Center Seminar on Mondays / Fall2019

Center for Advanced Studies
Seminar on Mondays at 14.30 at Skoltech

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

December 16, 2019
Anton Zorich
(Mathematics Inst. of Jussieu–Paris, Skoltech)
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