Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, October 25, 2016.

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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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Tuesday, October 25, 2016

11:00 am in 345 Altgeld Hall,Tuesday, October 25, 2016

#### Quantization of the Modular Functor and Equivariant Elliptic Cohomology

###### Nitu Kitchloo (Johns Hopkins)

Abstract: For a simple, simply connected compact Lie group G, let M be a compact G-space. I will describe a procedure that can be interpreted as the quantization of the category of parametrized positive energy representations of the loop group of G at a given level. This procedure is described in terms of dominant K-theory of the loop group parametrized over M. More concretely, I will construct a holomorphic sheaf over a universal elliptic curve with values in dominant K-theory of the loop space LM, and show that each stalk of this sheaf is a cohomological functor of M.I will also give compelling evidence that this theory is equivalent to equivariant elliptic cohomology of M as constructed by Grojnowski. In this talk, I will try to start from basics and give ample motivation.

12:00 pm in 243 Altgeld Hall,Tuesday, October 25, 2016

#### Character Varieties and the Hodge Monodromy Representation

###### Marina Logares (Oxford University)

Abstract: Character varieties have been studied thoroughly in relation with moduli spaces of Higgs bundles. Their topological invariants where studied via the non abelian Hodge correspondence, and recently attention has been focused on the so called e-polynomial, which provides algebraic information on them. I'm going to report on ongoing work on formalising the method to compute e-polynomials given in joint work with P. Newstead and V. Muñoz. This formalisation is intended for producing a 1+1 TQFT, this is a project with A. González and V. Muñoz

1:00 pm in 347 Altgeld Hall,Tuesday, October 25, 2016

#### Dispersive estimates for Dirac operators in dimension three

###### Ebru Toprak (UIUC Math)

Abstract: In this talk, we discuss $L^1\to L^\infty$ dispersive estimates for the three dimensional Dirac equation with a potential. We classify the structure of obstructions at the thresholds of the essential spectrum as being composed of a two dimensional space of resonances and finitely many eigenfunctions. We show that, as in the case of the Schr\"odinger evolution, the presence of a threshold obstruction generically leads to a loss of the natural $t^{-\frac32}$ decay rate. In this case we show that the solution operator is composed of a finite rank operator that decays at the rate $t^{-\frac12}$ plus a term that decays at the rate $t^{-\frac32}$.

1:00 pm in 345 Altgeld Hall,Tuesday, October 25, 2016

#### The subgroup structure of Richard Thompson's group F

###### Justin Moore (Cornell University)

Abstract: Consider the finitely generated subgroups of F, ordered by embeddability. How complex is it? Is it a well quasi-order? We show that it contains a strictly well ordered chain of length $\epsilon_0 +1$. This is joint work with Collin Bleak and Matt Brin.

2:00 pm in 241 Altgeld Hall,Tuesday, October 25, 2016

#### Some results related to the distribution of zeros of a family of Dirichlet series.

###### Paulina Koutsaki (UIUC )

Abstract: Let $G(s)=\sum\limits_{n=1}^{\infty} a_n \,n^{-s}$ be a Dirichlet series with coefficients bounded by $n^{\epsilon}$ for every $\epsilon>0$, and define $\beta_{k,G}$ to be the supremum of the real parts of zeros of combinations of $G$ and its $k$ first derivatives. In this talk, we give an asymptotic formula for the number $\beta_{k,G}$ and investigate in more detail the case of Dirichlet L-functions. We will also discuss an inverse-type problem for the Riemann-zeta function. In particular, we compute the degree of the largest derivative needed for such a combination to vanish at a given real number. For example, a combination that vanishes at $\beta=1,000,000$ will involve a derivative of order at least 2,178,301. This is joint work with A. Tamazyan and A. Zaharescu.

3:00 pm in 243 Altgeld Hall,Tuesday, October 25, 2016

#### A strange bilinear form on the space of automorphic forms

###### Jonathan Wang (University of Chicago)

Abstract: Let F be a function field and G a reductive group over F. We define a "strange" bilinear form B on the space of K-finite smooth compactly supported functions on G(A)/G(F). For G = SL(2), the definition of B generalizes to the case where F is a number field (and this is expected to be true for any G). The definition of B relies on the constant term operator and the standard intertwining operator. This form is natural from the viewpoint of the geometric Langlands program via the functions-sheaves dictionary. To see this, we show the relation between B and S. Schieder's geometric Bernstein asymptotics.

3:00 pm in 241 Altgeld Hall,Tuesday, October 25, 2016

#### Fractional Separation Dimension

###### Sarah Loeb (Illinois Math)

Abstract: Given a linear ordering $\sigma$ of $V(G)$, say that a pair of nonincident edges is separated by $\sigma$ if both vertices of one edge precede both vertices of the other. The separation dimension is the minimum size of a set of vertex orders needed to separated every pair of non-incident edges. The $t$-separation dimension $\pi_t(G)$ of a graph $G$ is the minimum size of a multiset of vertex orders needed to separate every pair of non-incident edges of $G$ $t$ times. The fractional separation dimension $\pi_f(G)$ of a graph $G$ is $\liminf_t \pi_t(G)/t$. We show that $\pi_f(G)\le 3$ for every graph $G$, with equality if and only if $K_4\subseteq G$. On the other hand, there is no sharper upper bound; we show $\pi_f(K_{m,m})=\frac{3m}{m+1}.$ This is joint work with Douglas West.

5:00 pm in 245 Altgeld Hall,Tuesday, October 25, 2016

#### IGL Mid-semester Meeting

Abstract: Mid-semester progress meeting for IGL project teams.