Department of

# Mathematics

Seminar Calendar
for events the day of Monday, November 6, 2017.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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Monday, November 6, 2017

3:00 pm in 141 Altgeld Hall,Monday, November 6, 2017

#### A Vietoris-Smale Mapping Theorem for the Homotopy of Hyperdefinable Sets

###### Elliot Kaplan   [email] (UIUC)

Abstract: I will discuss the paper $$A Vietoris-Smale mapping theorem for the homotopy of hyperdefinable sets$''$ by Alessandro Achille and Alessandro Berarducci $[\href{https://arxiv.org/abs/1706.02094}{arXiv}]$. In this paper, they compare the definable homotopy groups of a definable space $X$ (denoted $\pi_n(X)^\mathrm{def}$) with the (usual) homotopy groups of the quotient $X/E$, where $E$ is a certain kind of type-definable equivalence relation on $X$. Under certain assumptions, these groups are actually isomorphic. These types of quotients were first studied by Pillay in the case that $X$ is actually a definable group. No knowledge of model theory will be assumed. In particular definable,'' type-definable'' and o-minimal'' will all be defined.

3:00 pm in 243 AH,Monday, November 6, 2017

#### The partial compacti cation of the universal centralizer

###### Ana Balibanu (Harvard)

Abstract: Let $G$ be a semisimple algebraic group of adjoint type. The universal centralizer is the family of centralizers in $G$ of regular elements in $\text{Lie}(G)$, parametrized by their conjugacy classes. It has a natural symplectic structure, obtained by Hamiltonian reduction from the cotangent bundle $T^*G$. We consider a partial compactification of the universal centralizer, where each centralizer fiber is replaced by its closure inside the wonderful compactification of $G$. We show that the symplectic structure extends to a log-symplectic Poisson structure on the partial compactification, through a Hamiltonian reduction of the logarithmic cotangent bundle of the wonderful compactification.

4:00 pm in 245 Altgeld Hall,Monday, November 6, 2017

#### Asymptotically optimal shapes: the drum with lowest n-th frequency, and the ellipse enclosing the most lattice points

###### Rick Laugesen   [email] (Department of Mathematics, University of Illinois at Urbana-Champaign)

Abstract: What shape of domain minimizes the n-th energy level (frequency) of the Laplacian, for large n? How is that connected to lattice point counting in ellipses? And why do right triangles behave differently?

4:00 pm in 141 Altgeld Hall,Monday, November 6, 2017

#### Higher Categories: What's up with that?

###### Nima Rasekh

Abstract: Higher categories are very useful when we want to understand not only whether two objects are the same but how this "sameness" occurs. However, it is often difficult to motivate the abstract concepts necessary to define and use higher categories. The goal of this talk is to show some concrete examples in mathematics where higher categories naturally arise, without ever using the words "higher", "category" or "homotopy". Thus no background in these topics is assumed.

5:00 pm in 445 Altgeld Hall,Monday, November 6, 2017

#### Decay estimates for quantum systems

###### Marius Junge

Abstract: Recently the connection between hypercontractivity and decay estimates for quantum systems has become a hot topic in QIT. We present the first results by Temme and his collaborators.