Schedule

The periodic $q$-Whittaker measure is a probability measure on partitions defined in terms of $q$-Whittaker functions and an additional parameter $u$ known as the winding fugacity. I will explain a hidden distributional symmetry of this model which exchanges the $u$ and $q$ parameters, as well as related results on the periodic Hall-Littlewood measure. As a special case, we recover an identity of Imamura--Mucciconi--Sasamoto. This is joint work with Michael Wheeler.

Quantum group modules give rise to solvable two-dimensional square lattice models. Those lattice models in turn lead to interesting families of partition functions, including special functions in the local representation theory of algebraic groups and in the cohomology of flag varieties. This suggests the possibility of a quantum group (module) - partition function "dictionary" where one can look up special functions based on the required modules, and use the facility of lattice models to uncover their properties. We evaluate the potential for such a dictionary, first reviewing the basic constructions and then detailing examples of entries in the dictionary in joint work with Buciumas, Bump, and Gustafsson, and in work of Aggarwal, Borodin, Petrov, and Wheeler. Does the quantum group connection matter? Or is it more important that these lattice models arise from generalizations of the six-vertex free fermion point?

In this talk, we will be talking about Macdonald polynomials and why we care about them. Our motivation will come not only from different areas in mathematics, but also from physics. I will give an overview of some of the results in the literature, and dig more into them depending on the time we have.

We will discuss classical examples of how Young tableaux and Dyck path combinatorics have been used to answer questions in geometry and representation theory. In the 1980's, these examples inspired a more intricate framework centering around q,t-generalizations of Catalan numbers and Macdonald polynomials. We will talk about questions arising in this contemporary setting and about our approach using symmetric rational functions (called catalanimals) instead of polynomials. Subfamilies of catalanimals truncate to Macdonald polynomials and more general catalanimals enabled us to settle the longest standing conjecture in the area.

Joint work with Blasiak, Haiman, Pun, and Seelinger

Ginzburg and Peterson independently showed that the equivariant homology of the affine Grassmannian is Hopf-isomorphic to the functions on a family over the Cartan whose zero fiber is the centralizer of a principal nilpotent in Langlands dual type. These are (up to localization) isomorphic to equivariant quantum cohomology of G/B. We discuss a conjectural analogue of the Ginzburg-Peterson centralizer theorem in K-theory. Using ideas from integrable systems, in types A and C we have explicit realizations in symmetric functions Schubert bases, matrix entries, etc. This is joint ongoing work with T. Ikeda, S. Iwao, and K. Yamaguchi.

This talk introduces techniques for using the large deviations of interacting particle systems to study the large-N asymptotics of generalized Bessel functions. These functions arise from a versatile approach to special functions known as Dunkl theory, and they include as special cases most of the spherical integrals that have captured the attention of random matrix theorists for more than two decades. I will give a brief introduction to Dunkl theory and then present a result on the large-N limits of generalized Bessel functions, which unifies several results on spherical integrals in the random matrix theory literature. These limits follow from a large deviations principle for radial Dunkl processes, which are generalizations of Dyson Brownian motion. If time allows, I will discuss some further results on large deviations of radial Heckman-Opdam processes and/or applications to asymptotic representation theory. Joint work with Jiaoyang Huang.

Integrable lattice models usually arise from mathematical physics or from representation theory of quantum algebras. In this talk, I will discuss how integrable lattice models appear in representation theory of p-adic groups. In one instance, we’ll get colored fermionic models, in another — colored bosonic models. This approach provides a representation-theoretical interpretation of many results from integrable lattice models. I hope that this connection can lead to asymptotic representation theory of p-adic groups.

The concept of 'universality' motivates a wide variety of probability and mathematical physics problems, going back to the classical central limit theorem. Most recently, the Kardar--Parisi--Zhang universality class has been proven to have Tracy--Widom fluctuations in the long-time asymptotics. In this talk, I will present a new universality result about the long-time asymptotics of so--called ``dynamic'' processes. The asymptotic fluctuations are related to the Tracy--Widom distribution. The proof will utilize a duality of Markov processes, which is constructed using Drinfel'd twisters of the quantum group U_q(sl_2), viewed as a quasi-triangular quasi-Hopf algebra. The orthogonality of the duality functions allow for an asymptotic analysis.

We consider probability measures on GL(n,R) that are left-invariant under the orthogonal group O(n,R). For any such measure we consider the following two quantities: (a) the mean of the log of the absolute value of the eigenvalues of the matrices and (b) the Lyapunov exponents of random products of matrices independently drawn with respect to the measure. Our main result is a lower bound for (a) in terms of (b).

This lower bound was conjectured by Burns-Pugh-Shub-Wilkinson (2001), and special cases were proved by Dedieu-Shub (2002), Avila-Bochi (2003) and Rivin (2005). We give a proof in complete generality by using some results from the theory of spherical functions and Jack polynomials.

This is joint work with Diego Armentano, Gautam Chinta, and Michael Shub. (Ergodic Theory and Dynamical Systems, to appear)