue for all eigenfunctions. We will start with open questions about spherical harmonics and explain some methods to study nodal sets, which are zero sets of solutions of elliptic PDE.