BEGIN:VCALENDAR VERSION:2.0 PRODID:-//CERN//INDICO//EN BEGIN:VEVENT SUMMARY:School "Moduli Spaces\, Combinatorics and Integrable Systems" DTSTART;VALUE=DATE-TIME:20211115T140000Z DTEND;VALUE=DATE-TIME:20211126T161500Z DTSTAMP;VALUE=DATE-TIME:20260414T000120Z UID:indico-event-438@indico.eimi.ru DESCRIPTION:School "Moduli Spaces\, Combinatorics and Integrable Systems"\ n\nNovember 15 – 26\, 2021\n\nThe school "Moduli Spaces\, Combinatorics and Integrable Systems" is a part of the thematic program “Moduli Space s\, Combinatorics and Poisson Geometry” organized by EIMI in 2021-2022. The school consists of five lecture courses:\n\nModuli spaces\, planar cur ves and the Baker-Campbell-Hausdorff series (Video)\nAnton Alekseev • U niversity of Geneva\n\nGoncharov–Kenyon integrable systems (Video)\nVlad imir Fock • IRMA\n\nSuperintegrable systems on moduli spaces of flat co nnections over a surface (Video)\nNicolai Reshetikhin • Tsinghua Univer sity\n\nLarge genus asymptotic geometry of random square-tiled surfaces an d of random multicurves (Video\, Slides)\nAnton Zorich • Skoltech and Institut de Mathématiques de Jussieu\n\nThe Korteweg – de Vries hierarc hy: a system of PDEs from an algebraist’s point of view (Video)\nDimitri Zvonkine • Université de Versailles Saint-Quentin-en-Yvelines\n\n\nMo duli spaces\, planar curves and the Baker-Campbell-Hausdorff series\n\nAnt on Alekseev • University of Geneva\nNovember 15 – 18\, 18:15 – 19: 15\nVideo Recordings\n\nIn this mini-course\, we will cover several inter- related topics:\n\n\n \n Goldman Lie brackets and Turaev cobrackets define d in terms of intersections and self-intersections of curves on 2-dimensio nal oriented manifolds\;\n \n \n Symplectic\, Poisson and Batalin–Vilkov isky (BV) structures on moduli spaces of flat connections\;\n \n The Kashi wawra–Vergne problem on properties of the Baker–Campbell–Hausdorff ( BCH) series and its higher genus analogues.\n\n\nAmong other things\, we w ill explain the Fock–Rosly approach to Poisson structures on moduli spac es\, and we will discuss the Knizhnik–Zamolodchikov (KZ) flat connection on configuration spaces of points on the complex plane.\n\nThe mini-cours e will be essentially self-contained: in order to follow it\, no previous knowledge of the topics listed above is required!\n\n\nGoncharov–Kenyon integrable systems\n\nVladimir Fock • IRMA\nNovember 22 – 24\, 18:15  – 19:15\nVideo Recordings\n\nStarting from any integral convex polygon on a plane one can construct an integrable system with commuting continuo us 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 sys tem 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 p lanar curves with Jacobians as fibers.\n\nThese four points of view permit to describe these systems very explicitly in (cluster) coordinates (in te rms of tau-functions)\, give their solution (in terms of theta-functions) and establish relations to several other elementary problems.\n\n\nSuperin tegrable systems on moduli spaces of flat connections over a surface\n\nNi colai Reshetikhin • Tsinghua University\nNovember 15 – 17\, 19\, 17 :00 – 18:00\nVideo Recordings\n\nModuli space of flat connections over a surface admits a natural symplectic structure. This fact goes back to wor ks of Atiyah and Bott. Thus such moduli space can be regarded as a phase s pace of a classical Hamiltonian system. The goal of these lectures is to d escribe a natural family of superintegrable Hamiltonian systems on such mo duli spaces.\n\n\n The first lecture will be an overview of Liouville inte grability and superintegrability. An example of such a system\, the Kepler system will be discussed in details.\n In the second lecture we will focu s on the Atiyah-Bott symplectic structure on moduli spaces.\n In the third lecture superintegrable systems on moduli spaces of flat connections will be introduced and first examples will be given.\n The forth lecture will focus on examples.\n\n\nIf time permits\, quantization of these systems an d how it is related to representation theory will be discussed.\n\n\nLarge genus asymptotic geometry of random square-tiled surfaces and of random m ulticurves\n\nAnton Zorich • Skoltech and Institut de Mathématiques de Jussieu\nNovember 24 – 26\, 17:00 – 18:00\nVideo Recordings\, S lides\n\nConsider the prime decomposition of an integer number n taken ran domly 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 co unted without multiplicities tends to the normal distribution as the size N of the interval tends to infinity.\n\nTake 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 distribut ion of the number of cycles in such a random permutation also tends to the normal distribution as N tends to infinity.\n\nIn my lectures I plan to p resent our recent work with V. Delecroix\, E. Goujard and P. Zograf\, wher e we obtain analogous results for the distribution of the number of compon ents in a random multicurve on a surface of large genus and for the distri bution of the number of maximal horizontal cylinders on a square-tiled sur face 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 perfor med by A. Aggarwal and with the uniform large genus asymptotic formula for intersection numbers of psi-classes on the moduli spaces of complex curve s proved by A. Aggarwal.\n\n\nThe Korteweg – de Vries hierarchy: a syste m of PDEs from an algebraist’s point of view\n\nDimitri Zvonkine • U niversité de Versailles Saint-Quentin-en-Yvelines\nNovember 18\, 17:00 – 18:00\, November 19\, 18:15 – 19:15\, November 22\, 23\, 17:00 – 18:00\nVideo Recordings\n\nWe will describe three approaches to the KdV h ierarchy.\n\n\n The definition via pseudo-differential operators and Lax p airs allows one to construct the equations of the hierarchy and prove some of their properties.\n The approach via the infinite-dimensional Sato Gra ssmannian makes it possible to construct solutions of the hierarchy withou t even looking at the equations.\n Finally\, the most recent approach via the intersection theory on moduli spaces of curves\, allows one to constru ct the hierarchy\, its ”quantum” version\, and generalizes to many oth er integrable hierarchies.\n\n\n\nInstitutions participating in the organi zation of the event\n\n\n St. Petersburg Department of Steklov Mathematica l Institute of the Russian Academy of Sciences\n Leonhard Euler Internatio nal Mathematical Institute in St. Petersburg\n Chebyshev Laboratory at St. Petersburg State University\n\n\nThe event is financially supported by a g rant from the Government of the Russian Federation\, agreements 075-15-201 9-1619 and 075-15-2019-1620 and by a grant from Simons Foundation.\n\nht tps://indico.eimi.ru/event/438/ LOCATION:online/SPbU URL:https://indico.eimi.ru/event/438/ END:VEVENT END:VCALENDAR