Center for Advanced Studies

September 16, 2019 / Igor Krichever (Skoltech, HSE Univ., Columbia Univ.)= = = Bethe ansatz equations and integrable systems of particles September 23, 2019 / Leonid Rybnikov (HSE Univ., IITP)= = = Gaudin model and crystals September 30, 2019 / Misha Verbitsky (IMPA, Rio de Janeiro, HSE Univ.)= = = Hyperkahler structure on the space of quasi-Fuchsian representations October 7, 2019 / Kirill Polovnikov (Moscow State Univ., Skoltech)= = = Gaussian model of topologically stabilized polymer states October 14, 2019 / Albert Schwarz (Univ. of California, Davis)= = = Geometric approach to quantum theory October 21, 2019 at 11.30 / Florent Ygouf (Tel-Aviv Univ.)= = = Isoperiodic dynamics in rank 1 affines manifolds October 21, 2019 / Nikita Nekrasov (Simons Center for Geometry and Physics, Skoltech)= = = On recent progress in old problems: wavefunctions of quantum integrable systems from gauge theory and Bethe/gauge dual of N=4 super-Yang-Mills theory / Part II October 28, 2019 / Andrii Liashyk (Skoltech, HSE Univ.)= = = New symmetries of gl(N)-invariant Bethe vectors November 11, 2019 / Evgeny Feigin (Skoltech, HSE Univ.)= = = Arc schemes for Veronese curves November 18, 2019 / Oleg Ogievetsky (Aix Marseille Univ., Skoltech)= = = Fusion procedure for walled Brauer algebras November 25, 2019 / Victor Vassiliev (Steklov Inst., HSE Univ.)= = = Picard–Lefschetz theory, monodromy and applications December 2, 2019 / Alexei Rosly (Skoltech, ITEP, HSE Univ.)= = = Conformal properties of the self-dual YM theory |

**December 9, 2019
Maxim Kazarian** (Skoltech, HSE Univ.)

Topological recursion for generalized Hurwitz numbers

*Hurwitz numbers enumerate ramified coverings of two-dimensional sphere with prescribed ramification types. Generating functions for different kinds of Hurwitz numbers (simple, monotone, r-spin, orbifold Hurwitz numbers, Bousquet-Melou-Schaeffer numbers, numbers of maps and hypermaps, etc.) possess numerous integrable properties typical for models of mathematical physics and Gromov-Witten theory. Being rather elementary combinatorial objects, they provide, thereby, a convenient model for studying these properties. One of the most intriguing such properties is topological recursion that became so popular last years. In the talk, I will try to explain the true nature of topological recursion for the case of Hurwitz numbers*

Anton Zorich

Bridges between flat and hyperbolic enumerative geometry

*Square-tiled surfaces can be interpreted as integer points in the moduli spaces of Abelian and holomorphic quadratic differentials on complex curves. Such interpretation allows to express the Masur-Veech volume of these moduli spaces in terms of the asymptotic count of square-tiled surfaces.
The combinatorial geometry of a square-tiled surface is determined by several natural parameters. Together with Amol Aggarwal, Vincent Delecroix, Elise Goujard and Peter Zograf, we study volume contributions of square-tiled surfaces with specified combinatorial geometry. The resulting frequencies of square-tiled surfaces of different types describe geometry of a “random” square-tiled surface. Using Kontsevich’s count of metric ribbon graphs, we expressed the Masur-Veech volumes in terms of intersection numbers of psi-classes. Our formula allows to evaluate a frequency of square-tiled surfaces of given combinatorial type and, hence, to formalize a notion of a “random” square-tiled surface. We proved that our frequencies coincide with Mirzakhani’s frequencies of simple closed hyperbolic multicurves.
I will tell what we expect from a “random” square-tiled surface of large genus and from a “random” geodesic multicurve on a surface of large genus. Our results are conditional to two challenging conjectures on large genus asymptotics of certain intersection numbers for the Deligne-Mumford compactification of the moduli space of complex curves. A formula (published several days ago) of Chen-Moeller-Sauvaget expressing the Masur-Veech volumes in terms of the linear Hodge integrals allows to attack one of the two conjectures in a new way *

arXiv