Department of

Mathematics


Seminar Calendar
for events the day of Monday, October 2, 2017.

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Monday, October 2, 2017

3:00 pm in 141 Altgeld Hall,Monday, October 2, 2017

Why Higher Categories come into Topological Field Theories

Nima Rasekh   [email] (UIUC)

Abstract: In this talk we introduce topological field theories and give various examples. The eventual goal of the talk is to see why it makes sense to use higher categories when studying topological field theories. No knowledge of any of these subjects is assumed.

4:00 pm in 245 Altgeld Hall,Monday, October 2, 2017

How to recognize a metric space

Jeremy Tyson   [email] (Department of Mathematics, University of Illinois at Urbana-Champaign)

Abstract: An embedding of one metric space $X$ into another metric space $Y$ is a one-to-one continuous map with continuous inverse. We fix the target space $Y$, which may be a finite-dimensional Euclidean space or some other normed vector space. Natural classes of embeddings include isometric embeddings (distances are exactly preserved) and bi-Lipschitz embeddings (distances are distorted by at most a fixed multiplicative constant). In this talk, we'll discuss the existence of embeddings within a fixed class into a fixed target space. For instance, which metric spaces admit an isometric embedding into a finite-dimensional Euclidean space? To indicate the subtlety of the problem, note that every metric space with at most three points embeds isometrically in the plane, but there exist four-point metric spaces which do not embed isometrically into any Hilbert space. Rademacher's theorem on the almost everywhere differentiability of Lipschitz functions plays a starring role. We'll highlight the use of Rademacher-type differentiation theorems for Lipschitz mappings of metric measure spaces in the proof of bi-Lipschitz nonembeddability theorems. We'll also discuss several recent bi-Lipschitz embedding theorems proved by current and former graduate students in this department.