## 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, October 3, 2018 — 2:00 PM EDT

**Eli Shamovich, Pure Mathematics, University of Waterloo**

"Polynomials and rational functions"

In this talk, we will apply the theory of noncommutative function to two classes of examples, namely polynomials and rational functions. Polynomials for us are elements of the free algebra, but what are rational functions? We will show that an nc function given on each level by matrices of polynomials in the coordinates of uniformly bounded degree is, in fact, an element of the free algebra. We will define what are rational nc functions using ideas of P. M. Cohn and Amitsur and discuss their realization theory. This will allow us to discuss various results in free probability, algebra and even the theory of computations. In particular, we will discuss the result of Helton, Mai, and Speicher that allows one to evaluate rational functions on stably finite algebras and its application to random matrices.

Location

MC 5479

,

Canada

,

Canada

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.