Georgia Institute of Technology College of Sciences

Building on work of Jordan from 1870, in 1979 Harris showed that a geometric monodromy group associated to a problem in enumerative geometry is equal to the Galois group of an associated field extension. Vakil gave a geometric-combinatorial criterion that implies a Galois group contains the alternating group. With Brooks and Martin del Campo, we used Vakil's criterion to show that all Schubert problems involving lines have at least alternating Galois group. My talk will describe this background and sketch a current project to systematically determine Galois groups of all Schubert problems of moderate size on all small classical flag manifolds, investigating at least several million problems. This will use supercomputers employing several overlapping methods, including combinatorial criteria, symbolic computation, and numerical homotopy continuation, and require the development of new algorithms and software.

I will review the well known method (pushed mainly by Karlin and McGregor) to study birth-and-death processes with the help of orthogonal polynomials. I will then look at several extensions of this idea, including ¨poker dice¨ (polynomials in several variables) and quantum walks (polynomials in the unit circle).

An intesting class of bounded operators or algebras of bounded operators on Hilbert spaces, particularly on Hilbert spaces of holomorphic functions, have a natural interpretation in terms of concepts from complex geometry. In particular, there is an intrinsic hermitian holomorphic vector bundle and many questions can be answered in terms of the Chern connection and the associated curvature. In this talk we describe this setup and some of the results obtained in recent years using this approach. The emphasis will be on concrete examples, particularly in the case of Hilbert spaces of holomorphic functions such as the Hardy and Bergman spaces on the unit sphere in C^n.

Tom Church, Jordan Ellenberg and I recently discovered that the i-th Betti number of the space of configurations of n points on any manifold is given by a polynomial in n. Similarly for the moduli space of n-pointed genus g curves. Similarly for the dimensions of various spaces of homogeneous polynomials arising in algebraic combinatorics. Why? What do these disparate examples have in common? The goal of this talk will be to answer this question by explaining a simple underlying structure shared by these (and many other) examples in algebra and topology.