Analysis Student Seminar

Jacob Campbell, Department of Pure Mathematics, University of Waterloo

"Finite free convolutions"

In a remarkable series of papers from 2013-15, Marcus, Spielman, and Srivastava solved some important problems in graph theory, as well as the Kadison-Singer problem in operator algebras. A central role is played by expected characteristic polynomials of sums and products of randomly rotated matrices.
    
These random matrices are perennial objects of interest in free probability, since they asymptotically approximate free convolution. It turns out that looking at their expected characteristic polynomials, non-asymptotically, yields well-defined "finite free convolution" operations on polynomials, and in turn the beginnings of a "finite free probability" theory. One particular feature, which forms the connection with graph theory but is interesting in its own right, is what one might call a "quadrature" phenomenon: the continuous groups of rotations can be replaced with certain finite subgroups of reflections without changing the convolutions.
    
In joint work with Zhi Yin, we approach these finite free convolutions and quadrature results using techniques from combinatorial representation theory, namely Weingarten calculus and Schur/zonal functions. I will explain our approach, without assuming any particular knowledge of random matrices or combinatorics. Time permitting, I will mention some of the ways these ideas parallel well-established ones in free probability.

Zoom Meeting: https://us02web.zoom.us/j/87274747278?pwd=RG1Bak5lbk1GaHdIL0dtSzlBbjdiUT09