Alternatively have a look at the program.

## $q$-deformed Whittaker functions and Demazure modules

Whittaker functions are special functions on reductive groups, which are naturally arising in the theory of automorphic representations. The talk is devoted to recent results (joint with A. Gerasimov and D. Lebedev) on explicit construction of $q$-deformation of Whittaker functions for the group $GL(N,\mathbb{R})$. In the first part of my talk I will introduce two (integral) representations of $GL(N)$-Whittaker function, using two different limits of Macdonald polynomials. The first representation is a $q$-deformation of the classical Gelfand-Zetlin formula for the character.

## André-Quillen cohomology theory of an algebra over an operad

Following the ideas of Quillen and by means of model category structures, Hinich, Goerss and Hopkins have developped a cohomology theory for (simplicial) algebras over a (simplicial) operad. Thank to Koszul duality theory of operads, we describe the cotangent complex to make these theories explicit in the differential graded setting. We recover the known theories as Hochschild cohomology theory for associative algebras and Chevalley-Eilenberg cohomology theory for Lie algebras and we define the new case of homotopy algebras.

## The fundamental group of symplectic manifolds with Hamiltonian Lie group actions

## Six vertex model and enumerations of alternating-sign matrices

One more interaction between theoretical physics and mathematics will be discussed. There is a famous combinatorial problem to enumerate the so-called alternating-sign matrices, which are a generalization of the permutation matrices. This problem was solved by Doron Zeilberger in 1995. Much simpler solution was given by Greg Kuperberg in 1996. It was based on the one-to-one correspondence between the alternating-sign matrices and the states of the statistical six-vertex model.

## Moduli spaces of vector bundles over a Klein surface

A compact topological surface S, possibly non-orientable and with non-empty boundary, always admits a Klein surface structure (an atlas whose transition maps are dianalytic). Its complex cover is, by definition, a compact Riemann surface X endowed with an antiholomorphic involution which determines topologically the original surface S. In this talk, we relate dianalytic vector bundles over S and holomorphic vector bundles over X, devoting special attention to the implications this has for moduli spaces of semistable bundles over X.

## Layer cake and homotopy representations I: formal geometry approach

Representations up to homotopy of Lie algebras have attracted recently much attention. On the other hand J. Baez has introduced a way to build a homotopy Lie algebra out of a Lie algebra and an n-cocycle. We show in this work a common framework enabling to generalize both notions (replacing Lie algebras by homotopy Lie algebras) and extend them for other types of algebras (commutative and associative). The main tool is the language of homological vector fields on products of formal manifolds. This is a joint work with John Baez.

## Universal family for subgroups of an algebraic group

I describe the construction of a moduli space for the connected subgroups of an algebraic group, and of a universal family. I give a quick illustration of the notion of universal families, trough Grassmann varieties, then discuss in turn the construction of a moduli space and of a universal family, balancing general statements and examples.

## Triangle groups, finite simple groups and applications

In this talk we will discuss the following problem: Given a triple of integers (r,s,t), which finite simple groups are quotients of the triangle group T(r,s,t)? This problem has many applications, especially concerning Riemann surfaces and Beauville surfaces. In the talk we will focus on the group theoretical aspects of this problem.

## Geometry of Maurer-Cartan Elements on Complex Manifolds

The semi-classical data attached to stacks of algebroids in the sense of Kashiwara and Kontsevich are Maurer-Cartan elements on complex manifolds, which we call extended Poisson structures as they generalize holomorphic Poisson structures. A canonical Lie algebroid is associated to each Maurer-Cartan element. We study the geometry underlying these Maurer-Cartan elements in the light of Lie algebroid theory.

## Elliptic curves over imaginary quadratic fields

About 10 years ago the methods developed by A.Wiles, R. Taylor, and their collaborators led to the proof of the modularity of the elliptic curves defined over the field of rational numbers. In a recent work Dielefait, Gueberooff, and Pacetti developed a new method, allowing to compare two 2-dimensional l-adic Galois representations, and applied their method to prove modularity of three elliptic curves defined over an imaginary quadratic field.

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