Title: Geometry and Topology of Spaces of Structured Matrices
Speaker: Tom Needham (Florida State University)
Speaker's URL: https://sites.google.com/site/tneedhammath
Abstract: A finite unit norm tight frame (FUNTF) is a spanning set of unit vectors in a finite-dimensional Hilbert space such that the spectrum of singular values of an associated operator is constant. In signal processing applications, it is desirable to use FUNTFs to encode signals, as such representations are proven to optimally robust to noise. This naturally gives rise to questions about the geometry and topology of the space of FUNTFs. For example, the conjecture that every space of FUNTFs is connected was open for 15 years, and variants of this problem still remain open. I will discuss recent work with Clayton Shonkwiler, where we answer several questions about random matrix theory and optimization in spaces of structured matrices, using tools from symplectic geometry and geometric invariant theory. I will also describe how similar ideas can be used to understand the geometry of the space of normal matrices, with applications to balancing directed graphs, and the (non-)existence of topological obstructions to extending these techniques to fields of linear operators on vector bundles (the latter is joint work with Samuel Ballas and Clayton Shonkwiler).
URL associated with Seminar: https://tgda.osu.edu/
