## Lectures

Speakers

Claude Sabbah, Jeng-Daw Yu

Dates

2024, January 29, 14:00--16:00

January 30, 10:00--12:00

January 30, 14:00--16:00

January 31, 10:00--12:00

Room

RIMS, Rm 111

Title

**
Bessel and Airy moments: arithmetic, periods and Hodge theory
**

Abstracts

1. General introduction (C. Sabbah)

After presenting the general framework envisioned by the physicists Broadhurst and Roberts concerning the moments of Kloosterman sums, we will summarize the known results within their program and the methods that lead to such results. We will also consider moments of cubic sums, for which similar arithmetic results are less advanced. We will finally emphasize open questions.

2. Arithmetic (J.-D. Yu)

The main arithmetic result is the proof of a functional equation for the modified L-function attached to moments of Kloosterman sums. The objective of this talk is to explain the methods that enable one to make use of a theorem of Patrikis and Taylor in the theory of automorphic forms. It relies on the analysis of a pure motive attached to a toric hypersurface acted on by a finite group.

3. Exponential Hodge theory (C. Sabbah)

Broadhurst and Roberts computed experimentally Hodge numbers related to some symmetric powers of Bessel differential equations and conjectured a general formula for these numbers. One of the main question was to construct a pure motive that would produce these numbers. We will explain how the exponential Hodge theory of Kontsevich and Soibelman makes easier such a computation and how it can be adapted to the case of symmetric powers of Airy differential equations, where irregular Hodge numbers appear.

4. Periods (J.-D. Yu)

Physicists have been working with integrals and sums that are related by an underlying motive. Bessel and Airy moments are periods attached to a motive related to moments of Kloosterman and cubic sums. These periods satisfy quadratic relations that can be made more explicit by using an exponential motive rather than a classical motive. For moments of Kloosterman sums, this method enables us to check that a conjectural formula of Broadhurst and Roberts for the critical values of the L-function is in accordance with a general conjecture by Deligne.

Speaker

Boris Tsygan (Northwestern University)

Date

November 28 (Thu), 14:00--16:00, 2019

Room

RIMS, Rm 110

Title

**
On noncommutative geometry of algebras in positive characteristic
**

Abstract

Noncommutative geometry is a theory that takes an associative algebra and assigns to it invariants in such a way that, if the algebra is question is an algebra of functions on a variety, one gets geometric/topological invariants of this variety. For a variety over complex numbers one can recover much of the usual calculus of differential forms, as well as De Rham cohomology. We will show how to generalize this to algebras over a field of positive characteristic. The talk is an introductory review of recent works of Kaledin, Petrov-Vologodsky, and myself.

Speaker

Boris Tsygan (Northwestern University)

Date

November 25 (Mon), 14:00--16:00, 2019

Room

RIMS, Rm 110

Title

**
Microlocal invariants of symplectic manifolds
**

Abstract

I will discuss how to construct a category starting with a symplectic manifold using microlocal methods, following works of Tamarkin, Nadler-Zaslow, and myself. The talk will be very introductory and concentrate mainly on examples of the plane, cylinder, and 2-torus.