Abstract: What representation theorists usually do, they try to compute characters (so Representation theory is related to Psychology). Unlike in Psychology, we usually care about characters of irreducible representations of algebraic objects which usually originate in Lie theory. I'll review various developments in the subject starting 1900 or so and concentrate on algebraic objects that have to do with the general linear group GL(n).

Abstract: Generalized theta functions are a non-abelian generalization for the classical theta functions. In this talk, we study the space of generalized theta functions for odd orthogonal bundles with nontrivial Stiefel-Whitney class and the associated space of twisted spin bundles. We will present a Verlinde type formula that was conjectured by Oxbury-Wilson and address the issue of strange duality for odd orthogonal bundles. This is a joint work with Richard Wentworth. 

Abstract: Infinite–dimensional quantum groups precede historically their finite–dimensional counterparts, and were discovered during 1970’s in the study of exactly solved mod- els of statistical mechanics. By now their structures and representation theories are quite well understood, while a lot of questions still remain open.

In this talk, I will explain how the monodromy of difference equations can be used to answer a few of these questions. The use of difference equations in the theory of affine quantum groups is nothing new. However the family of equations we shall use seems to be. We will exploit this new technique to find explicit connections between various quantum groups, and relating their tensor structures. This talk is based on my joint research with V. Toledano Laredo.

Abstract: In the 1980's, Eisenbud and Harris developed the theory of limit linear series to study the behavior of linear series on curves under degenerations. They applied their theory to prove new results on imbeddings of curves in projective spaces, on existence of Weierstrass points, and on the birational geometry of moduli spaces of curves of high genus. Since then, limit linear series have remained the primary tool in the field. However, foundational limitations have until very recently imposed some restrictions on the sorts of arguments that could be made via limit linear series. We will describe recent advances in the foundations, and survey some of the resulting applications to statements about smooth curves.

Abstract: Suppose you have a finite group G and you want to study certain related structures (e.g., random walks, Cayley graphs, word maps, etc.). In many cases, this might be done using sums over the characters of G. A serious obstacle in applying these Fourier type formulas is lack of knowledge on the low dimensional representations of G. In fact, numerics shows that the “small" representations tend to contribute the largest terms to these sums, so a systematic knowledge on them might assist in the solution of important problems.

In this talk I will discuss a joint project (see arXiv:1609.01276) with Roger Howe (Yale/Texas AM). We introduce a language to speak about “size” of a representation, and we develop a method (called “eta correspondence") that produces analytic information on (conjecturally all the) “small" representations of finite classical groups.

The talk should be accessible to anyone with basic linear algebra knowledge. I will illustrate our theory with concrete motivations and numerical data obtained with John Cannon (MAGMA, Sydney) and Steve Goldstein (Scientific computing, Madison).

Abstract: We describe a method for constructing the generators, and their commutation relations, for the finite W-algebras of type A. We also see how the analogue result in the classical affine case can be used to construct integrable Hamiltonian hierarchies of Lax type.

Abstract: Classical Luna's slice theorem tells that if you have a conical resolution of singularities then any non-central point of the base admits an etale slice to the orbit of the G_m-action. Resolution with conical slices is (roughly speaking) a resolution where for any point of the base one can find the slice itself to be conical. I will talk about joint work with R.Travkin where we prove the descent for Brauer group classes of certain central reductions of the algebra of differential operators in characteristic p for a generic reduction of a resolution with conical slices. In the case when the resolution is symplectic this question is related to the construction of non-commutive resolutions of the corresponding singularity and derived equivalences between them.

Abstract: We revisit the classical linearization problem of non-resonant germs of diffeomorphisms in one complex dimension, which contains the well-known difficulties due to the so-called small divisor phenomenon. Using a small part of J. Ecalle's ``mould formalism'', we obtain explicit tree -- indexed formulas for the transformations involved, which yield Yoccoz's lower bound for the radius of convergence of the linearization; moreover, we reach a new global regularity result with respect to the multiplier (C^1 holomorphy, with quasianalyticity properties of monogenic character).

[Joint work with David Sauzin (CNRS Paris and Pisa) and Frederic Menous (Orsay Univ.); Link , to appear in Bulletin de la Soc. Math. de France]