##### Department of Mathematics,

University of California San Diego

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### Math 278C - Optimization Seminar

## Eric Evert

#### UCSD

## Extreme points of matrix convex sets

##### Abstract:

The solution set of a linear matrix inequality (LMI) is known as a spectrahedron. Free spectrahedra, obtained by substituting matrix tuples instead of scalar tuples into an LMI, arise canonically in the theory of operator algebras, systems and spaces and the theory of matrix convex sets. Indeed, free spectrahedra are the prototypical examples of matrix convex sets, set with are closed with respect to taking matrix convex combinations. They also appear in systems engineering, particularly in problems governed by a signal flow diagram. Extreme points are an important topic in convexity; they lie on the boundary of a convex set and capture many of its properties. For matrix convex sets, it is natural to consider matrix analogs of the notion of an extreme point. These notions include, in increasing order of strength, Euclidean extreme points, matrix extreme points, and Arveson boundary points. This talk will, in the context of matrix convex sets over $\mathbb{R}^g$, provide geometric unified interpretations of Euclidean extreme points, matrix extreme points, and Arveson boundary points. Additionally, methods for computing Arveson boundary points of free spectrahedra will be discussed.

Organizer: Jiawang Nie

### March 1, 2017

### 3:00 PM

### AP&M 7321

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