# Integrable random systems, representation theory and geometry of Lie groups 2017

This workshop will take place in Les Diablerets on January 22-27, 2017.

Organizers: Anton Alekseev (UniGe), Thierry Lévy (Paris 6), Maria Podkopaeva (UniGe).

**Confirmed speakers:**

**Minicourses:**

**Reda Chhaibi **(Toulouse), "From random dynamics to combinatorial and geometric crystals"

*1. Panorama of different models in mathematical physics: random matrices, last passage percolation, directed polymers.
In this first talk, I want to give an informal panorama of the three aforementioned problems in mathematical physics. The only promise is that all three of them will be related to representation theory.
The goal is to explain these models, as well as list the known results.*

*2. 3. Integrable models arising from representation theory in general and crystal combinatorics in particular.
In the second and third talk, I want to explain that in certain cases, our favorite models are very tied to the representation theory of GL_N. - Dyson's Brownian motion as a (limit of) tensor product dynamic.
- Last passage percolation and the RS correspondence / Littelmann path model.
- Brownian Directed polymers as a geometric lifting, and geometric crystals*

*4. The probabilistic approach in order to unveil superpotentials: the fully worked out case of rank 1.*

**Sergey Fomin** (University of Michigan), "Quivers, clusters, and beyond"

*This mini-course will provide an elementary introduction to the basic theory of quiver mutations and cluster transformations, and review some of its applications.*

**Talks:**

**Arkady Berenstein **(Oregon), "Hecke-Hopf algebras and new solutions of QYBE"

*Hecke algebras H_q(W) of Coxeter groups W first emerged in the study of Chevalley groups in mid sixties and since then became central objects in Representation Theory of Coxteter groups and semisimple Lie groups over finite fields. In particular, as a one-parameter deformation of the group algebra kW of W, the Hecke algebra H_q(W) helps to classify representations of W and to equip each simple kW-module with the canonical Kazhdan-Lusztig basis.*

*Unfortunately, unlike the group algebra kW, the Hecke algebra H_q(W) lacks a Hopf algebra structure, that is, it is not clear how to tensor multiply H_q(W)-modules. Moreover, there is a general consensus that a naive Hopf structure on H_q(W), if exists, would essentially coincide with that on kW, so we would not gain any new information.*

*In may talk (based on joint work with D. Kazhdan) I suggest a roundabout: instead of forcing a naive Hopf structure on H_q(W), we find a "reasonably small" Hopf algebra H(W) (we call it Hecke-Hopf algebra of W) that "naturally" contains H_q(W) as a coideal subalgebra. The immediate benefit of this enlargement of H_q(W) is that each representation of H(W) and each representation of H_q(W) can be tensor multiplied into a new representation of H_q(W), thus allowing to create infinitely many new H_q(W)-modules out of a single one.*

*Hecke-Hopf algebras have some other applications, most spectacular of which is the construction of new infinite families of solutions to the quantum Yang-Baxter equation. *

**Cédric Boutillier **(Paris 6), "Integrable statistical mechanics on isoradial graphs"

*Isoradial graphs are planar graphs embedded in such a way that all bounded faces are inscribed in a circle of radius 1. After a presentation of some properties of these graphs, we will introduce a one-parameter family of massive Laplacians on these graphs.
We will discuss some integrability properties of these operators (in the geometric sense) and the connection with integrability of related models of statistical mechanics, namely spanning forests and the Ising model.*

**Roland Friedrich** (Saarland University), "Operadic Aspects of Stochastic Calculus"

*In this talk we shall discuss recently discovered algebraic structures of an operadic nature underlying stochastic calculus. This development nicely ties in with previous and upcoming results related to the algebraic structure stochastic calculus and stochastic differential equations have. *

**Volker Genz** (University of Cologne), "Crystal combinatorics and mirror symmetry of cluster varieties"

*tba*

**Gleb Koshevoy** (Central Institute of Economics and Mathematics, Moscow), "Combinatorics of crystals and superpotentials"

*tba*

**Thierry Lévy** (Paris 6), "Two-dimensional Yang-Mills measure and the master field"

*tba*

**Pierre-Loïc Méliot** (Paris-Sud), "Spectral properties of random geometric graphs on symmetric spaces"

*We study the spectral measure of a geometric graph obtained from random points taken in a compact Riemannian manifold. In the special case of rank one manifolds, we compute the exact asymptotics of this spectrum, by using the spherical functions of the space. *

**Konstanze Rietsch** (King's College London), "Grassmannians and Polytopes" (to be confirmed)

*tba*

**Jacinta Torres** (University of Cologne & MPIM Bonn), "Non-Levi branching rules and Littelmann paths"

*In recent work with Schumann we have proven a conjecture of Naito-Sagaki giving a branching rule for the decomposition of the restriction of an irreducible representation of the special linear Lie algebra to the symplectic Lie algebra, therein embedded as the fixed-point set of the involution obtained by the folding of the corresponding Dyinkin diagram. This conjecture had been open for over ten years, and provides a new approach to branching rules for non-Levi subalgebras in terms of Littelmann paths. In this talk I will introduce the path model, explain the setting of the problem, our proof, and provide some examples of other non-Levi branching situations. *

**Jonathan Weitsman **(Northeastern University), "Quantization of (some) Poisson manifolds"

*tba*