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for events the day of Friday, September 14, 2018.

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Friday, September 14, 2018

1:00 pm in 2 Illini Hall,Friday, September 14, 2018

Dynamics and Rigidity 3: Mostow Rigidity Theorem

Cameron Rudd

2:00 pm in 343 Altgeld Hall,Friday, September 14, 2018

Covering Lemmas and Differentiation

Chris Gartland (Illinois Math)

Abstract: The classical Lebesgue density theorem states that for any Lebesgue measurable $E \subset [0,1]$ and $\mathcal{L}$-almost every $x \in E$, $\lim_{r \to 0} \frac{ \mathcal{L}(E \cap B_r(x))}{\mathcal{L}(B_r(x))} = 1$. A typical way to prove this uses a maximal inequality, which in turn uses a weak Vitali covering lemma and that fact that $\mathcal{L}$ is doubling, meaning $\sup_{x \in [0,1]} \sup_{r > 0} \frac{\mathcal{L}(B_{2r}(x))}{\mathcal{L}(B_r(x))} < \infty$. The statement of the density theorem has a clear generalization to any metric measure space and can be proven true in any doubling space by proving a stronger Vitali covering lemma. In this talk, we'll work only with measure spaces and won't consider any metric or topological structure. The sets $\{B_r(x)\}_{r >0}$ willbe generalized to nets of measurable sets $\{B_\alpha(x)\}_{\alpha \in A}$ that "converge" to $x$. We then show that the stronger Vitali covering lemma is actually equivalent to the density theorem in this setting. An application will include an alternate proof of the almost sure convergence of uniformly bounded martingales.

3:00 pm in 341 Altgeld Hall,Friday, September 14, 2018

Examples of amenable, non-unitarizable quantum groups

Michael Brannan (Texas A&M)

Abstract: A well-known theorem of Day and Dixmier from around 1950 states that if G is an amenable locally compact group, then any uniformly bounded representation of G on a Hilbert space is similar to a unitary representation. In short, amenable groups are ``unitarizable''. In this talk, I will focus on the question of whether a version of the Day-Dixmier unitarizability theorem holds in the more general framework of locally compact quantum groups. It turns out that the answer to this question is no: We show that many amenable quantum groups (including all Drinfeld-Jimbo-Woronowicz q-deformations of classical compact groups) admit non-unitarizable uniformly bounded representations. (Joint work with Sang-Gyun Youn.)

3:00 pm in 345 Altgeld Hall,Friday, September 14, 2018

A Structural Model of Cyber Risk Aggregate Loss Distribution of Medium Size Enterprises

Petar Jevtic (Arizona State University)

4:00 pm in 345 Altgeld Hall,Friday, September 14, 2018

Introduction to continuous logic, continued.

Erik Walsberg

Abstract: I will continue to introduce continuous logic. This time I will try to cover the syntax and some examples of continuous structures. I will assume basic knowledge of first order model theory, and that you came to my previous talk.

4:00 pm in 245 Altgeld Hall,Friday, September 14, 2018

Alan Turing and why the apps on your phone crash

Philipp Hieronymi (Mathematics)

Abstract: Ever been annoyed by crashing/freezing apps on your phone? So here is my business idea: Let's write an app together that checks whether a given app on a given (user) input crashes or not. Sounds good? Great, let's get start up money, create such an app and get rich! Well, the problem is that the British logician Alan Turing proved in the 1930's that it is impossible to create such an app. Turing's brilliant solution of this problem (which is called the halting problem) is one of most beautiful and important mathematical results for the 20th century, substantially limiting the things a computer can possibly do. We will discuss the proof of this result and then turn our attention to more restricted situations where can actually prove whether or not a system crashes.

4:00 pm in 241 Altgeld Hall,Friday, September 14, 2018

3 invariants of manifolds you won’t believe are the same!

Hadrian Quan

Abstract: We’ll start by discussing everyone’s favorite invariant: the determinant of a linear map. After generalizing this to an invariant of a chain complex, we’ll talk about three different different ways to get a number from a representation of $\pi_1(M)$: topological, analytic, and dynamical. Number 3 might surprise you!