## 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

Monday, October 26, 2015 — 4:00 PM EDT

**Florent Benaych-Georges, University Paris 5 **

“The Single Ring Theorem”

The singular values of a complex matrix are the eigenvalues of the positive part of its polar decomposition. While the eigenvalues of the matrix are usually related to its algebraic properties, the singular values are related to its analytic properties (the largest one is its operator norm, the smallest one, when positive, is the inverse of the norm of the inverse of the matrix, etc...).

Though there are relationships between eigenvalues and singular values, neither are the singular values determined by the eigenvalues, nor are the eigenvalues determined by the singular values. How- ever, if one considers large isotropic random matrices (isotropic matrices are random non-Hermitian matrices which are invariant, in law, under the left and right actions of the unitary group), then the Single Ring Theorem, proved by Guionnet, Krishnapur and Zeitouni in 2012 (but conjectured for long by physicists), gives an asymptotic relation between the singular values and the eigenvalues, partly based on free probability theory. In this talk, we will state this result, explain some ideas used in its proof, and then present some of its recent extensions, especially on the boundary of the spectrum and at local level in the bulk of the spectrum.

MC 5501

*Refreshments will be served in MC 5403 at 3:30 p.m. Everyone is welcome to attend. *

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

The University of Waterloo acknowledges that much of our work takes place on the traditional territory of the Neutral, Anishinaabeg and Haudenosaunee peoples. Our main campus is situated on the Haldimand Tract, the land granted to the Six Nations that includes six miles on each side of the Grand River. Our active work toward reconciliation takes place across our campuses through research, learning, teaching, and community building, and is centralized within our Office of Indigenous Relations.