Title: Orthogonal polynomial dualities of multi-species ASEP and related processes via the *-bialgebra structure of quantum groups

Abstract: In this talk, I present a general method to produce orthogonal polynomial dualities from the *--bialgebra structure of Drinfeld--Jimbo quantum groups. In the case of the quantum group Uq (gl _{n+1}), the result is a nested multivariate q--Krawtchouk duality for the n--species ASEP(q,θ). The method also applies to other quantized simple Lie algebras and to stochastic vertex models. As a probabilistic application of the duality relation found, we provide the explicit formula of the q-shifted factorial moments for the two--species q--TAZRP.

Title: Introduction to large deviation theory

Abstract: Large deviation theory is one of the pillars of probability theory. It concerns the asymptotic behavior of remote tails of sequences of probability distributions. In this talk, I will talk about the large deviations of various random objects. I will also explain its connection to a notion called “most probable shape”.

Title: The Kronecker and Littlewood-Richardson Coefficients

Abstract: Algebraic combinatorics is an area of mathematics that studies discrete objects frequently originating in representation theory, abstract algebra, algebraic geometry and number theory using combinatorial techniques. One of the main open problems in the field is to obtain a combinatorial interpretation for the Kronecker coefficients. The Kronecker coefficients, introduced by Murnaghan in 1938, are the multiplicities of irreducible representation in the decomposition of the tensor product of two irreducible representations of the symmetric group. As such they are naturally nonnegative numbers, yet to this day we have no positive combinatorial formula. The Littlewood-Richardson coefficients, which describe the decomposition of tensor products of representations of the general linear group, are a special case of the Kronecker coefficients, which have a nice combinatorial interpretation. They both play a crucial role in algebraic combinatorics and related fields; the Kronecker coefficients have also played a special role in Geometric Complexity Theory.

In this talk, I will introduce the Littlewood-Richardson and Kronecker coefficients and give a brief overview of some recent developments on the Kronecker coefficients. This talk will be aimed at a general audience, with no prior background in algebraic combinatorics necessary.