"Moduli Spaces, Combinatorics and Poisson Geometry"
November 2021 – August 2022
Moduli spaces have many non-trivial connections to other areas of mathematics: combinatorics, dynamics, integrable systems and Poisson geometry, to name a few. Among most celebrated results over the last 30 years one can mention several proofs of Witten’s conjecture about intersection numbers of ψ-classes (Kontsevich, Mirzakhani and others), computation of Euler’s characteristics of moduli spaces (Harer-Zagier), development of the higher Teichmüller theory (Fock, Goncharov) and its links with cluster algebras and associated Poisson structures (Fomin, Zelevinsky).
The research problems central for the program are:
- Establishing a relationship between meandric systems (pairs of transversal multicurves) on higher genus surfaces and square-tiled surfaces.
- Study of the large genus asymptotics of the numbers of meandric systems of given topological type.
- Computation of Masur-Veech volumes of lower dimensional strata in the moduli space of quadratic differentials.
- Obtaining a relation of the distribution of geodesic multicurves to Masur-Veech volumes.
- Establishing a relation between Joyce’s structures by Bridgeland to Frobenius manifolds and topological recursion formalism.
- Description the complete WKB expansion of the generating function of monodromy symplectomorphism for second order differential equations on Riemann surfaces with second order poles and establishing the link to topological recursion formalism.
- Application of the WKB formalism to general isomonodromic tau-function and embed it into the topological recursion framework. Generalization to higher genus using the formalism of Krichever and Bertola-Malgrange.
- Construction of the dilogarithm line bundle over SL(2, R) cluster variety associated to the canonical symplectic form over the moduli spaces of bordered Riemann surfaces; description of the Bohr-Sommerfeld symplectic leaves and their quantization.
The following activities will be organised during the program:
- school "Moduli Spaces, Combinatorics and Integrable Systems", November 15 – 26, 2021
- conference "Combinatorics of Moduli Spaces, Cluster Algebras and Topological Recursion", June 13 – 17, 2022
- conference "Geometry and Dynamics of Moduli Spaces", August 1 – 5, 2022
- a number of minicourses given by invited visitors
Tentative list of minicourse lecturers includes:
|Amol Aggarwal, Harvard University||Nicolai Reshetikhin, University of California,|
|Anton Alekseev, University of Geneva||Berkeley Michael Shapiro, Michigan State University|
|Gaëtan Borot, Max Planck Institute for Mathematics||Leon Takhtajan, Stony Brook University and EIMI|
|Vladimir Fock, University of Strasbourg||Richard Wentworth, University of Maryland|
|Sergey Fomin, University of Michigan||Don Zagier, Max Planck Institute for Mathematics|
|Martin Möller, Goethe University||Anton Zorich, Skoltech and IMJ – PRG|
|Alexey Rosly, Skoltech||Dimitri Zvonkine (CNRS) (to be confirmed)|
- Dmitry Korotkin, Concordia University and Centre de Recherches Mathématiques
- Peter Zograf, PDMI RAS and St. Petersburg University
Institutions participating in the organization of the event:
- St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences
- Leonhard Euler International Mathematical Institute in Saint Petersburg
- Chebyshev Laboratory at St.Petersburg State University
The program is supported by a grant from the Government of the Russian Federation, agreements 075-15-2019-1619 and 075-15-2019-1620, and by a grant from Simons Foundation.