Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, December 8, 2015.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
    November 2015          December 2015           January 2016
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1  2  3  4  5  6  7          1  2  3  4  5                   1  2
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Tuesday, December 8, 2015

11:00 am in 241 Altgeld Hall,Tuesday, December 8, 2015

#### Bad reduction of Hilbert modular varieties and arithmetic application

###### Yifeng Liu (Northwestern University)

Abstract: In this talk, we will study the reduction of Shimura varieties attached to certain quaternion algebras at some ramified prime. We explain how the global structure of the bad reduction is related to the level raising phenomenon for modular forms. As an application, we will use this to bound the Selmer groups of certain motives of high rank.

11:00 am in 243 Altgeld Hall,Tuesday, December 8, 2015

#### Effective field theories and elliptic cohomology

###### Daniel Berwick-Evans (UIUC)

Abstract: I will describe a geometric model for elliptic cohomology at the Tate curve whose cocycles are a class of 2-dimensional effective field theories. A geometrically-motivated modularity condition singles out cocycles whose Chern characters take values in the complexification of topological modular forms (TMF). The Witten genus of a string manifold and the moonshine module furnish examples.

12:00 pm in 345 Altgeld Hall,Tuesday, December 8, 2015

#### Inifinite unicorn paths and Gromov boundaries

###### Witsarut Pho-On (UIUC Math)

Abstract: I will provide direct elementary proofs of results of Klarreich and Schleimer identifying the Gromov boundaries of the arc and curve graph and the arc graph, respectively. The proofs use the tool called unicorn paths, developed by Hensel, Przytycki and Webb in their elementary proofs of hyperbolicity of the arc and curve graph and the arc graph. More precisely, I extend the notion of unicorn paths between two arcs to the case where one arc is replaced by a bi--infinite geodesic asymptotic to a lamination. Using these modified unicorn paths, I define homeomorphisms from some spaces of laminations to the Gromov boundaries of the arc and curve graph and the arc graph which are also equivariant under mapping class groups.

1:00 pm in 347 Altgeld Hall,Tuesday, December 8, 2015

#### Weyl's Law and Pólya's Conjecture

###### Neal Coleman   [email] (Indiana University)

Abstract: I'll introduce and define eigenvalues of the Dirichlet and Neumann Laplacians. These eigenvalues are the frequencies of standing waves, and have an intriguing, elusive relationship with the geometry of the domain. In 1911, Weyl proved a result now known as "Weyl's law," showing that the asymptotic distribution of these frequencies encodes the volume of the domain. In 1961, Polya proved a much sharper result for domains that tile Euclidean space, bounding individual eigenvalues in terms of the volume of the domain, and conjectured that this bound holds for general domains. I'll state Weyl's law and sketch its proof, then spend most of the talk sketching a proof of Polya's theorem. Then I will state my recent result generalizing Polya's theorem and indicate its proof. Finally, I will discuss interesting open questions, including an analogue for Neumann eigenvalues and Polya's conjecture.

2:00 pm in 347 Altgeld Hall,Tuesday, December 8, 2015

#### A Introduction of Stochastic Delay Model for Corporate Claims Value Pricing

###### Isabelle Kemajou-Brown (Morgan State University)

Abstract: In this talk, I will start with an introduction of stochastic differential equation and its applications to finance, then present a non linear stochastic delay model for pricing a firm value. A brief introduction of the classical Euler Maruyama scheme and Monte Carlo simulation method will be presented. I will show and explain the use of semi-implicit Euler Maruyama scheme to provide a numerical approximation of the value of a firm. Further, I test the delay model against real data using Monte Carlo simulation method, and compare it with the corresponding non delay model for the same firm value.

3:00 pm in Altgeld Hall 241,Tuesday, December 8, 2015

#### Extremal Results for Semi-Algebraic Hypergraphs

###### Andrew Suk   [email] (UIC Math)

Abstract: Turán and Szemerédi-type theorems prove the existence of certain well-behaved patterns in dense graphs and hypergraphs. Here we show much stronger/larger patterns exist for graphs and hypergraphs that arise from geometry or algebra. The talk is based on recent joint works with Jacob Fox, János Pach, Adam Sheffer, and Josh Zahl.

4:00 pm in 245 Altgeld Hall,Tuesday, December 8, 2015

#### Quantization, reduction mod p, and Autoequivalences of the Weyl algebra

###### Christopher Dodd (Perimeter Institute for Theoretical Physics, Waterloo, Ontario)

Abstract: The Weyl algebra of polynomial differential operators is a basic object which appears in algebraic geometry, representation theory, and mathematical physics. In this talk, I will discuss my recent proof of some conjectures of A. Belov-Kanel and M. Kontsevich concerning the structure of the automorphism group of the Weyl algebra. The question turns out to be related to defining an appropriate notion of "support cycle" for a differential equation, which, in turn, involves techniques from positive characteristic. In particular, we shall explain a "quantization correspondence" which is based on reducing differential equations to finite characteristic. View talk at https://youtu.be/cKNa6rQW1RQ