## Hecke eigen-functions for curves over local non-archimedian fields

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.

## Classical real local Langlands for GL_2(R)

## Computing differentiable stack cohomology via multiplicative Ehresmann connections

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## Stability problems and differential graded Lie algebras

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.

## Proper sheaves on Bun_G

## IMPRS seminar on various topics: single talk

## On the distribution of supersingular primes for abelian surfaces

In 1976, Lang and Trotter conjectured the asymptotic growth for the number of primes p up to x for which the reduction of a non-CM elliptic curve E/Q at p is supersingular. Though the conjecture is still open, we now have unconditional upper and lower bounds thanks to the work of several mathematicians. However, less has been studied for the distribution of supersingular primes for abelian surfaces (even conjecturally).

## The equation $c_1x_1^k+...+c_sx_s^k=0$

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## Teleparallel Newton—Cartan Gravity

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## First order rigidity of manifold homeomorphism groups

Two groups are elementarily equivalent if they have the same sets of true first order group theoretic sentences. We establish that if the homeomorphism groups of two compact connected manifolds are elementarily equivalent, then the manifolds are homeomorphic. This generalizes Whittaker's theorem on isomorphic homeomorphism groups (Ann. of Math. 1963) without relying on it. Our result also confirms a conjecture of M. Rubin (1989). We also show the analogous results for measure-preserving homeomorphism groups.

## Hard Lefschetz theorem and Hodge-Riemann relations for convex valuations

The Alexandrov-Fenchel inequality, a fundamental result in convex

geometry, has recently been shown to be one component within a broader

'Kahler package'. This structure was observed to emerge in different

areas of mathematics, including geometry, algebra, and combinatorics,

and encompasses Poincare duality, the hard Lefschetz theorem, and the

Hodge-Riemann relations. After unpacking these statements within the

context of this talk, I will explain where complex geometry intersects

with convex geometry in the proofs.

## Finite braid group orbits on character varieties

The moduli space of rank n local systems on a topological surface admits an action of a huge group, namely the mapping class group. The finite orbits of this action then correspond to very special points in the moduli space. This is a generalization of the classical question of finding algebraic solutions to the Painlevé VI equation, which corresponds to n = 2 and the surface being the 4-punctured sphere.

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