Information, Geometry, and Physics Seminar

Convex sets are often easier to study than more general spaces in mathematics; it is therefore of interest to understand maps that can transform a general set into a convex one. This talk will present a topological property of a continuous map from a general topological space into Euclidean space has a convex image. In particular, for any given map f, we associate an object called the `Lagrangian dual bundle', and we show that the existence of a section of this bundle implies that the image of f is convex (up to some technical conditions). We will present examples arising from homogeneous spaces, such as the set of Hermitian matrices with prescribed eigenvalues or the set of matrices with prescribed singular values, and give connections to the well known Kostant convexity theorem. One aim of this talk is to give a complete proof of the fact that the image of a sphere of under any Hermitian quadratic map into R^3 is convex. We will also aim to illustrate how these techniques can also be used in the context of convex optimization to give exactness results for semidefinite programming relaxations of natural optimization problems. Generally, this talk will highlight broad connections between convex geometry, algebraic topology, and algebra.