## 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

Friday, October 19, 2018 — 3:00 PM EDT

**Ben Passer, Department of Pure Mathematics, University of Waterloo**

"Containment of Matrix Convex Sets"

This talk contains joint work with Orr Shalit and Baruch Solel.

A set $S \subseteq \bigcup_{n=1}^\infty M_n(\mathbb{C})^d$ is called matrix convex if it is closed under direct sums and the application of unital completely positive maps. We may view $S$ as a disjoint union of its levels $S_n \subseteq M_n(\mathbb{C})^d$, each of which is convex and is closed under simultaneous unitary conjugations. Following work of Davidson, Dor-On, Shalit, and Solel, we seek to determine how much information about $S$ is contained in its first level, $S_1 = K$.

If $K$ is a compact convex set in $\mathbb{C}^d$, then there could be many matrix convex sets which have $K$ as the first level. However, there are always two extremal options $\mathcal{W}^\text{min}(K)$ and $\mathcal{W}^{\text{max}}(K)$. We study the disparity between these two matrix convex sets by searching for an optimal constant $C$ with

\begin{equation*}

\mathcal{W}^{\text{max}}(K) \subseteq C \cdot \mathcal{W}^{\text{min}}(K).

\end{equation*}

Among our results, we find an optimal scaling constant when $K$ is an $\ell^p$ ball, and we completely characterize when the constant $C = 1$ may be achieved. In doing so, we make use of theorems in convex geometry, and we find that it suffices to check a single matrix level $n = 2^{d-1}$.

Time permitting (which occurs with probability zero), I will discuss related/generalized problems concerning products of matrix convex sets, dilation theory, and extreme points.

MC 5417

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.