## Contact Info

Pure MathematicsUniversity of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x43484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

Wednesday, August 11, 2021 — 3:00 PM EDT

**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

Event tags

University of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x43484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

University of Waterloo

University of Waterloo

43.471468

-80.544205

200 University Avenue West

Waterloo,
ON,
Canada
N2L 3G1

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