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Gaëtan Borot (MPIM) "Geometric and topological recursion"

Initially inspired by the separate works of Mirzakhani and Eynard, the geometric recursion is a recipe proposed last year together with Andersen and Orantin, which aims at constructing mapping class group invariants for surfaces of arbitrary topologies, from a small amount of initial data, by induction on the Euler characteristic. It can be used, for instance, to construct certain functions on the moduli space of bordered Riemann surfaces, which are such that their integration over the moduli space gives functions of boundary lengths that satisfy a variant of the topological recursion of Chekhov, Eynard and Orantin. One can produce in this way functions which are relevant in enumerative geometry and hyperbolic geometry, such as the constant function 1 (which yield, after integration, the Weil-Petersson volumes of the moduli space) and linear statistics of the hyperbolic length of simple closed curves. Besides, any initial data for the topological recursion can be lifted to an initial data for the geometric recursion, so that the topological recursion amplitudes are recovered by integration of the corresponding geometric recursion amplitude. This gives a refined and an hyperbolic perspective on certain classical problems of enumerative geometry. The meaning of this refined information and the perspectives opened by this construction are still to be explored.

In this lecture, I will give an introduction to the geometric recursion and its main properties. As illustration, I will revisit some problems solved by the topological recursion in its light, explain new applications in the context of hyperbolic geometry, and indicate natural questions in geometry and topological field theories that one can think of addressing via his formalism.

Michael Gekhtman (Notre Dame) "Cluster Integrable Systems"

This will be an overview of discrete integrable systems arising in the context of cluster algebras. Topics to be discussed include: - realization of r-matrix Poisson brackets on Poisson-Lie groups via Poisson geometry of directed networks on surfaces; - distinguished sequences of cluster transformations as discrete dynamical systems; - examples of cluster integrable systems: discrete Coxeter-Toda lattices, Q-systems, T-systems, the pentagram map and its generalizations.

Nicolai Reshetikhin (Berkeley) "Factorizable Lie groups and the construction of Integrable systems"

The lectures will start with the definition of factorizable Poisson Lie groups and with several constructions of integrable systems on their symplectic leaves. Examples will include Coxeter-Toda systems and classical spin chains. Then we will discuss the construction of solutions to equations of motion in terms of factorization on the underlying Poisson Lie group and the relation to spherical functions.

Hugh Thomas (UQAM) " Introduction to cluster algebras"

I will give an introduction to cluster algebras, taking the cluster algebras arising from surfaces as paradigmatic examples. I will also present the cluster algebra structure on Grassmannians. I will discuss work of Kodama and Williams on soliton solutions to the KP equation. Time permitting, I will say something about the connection to scattering amplitudes for planar N=4 super Yang-Mills and the amplituhedron.