Talk

MPI-Oberseminar jointly with  "Arithmetic Geometry, A conference in Honor of Hèléne Esnault on the occasion of Her 70th Birthday" at IHES:
https://indico.math.cnrs.fr/event/11114/
 

Abstract: Prismatic cohomology is a unifying p-adic cohomology of p-adic formal schemes. Motivated by questions on locally analytic representations of p-adic groups and the p-adic Simpson correspondence, an extension of prismatic cohomology to rigid-analytic spaces (over Q_p or over F_p((t))) has been sought. We will explain what form this should take, and our progress on realizing this picture. This includes a degeneration from the analytic Hodge-Tate stack underlying the p-adic Simpson correspondence to a similar (analytic) stack related to the Ogus-Vologodsky correspondence in characteristic p. This is joint work in progress with Johannes Anschütz, Arthur-César le Bras and Juan Esteban Rodriguez Camargo.

We give a survey of some recent constrictions and results concerning Hecke operators for the moduli space of G-bundles on a smooth projective curve X defined over a local non-archimedian field F (possibly with level structures). These operators are somewhat analogous to the usual Hecke operators in the case when F is a finite field but describing their common eigen-values is a much more difficult task. We'll discuss on what space the Hecke operators act, formulate some general conjectures about eigen-functions and consider some examples. In the end (if time permits) I would like to discuss the example when X is of genus 0, and we study bundles with triviliazation at two points. This example is closely related to  the usual representation theory of G(F). 

I will start by discussing 'frequently occurring congruences' and the related notion of congruence sheaves.  I will then introduce
a method to construct lifts of differential equations by using generalized Taylor's formula in a systematic way.

The talk is based on joint work with Ilia Gaiur.

Stability problems appear in various forms throughout geometry and algebra. In differential geometry, given a vector field on a manifold that vanishes in a point, one can ask when nearby vector fields also vanish in a point. In algebra, given a Lie algebra and a subalgebra, one can ask when all deformations of the ambient Lie algebra also admit a Lie subalgebra of the same dimension. I will show that both questions are instances of a general question about differential graded Lie algebras. Under a finite-dimensionality assumption which is satisfied in the examples above, I will give a sufficient condition for a positive answer to the general question. I will then discuss some applications.