School "Moduli Spaces, Combinatorics and Integrable Systems"
November 15 – 26, 2021
The school "Moduli Spaces, Combinatorics and Integrable Systems" is a part of the thematic program “Moduli Spaces, Combinatorics and Poisson Geometry” organized by EIMI in 20212022. The school consists of five lecture courses:
Moduli spaces, planar curves and the BakerCampbellHausdorff series
Anton Alekseev • University of Geneva
Goncharov–Kenyon integrable systems
Vladimir Fock • IRMA
Large genus asymptotic geometry of random squaretiled surfaces and of random multicurves
Anton Zorich • Skoltech and Institut de Mathématiques de Jussieu
The Korteweg – de Vries hierarchy: a system of PDEs from an algebraist’s point of view
Dimitri Zvonkine • Université de Versailles SaintQuentinenYvelines
Superintegrable systems on moduli spaces of flat connections over a surface
Nicolai Reshetikhin • Tsinghua University
Moduli spaces, planar curves and the BakerCampbellHausdorff series
Anton Alekseev • University of Geneva
November 15 – 18, 18:15 – 19:15
In this minicourse, we will cover several interrelated topics:

Goldman Lie brackets and Turaev cobrackets defined in terms of intersections and selfintersections of curves on 2dimensional oriented manifolds;

Symplectic, Poisson and Batalin–Vilkovisky (BV) structures on moduli spaces of flat connections;
 The Kashiwawra–Vergne problem on properties of the Baker–Campbell–Hausdorff (BCH) series and its higher genus analogues.
Among other things, we will explain the Fock–Rosly approach to Poisson structures on moduli spaces, and we will discuss the Knizhnik–Zamolodchikov (KZ) flat connection on configuration spaces of points on the complex plane.
The minicourse will be essentially selfcontained: in order to follow it, no previous knowledge of the topics listed above is required!
Goncharov–Kenyon integrable systems
Vladimir Fock • IRMA
November 22 – 24, 18:15 – 19:15
Starting from any integral convex polygon on a plane one can construct an integrable system with commuting continuous and discrete flows. The simplest example of such system is the Poncelet porism, but this scheme include a very large class of integrable systems (so far except Hitchin and Calogero systems). The phase space of this system can be viewed either as a configuration space of flags in an infinite dimensional space, or as weight space of a dimer model on a torus or as a symplectic leaf of an affine Lie group or as a bundle over the space of planar curves with Jacobians as fibers.
These four points of view permit to describe these systems very explicitly in (cluster) coordinates (in terms of taufunctions), give their solution (in terms of thetafunctions) and establish relations to several other elementary problems.
Superintegrable systems on moduli spaces of flat connections over a surface
Nicolai Reshetikhin • Tsinghua University
November 15 – 17, 19, 17:00 – 18:00
Moduli space of flat connections over a surface admits a natural symplectic structure. This fact goes back to works of Atiyah and Bott. Thus such moduli space can be regarded as a phase space of a classical Hamiltonian system. The goal of these lectures is to describe a natural family of superintegrable Hamiltonian systems on such moduli spaces.
 The first lecture will be an overview of Liouville integrability and superintegrability. An example of such a system, the Kepler system will be discussed in details.
 In the second lecture we will focus on the AtiyahBott symplectic structure on moduli spaces.
 In the third lecture superintegrable systems on moduli spaces of flat connections will be introduced and first examples will be given.
 The forth lecture will focus on examples.
If time permits, quantization of these systems and how it is related to representation theory will be discussed.
Large genus asymptotic geometry of random squaretiled surfaces and of random multicurves
Anton Zorich • Skoltech and Institut de Mathématiques de Jussieu
November 24 – 26, 17:00 – 18:00
Consider the prime decomposition of an integer number n taken randomly in a large interval [1,N]. The Erdos–Kac theorem proves that the centered and rescaled distribution of the number of prime divisors of n counted without multiplicities tends to the normal distribution as the size N of the interval tends to infinity.
Take a random permutations in the symmetric group of N elements endowed with the uniform probability measure. The theorem of Goncharov proves that the centered and rescaled distribution of the number of cycles in such a random permutation also tends to the normal distribution as N tends to infinity.
In my lectures I plan to present our recent work with V. Delecroix, E. Goujard and P. Zograf, where we obtain analogous results for the distribution of the number of components in a random multicurve on a surface of large genus and for the distribution of the number of maximal horizontal cylinders on a squaretiled surface of large genus. These results are based on our formula for the Masur–Veech volume of the moduli space of holomorphic quadratic differentials combined with deep large genus asymptotic analysis of this formula performed by A. Aggarwal and with the uniform large genus asymptotic formula for intersection numbers of psiclasses on the moduli spaces of complex curves proved by A. Aggarwal.
The Korteweg – de Vries hierarchy: a system of PDEs from an algebraist’s point of view
Dimitri Zvonkine • Université de Versailles SaintQuentinenYvelines
November 18, 17:00 – 18:00, November 19, 18:15 – 19:15, November 22, 23, 17:00 – 18:00
We will describe three approaches to the KdV hierarchy.
 The definition via pseudodifferential operators and Lax pairs allows one to construct the equations of the hierarchy and prove some of their properties.
 The approach via the infinitedimensional Sato Grassmannian makes it possible to construct solutions of the hierarchy without even looking at the equations.
 Finally, the most recent approach via the intersection theory on moduli spaces of curves, allows one to construct the hierarchy, its ”quantum” version, and generalizes to many other integrable hierarchies.
Institutions participating in the organization of the event
 St. Petersburg Department of Steklov Mathematical Institute of the Russian Academy of Sciences
 Leonhard Euler International Mathematical Institute in St. Petersburg
 Chebyshev Laboratory at St.Petersburg State University
The event is financially supported by a grant from the Government of the Russian Federation, agreements 0751520191619 and 0751520191620 and by a grant from Simons Foundation.