The lecture announcements will be
continually updated. The arrangement of the upcoming lectures is as follows:
Quantum integrable systems and Symplectic Field Theory
Speaker: Paolo Rossi (University of Padua, Italy)
Time：2020-05-29 17:00 Beijing time (12:00 Moscow time, 16:00 Novosibirsk time)
Eliashberg, Givental and Hofer's Symplectic Field Theory is a large project aiming to subsume under a unified topological field theoretical approach several techniques from symplectic topology (Floer homology, contact homology and more). Similarly to what happens in Gromov-Witten theory, at its core we find holomorphic curve counting. The general target manifold considered in SFT is a symplectic cobordism between contact manifolds (or more generally between stable Hamiltonian structures). When the cobordism is just a cylinder from a contact manifold to itself, the corresponding operator in SFT is, in particular, a collection of mutually commuting quantum Hamiltonians in a Weyl algebra.
These ideas were behind the introduction, by Buryak and myself, of the quantum double ramification hierarchy, which can be seen as a transposition of the SFT approach to the algebraic category together with several enhancements. I will introduce the double ramification hierarchy with an eye to its origins in Symplectic FIeld Theory and showcase some examples that we were able to fully compute.
intersection theory and the open analog of Witten’s r-spin conjecture.
Speaker: Ran Tessler (Weizmann Institute of Science)
Time：2020-06-05 17:00 Beijing time (12:00 Moscow time, 16:00 Novosibirsk time)
We will describe the moduli of r-spin disks and its associated vector bundles. We will then define intersection theory on the moduli of r-spin disks, and relate its potential to the r-KdV hierarchy. We will also make a high genus conjecture, generalizing Witten’s r-spin conjecture to the open setting. Based on joint works with A. Buryak and E. Clader.
Flat F-manifolds in higher genus and
Time: 2020-04-24 17:00
Speaker: Alexandr Buryak (National Research
University Higher School of Economics)
Abstract: By Dubrovin--Zhang theory, there
is a deep relation between dispersive deformations of the hierarchies of
hydrodynamic type corresponding to Frobenius manifolds and the geometry of the
moduli spaces of stable algebraic curves. I will talk about a generalization of
some of the results of the Dubrovin--Zhang theory for flat F-manifolds, which
we obtained in joint works with A. Arsie, P. Lorenzoni and P. Rossi.
Stokes phenomenon, reflection equations and
Time: 2020-05-01 17:00
Speaker: Xu Xiaomeng (Peking
Abstract: Reflection equations, arsing from
quantum integrable systems with boundary conditions, are the analog of
Yang-Baxter equations on a half line. Geometrically, they encode the cylinder
braid groups. Algebraically they are closely related to quantum homogenous
spaces. In this talk, we first give an introduction to the Stokes phenomenon of
an ODE with irregular singularities. We then prove that the Stokes matrices of
cyclotomic Knizhnik–Zamolodchikov (KZ) equations give universal solutions to
reflection equations. As an application, we show that the isomonodromy
deformation of the KZ equations is a quantization of the Dubrovin connections
of Frobenius manifolds from various aspects.
Slides Download: https://disk.pku.edu.cn:443/link/9E687F9588A75F6DC28A5BF16C134A18
Topological recursion and KP tau-functions
Time: 2020-05-08 17:00
Speaker: Sergey Shadrin (University
of Amsterdam, Netherlands)
We would like to recall some basic
definitions of the so-called Chekhov-Eynard-Orantin theory of topological
recursion. Originally it was developed to compute the cumulants for a class of
matrix model, but since then it has evolved to one of the key tools on the edge
between combinatorics and algebraic geometry that helped to resolve some famous
open conjectures. In particular, it has appeared that the topological recursion
can be proved for a large class of KP tau-functions from the Orlov-Scherbin
family. We'll explain what extra properties of these tau-functions can be
derived this way.An example of a direct application of this circle of ideas is
a recent proof (our joint work with Dunin-Barkowski, Kramer, and Popolitov) of
the so-called r-ELSV formula conjectured by Zvonkine in mid 2000's. We'll try
to explain that formula, and, if time permits, sketch the main steps of the
Spaces with indefinite metrics and the spectral theory of singular Schrodinger operators
Time: 2020-05-15 17:30
Speaker: P.G. Grinevich (Steklov Mathematical Institute)
Abstract: The famous Korteweg- de Vries (KdV) equation admits important singular solutions, but only very special singularities are compatible with the KdV dynamics. We show, that for the Schrodinger operators from the KdV Lax pair with such special singularities the spectral theory can be naturally formulated in terms of pseudo-Hilbert spaces with indefinite metrics. IN particular, the number of negative squares in this metric provides a new conservation law for such solutions. The talk is based on joint works with S.P. Novikov.
Geometrization, integrability and knots.
Speaker: A.P. Veselov (Loughborough, UK and Moscow, Russia)
Time：2020-05-22 17:30 Beijing time (16:30 Novosibirsk time)
I will discuss the coexistence of the chaos and Liouville integrability in relation with Thurston’s geometrization programme, using as the main example the geodesic flows on the 3-folds with SL(2,R)-geometry.
A particular case of such manifold SL(2,R)/SL(2,Z) is known after Milnor and Quillen to be topologically equivalent to the complement of the trefoil knot in 3-sphere. I will explain that the remarkable results of Ghys about modular and Lorenz knots can be naturally extended to the integrable region, where these knots are replaced by the cable knots of trefoil.
The talk is based on a joint work with Alexey Bolsinov and Yiru Ye.
◆ Huijun FAN
◆ A.E. MIRONOV
◆ I.A. TAIMANOV
◆ Youjin ZHANG