Department of

Mathematics


Seminar Calendar
for events the day of Thursday, September 20, 2018.

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Thursday, September 20, 2018

11:00 am in 241 Altgeld Hall,Thursday, September 20, 2018

Maass forms and the mock theta function f(q)

Scott Ahlgren (Illinois Math)

Abstract: Let f(q) be the well-known third order mock theta function of Ramanujan. In 1964, George Andrews proved an asymptotic formula for the Fourier coefficients of f(q), and he made two conjectures about his asymptotic series (these coefficients have an important combinatorial interpretation). The first of these conjectures was proved in 2009 by Bringmann and Ono. Here we prove the second conjecture, and we obtain a power savings bound in Andrews’ original asymptotic formula. The proofs rely on uniform bounds for sums of Kloosterman sums which follow from the spectral theory of Maass forms of half integral weight and in particular from a new estimate which we derive for the Fourier coefficients of such forms. This is joint work with Alexander Dunn.

12:00 pm in 243 Altgeld Hall,Thursday, September 20, 2018

The generalization of the Goldman bracket to three manifold and its relation to Geometrization

Moira Chas (Stony Brook)

Abstract: In the eighties, Bill Goldman discovered a Lie algebra structure on the free abelian group with basis the free homotopy classes of closed oriented curves on an oriented surface S. In the nineties, jointly with Dennis Sullivan, we generalized this Lie algebra structure to families of loops (defining the equivariant homology of the free loop space of a manifold). This Lie algebra, together with other operations in spaces of loops is now known as String Topology. The talk will start with a discussion of the Goldman Lie bracket in surfaces, and how it "captures" the geometric intersection number between curves. It will continue with the description of the string bracket, which generalizes of the Goldman bracket to oriented manifolds of dimension larger than two, and the space of families of loops where the string bracket is defined. The second part of the lecture describes how this structure in degrees zero and one plus the power operations in degree zero recognizes key features of the Geometrization, the above mentioned joint work. The lions share of effort concerns the torus decomposition of three manifolds which carry mixed geometry. This is joint work with Siddhartha Gadgil and Dennis Sullivan.

2:00 pm in 243 Altgeld Hall,Thursday, September 20, 2018

Three and a half asymptotic properties

Ryan Causey (Miami University Ohio)

Abstract: We introduce several isomorphic and isometric properties related to asymptotic uniform smoothness. These properties are analogues of p-smoothability, martingale type p, and equal norm martingale type p. We discuss distinctness, alternative characterizations, and renorming theorems for these properties.

3:00 pm in 345 Altgeld Hall,Thursday, September 20, 2018

Wreath Macdonald polynomials as eigenstates

Joshua Wen (University of Illinois)

Abstract: Proposed by Haiman, wreath Macdonald polynomials are distinguished bigraded characters of the wreath product $\Sigma_n\wr \mathbb{Z}/\ell\mathbb{Z}$ generalizing the usual (transformed) Macdonald polynomials. Their existence was proved in 2014 by Bezrukavnikov-Finkelberg via a generalization of Haiman’s proof of Macdonald positivity. Little else is known about them, and thus anyone trying to develop analogues for the rest of the 'Macdonald package’ (Macdonald operators, DAHA, Pieri rules, evaluation formulas, refined topological vertex, refined knot invariants, etc.) is in the strange position of only having Macdonald positivity as a starting point. I’ll present work on a necessary ingredient for many of these structures: that the wreath Macdonald polynomials diagonalize something. This something is a commutative subalgebra of the quantum toroidal algebra of $\mathfrak{sl}_\ell$. While the proof is still incomplete, it already involves a wide range of techniques from quantum algebra and partition combinatorics that might be of independent interest.

4:00 pm in 245 Altgeld Hall,Thursday, September 20, 2018

Computer Driven Questions, Theorems and Pre-theorems in Low Dimensional Topology

Moira Chas (Stony Brook University)

Abstract: Consider an orientable surface S with negative Euler characteristic, a minimal set of generators of the fundamental group of S, and a hyperbolic metric on S. Then each unbased homotopy class C of closed oriented curves on S determines three numbers: the word length (that is, the minimal number of letters needed to express C as a cyclic word in the generators and their inverses), the minimal geometric self-¬intersection number, and finally the geometric length. Also, the set of free homotopy classes of closed directed curves on S (as a set) is the vector space basis of a Lie algebra discovered by Goldman. This Lie algebra is closely related to the intersection structure of curves on S. These three numbers, as well as the Goldman Lie bracket of two classes, can be explicitly computed (or approximated) using a computer. These computations led us to counterexamples to existing conjectures, to formulate new conjectures and (sometimes) to subsequent theorems.