Saturday, April 23, 2022 at the University of Maryland

 

Vesselin Dimitrov (Toronto)

Title: Arithmetic algebraization: holonomy bounds and applications.
Abstract:  The arithmetic algebraization method uses ideas from Diophantine approximations and Arakelov geometry in order to recognize when a formal-analytic scheme or morphism is in fact algebraic. The first part of the talk will be an introductory talk to set up the scene and then prove a basic ‘arithmetic holonomy bound’ from a joint recent work with Frank Calegari and Yunqing Tang.

In the second part, we will look into some applications of arithmetic holonomy bounds to number theory. Our two case study examples will be the solutions of the unbounded denominators conjecture on noncongruence modular forms (joint with Calegari and Tang), and the Hall-Ruzsa conjecture on pseudo polynomials.

Dennis Gaitsgory (Harvard)

Title: Restricted geometric Langlands, and connection to the classical theory via the categorical trace of Frobenius.
Abstract: The two existing forms of the geometric Langlands conjecture say that (1) In the de Rham setting, the category of (ind)-coherent sheaves on the stack of Ǧ-local systems is equivalent to the category of D-modules on Bun_G and (2) In the de Betti setting, the category of (ind)-coherent sheaves on the stack of Ǧ-local systems is equivalent to the category of Betti sheaves on Bun_G with nilpotent singular support.

We propose yet another version, which makes sense in an arbitrary constructible sheaf-theoretic context: the category of (ind)-coherent sheaves on the stack of Ǧ-local systems with restricted variation is equivalent to the category of ind-constructible sheaves on  Bun_G with nilpotent singular support.

It turns out that (when working over a finite field), the latter category can be directly related to the classical space of unramified automorphic functions via the operation of categorical trace of Frobenius. 

Combined with the above version of the geometric Langlands conjecture,this gives an explicit description of the space of automorphic functions in terms of Langlands parameters.

Giulia Saccà (Columbia)

Title: Hyper-Kähler manifolds and Lagrangian fibrations.
Abstract: Compact hyper-Kähler manifolds are one of the building blocks of compact Kähler manifolds with trivial first Chern class and are the natural higher dimensional analogue of K3 surfaces. Lagrangian fibrations are the natural generalization of elliptic K3 surfaces. In the first talk I will give an introduction to hyper-Kähler manifolds and Lagrangian fibrations, while in the second talk I will discuss some recent results on the topic.