Beijing-Novosibirsk seminar on geometry and mathematical physics (online seminar)

TIME:April 24 - ,2020

    The lecture announcements will be continually updated. The arrangement of the upcoming lectures is as follows:


    Lecture VI

    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)

    Zoom Meeting(ID: 913 5852 8768)(Link: https://zoom.us/j/91358528768 )


    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.

    Lecture VII

    Open r-spin 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)

    Zoom MeetingID: 690 2810 1557 PW: 864226)(Link: https://zoom.com.cn/j/69028101557?pwd=UWtqSUNueWh2UEEwOHZ2WmNOU0pLdz09 )


    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.


    Previous Lectures:

    Lecture I

    Flat F-manifolds in higher genus and integrable hierarchies

    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.

    Lecture II

    Stokes phenomenon, reflection equations and Frobenius manifolds

    Time: 2020-05-01 17:00

    Speaker:  Xu Xiaomeng (Peking university)

    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

    Lecture III

    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 proof.

    Recording: https://disk.pku.edu.cn:443/link/B52726DF75273ED94D26B14311F61117

    Lecture IV

    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.

    Recording: https://disk.pku.edu.cn:443/link/1B4697F456A8F83AB5E1255ED6D8B048

    Lecture V

    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. 

    Recording: https://disk.pku.edu.cn:443/link/6B3CAF4404B1DE765A819566FA18856A

    or https://cloud.mail.ru/stock/kv7Re1gF8pM3Jk2us6F9CNS8 

    Organization Committee 

    ◆ Huijun FAN 

    ◆ A.E. MIRONOV


    ◆ Youjin ZHANG

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