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

TIME:April 24 -July 10 ,2020 (Every Friday, 17:00 Beijing time, 12:00 Moscow time, 16:00 Novosibirsk time)


    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/7C9E63FCCD7F4AD1C2F5D6C36580D771

    ExpirationTime:2021-07-31 23:59

    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.

    Video: https://disk.pku.edu.cn:443/link/E4C16B2939A52002C336A09B9DB7382B

    ExpirationTime:2021-07-31 23:59

    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.

    Video: https://disk.pku.edu.cn:443/link/7260D810407C0116778F69460F77844A
    ExpirationTime:2021-07-31 23:59

    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. 

    or https://disk.pku.edu.cn:443/link/1DA7AC972BA1CD9240628FFC9E4B6C6F
    ExpirationTime:2021-07-31 23:59

    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)

    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.

    Video: https://disk.pku.edu.cn:443/link/F4B5DF6778C07DF59EF566A8F4693A8C

    ExpirationTime:2021-07-31 23:59

    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)

    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.

    ExpirationTime:2021-07-31 23:59

    Lecture VIII
    Gamma conjecture I for del Pezzo surfaces 
    Time: 2020-06-12 17:00 Beijing time (12:00 Moscow time, 16:00 Novosibirsk time)
    Speaker: Li Changzheng (Sun Yat-Sen University)
    Gamma conjectures were proposed to relate the quantum cohomology of a Fano manifold and the Gamma class interms of differential equations. Gamma conjectures consist of the underlying conjecture O and Gamma conjecture I and II. In this talk, I will first introduce the conjecture O for del Pezzo surfaces, then I will talk about the Gamma conjecture I for del Pezzo surfaces. This talk is based on a joint work with Jianxun Hu, Hua-Zhong Ke and Tuo Yang.

    Lecture IX

    Mirror Symmetry for quasi-smooth Calabi-Yau hypersurfaces in weighted projective spaces 
    Time: 2020-06-19 17:00 Beijing time (12:00 Moscow time, 16:00 Novosibirsk time)
    Speaker: Victor Batyrev (University of Tubingen)
    In the talk based on my joint work with K.Schaller I will explain a general combinatorial framework for constructing mirrors of d-dimensional Calabi-Yau orbifolds defined by arbitrary non-degenerate weighted homogeneous polynomials W. Our mirror construction generalizes the one of Berglund-Huebsch-Krawitz in the case of invertible polynomials W. 

    Lecture X

    Derived categories and Chow theory of Quot-schemes of Grassmannian type.
    Time: 2020-06-26 17:00 Beijing time (12:00 Moscow time, 16:00 Novosibirsk time)
    Speaker: Jiang Qingyuan (University of Edinburgh)
    Quot-schemes of Grassmannian type naturally arise as resolutions of degeneracy loci of maps between vector bundles over a scheme. In this talk we will discuss the relationships of the derived categories and Chow groups among these Quot-Schemes. This provides a unified way to understand many known formulae such as blowup formula, Cayley's trick, projectivization formula, Grassmannian bundles formula and formula for Grassmannain type flops and flips, as well as provide new phenomena such as virtual flips. We will also discuss applications to the study of moduli of linear series on curves, blowup of determinantal ideals, generalized nested Hilbert schemes of points on surfaces, and Brill--Noether problem for moduli of stable objects in K3 categories.

    Download Slides: https://disk.pku.edu.cn:443/link/E89CA08FEABF9D84D2569EE6B5529037
    Valid Until: 2020-07-31 23:59

    Lecture XI

    Kostant, Steinberg, and the Stokes matrices of thett*-Toda equations
    Time: 2020-07-03 17:00 Beijing time (12:00 Moscow time, 16:00 Novosibirsk time)
    Speaker: Ho Nan-Kuo (Department of Mathematics, NTHU)
    We propose a Lie-theoretic definition of the tt*-Toda equations for anycomplex simple Lie algebra, based on the concept of topological-antitopological fusion which was introduced by Cecotti and Vafa. Our main result concerns the Stokes dataof a certain meromorphic connection, whose isomonodromic deformations are controlled by these equations. First, by exploiting a framework introduced by Boalch,we show that this data has a remarkable structure. It can be described using Kostant’stheory of Cartan subalgebras in apposition and Steinberg’s theory of conjugacy classesof regular elements, and it can be visualized on the Coxeter Plane. Second, we compute canonical Stokes data for a certain family of solutions of the tt*-Toda equationsin terms of their asymptotics.This is joint work with Martin Guest.


    Valid Until: 2025-08-31 23:59

    Lecture XII
    The Landau-Ginzburg/Calabi-Yau correspondence for the quintic threefold
    Time: 2020-07-10 17:00 Beijing time (12:00 Moscow time, 16:00 Novosibirsk time)
    Speaker: Prof. Guo Shuai, Department of Mathematics, School of Mathematical Sciences, Peking University
    Abstract: In this talk, we will first introduce the physical and mathematical versions of the Landau-Ginzburg/Calabi-Yau correspondence conjecture for the Calabi-Yau threefolds. Then we will explain our approach to prove this conjecture for the most simple Calabi-Yau threefold - the quintic threefold. This is a work in progress joint with Felix Janda and Yongbin Ruan. 

    Valid Until: 2025-08-31 23:59

    Organization Committee 

    ◆ Huijun FAN 

    ◆ A.E. MIRONOV


    ◆ Youjin ZHANG

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