Mathematics

Toeplitz Operators and Bott Periodicity

Timothy Drake (UIUC)

Abstract: The Toeplitz operators are a class of bounded operators on the Hardy space $H^2$, which may be thought of as infinite-dimensional analogues of Toeplitz matrices. They form a $C^*$-algebra known as the Toeplitz algebra, which plays a central role in the theory of $C^*$-algebras as the "universal $C^*$-algebra generated by a single isometry". I will discuss the properties of this algebra and its relation to $K$-theory, and use it to sketch a proof of the Bott Periodicity Theorem for $C^*$-algebras.

Generalized notions of sparsity and restricted isometry property: A unified framework and applications

Kiryung Lee (Georgia Tech)

Abstract: The restricted isometry property (RIP) has been an integral tool in the analysis of various inverse problems with sparsity models in signal processing and statistics. We propose generalized notions of sparsity and provide a unified framework for the RIP of structured random measurements given by isotropic group actions. Our results extend the RIP for partial Fourier measurements by Rudelson and Vershynin to a much broader context and identify a sufficient number of group structured measurements for the RIP on generalized sparsity models. We illustrate the main results on an infinite dimensional example, where the sparsity represented by a smoothness condition approximates the total variation. We also discuss fast dimensionality reduction on generalized sparsity models. In generalizing sparsity models, the parameter accounting for the level of sparsity becomes no longer sub-additive. Therefore, the RIP does not preserve distances between sparse vectors. We show a weaker version with additive distortion, which is similar to analogous property arising in the 1-bit compressed sensing problem. This is a joint work with Marius Junge.

Distal and non-distal ordered abelian groups

Allen Gehret (UCLA Math)

Abstract: I will discuss various things we know about distal and non-distal ordered abelian groups. This is joint work with Matthias Aschenbrenner and Artem Chernikov.

H-principle for Symplectic structure

Venkata Sai Narayana Bavisetty (UIUC)

Abstract: H-principle is a tool which helps reduce problems involving analysis and geometry into problems involving just geometry. I will start out by motivating the basic idea of H-principle and then use the idea of Holonomic approximation to sketch a proof of H-principle for symplectic structure. This will be an introductory talk and even though the title sounds esoteric, I hope to convince you that this is a central theme in Geometry.