Mathematics

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

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 $t^{-\frac32}$.

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.

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.

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.

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.

IGL Mid-semester Meeting

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