Description
The lecture announcements will be continually updated. The arrangement of the upcoming lectures is as follows:
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Lecture XXV - June 10th，2021
Mirror symmetry for a cusp polynomial Landau-Ginzburg orbifold
Speaker: Basalaev Alexey Andreevich (HSE)
Time: 2021-06-10 17:00 Beijing time, 16:00 Novosibirsk time, 12:00 Moscow time
Abstract:
We will establish mirror symmetry between the cusp polynomials considered with a nontrivial symmetry group and Geigle-Lenzing orbifold projective lines. In particular, we will introduce Dubrovin-Frobenius manifold of equivariant Saito theory of any cusp polynomial and show that it is isomorphic to Dubrovin-Frobenius manifold of the respective Geigle-Lenzing orbifold.
We will also show that in the case of simple-elliptic singularities this mirror isomorphism is equivalent the certain relations in the ring of modular forms.
This is a joint work with A.Takahashi (Osaka).
Video: https://disk.pku.edu.cn:443/link/038A4056A5DFA2E111A615470EFAA6B5
ExpirationTime：2026-06-01
23:59
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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/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)
Abstract：
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)
Abstract：
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)
Abstract：
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)
Abstract：
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）
Abstract:
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)
Abstract:
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)
Abstract:
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)
Abstract:
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.
Video: https://disk.pku.edu.cn:443/link/F3047C92A5299C446000061029BB1915
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.
Video:
https://disk.pku.edu.cn:443/link/F54316E400E92E93AEA5D31C6DE880B2
Valid Until: 2025-08-31 23:59
Lecture XIII
Video:https://disk.pku.edu.cn:443/link/DE42FB3F2DD64F94C7A966B6B8AC193B
Video Until:2025-01-01
Transposed Poisson algebras
Time: 2020-10-15 17:00 Beijing time (16:00 Novosibirsk time)
Speaker: Prof. Bai Chengming (Nankai Institute)
Abstract：
We introduce a notion of transposed Poisson algebra which is a dual notion of the Poisson algebra by exchanging the roles of the two binary operations in the Leibniz rule defining the Poisson algebra. We interpret the close relationships between it and some structures such as Novikov-Poisson and pre-Lie Poisson algebras including the example given by a commutative associative algebra with a derivation, and 3-Lie algebras.
Lecture XIV
Video:https://disk.pku.edu.cn:443/link/DE42FB3F2DD64F94C7A966B6B8AC193B
Video Until: 2025-01-01
Title: Logarithmic GLSM and its applications
Speaker: Prof.Ruan Yongbin，IAS，Zhejiang University,Hangzhou.
Time: 2020-10-29 17:00-18:00
Abstract: In early 2010, a mathematical theory of GLSM was proposed by Fan-Jarvis-Ruan to generalize both Gromov-Witten theory and FJRW-theory. The mathematical GLSM theory produced an open moduli space, in contrast to the traditional moduli theory where the compactness is required. Then, a cosection (constructed out of superpotential) localized the theory to the critical locus. The above theory is theoretically beautiful, but not so useful in computation. Recently, a delicate compactification of GLSM (logarithmic GLSM) was constructed to remedy the above defect. Its localization formula is proved to be extremely effective to solve many outstanding problems in the subject of Gromov-Witten theory, including BCOV axioms of higher genus Gromov-Witten theory of quintic 3-fold, r-spin conjecture relating r-spin virtual cycle and locus of holomorphic differential, modularity of Gromov-Witten theory of elliptic fibration and so on. In the talk, we will survey the above developments.
These are joint works with Shuai Guo, Felix Janda and Qile Chen.
Lecture XV
Video:https://disk.pku.edu.cn:443/link/74EB24823ABACB010B4349C229404068
Video Until:2025-01-01
Higher, Super, and Quantum
Speaker：Vincent Bouchard (University of Alberta, Саnada)
Time：2020-11-12 17:00
Abstract: Kontsevich and Soibelman recently introduced the concept of quantum Airy structures, which may be understood as generalizations of Virasoro constraints in enumerative geometry. In this talk I will present two broad generalizations, namely higher and super quantum Airy structures. I will explain how many examples of these structures can be constructed as modules of vertex operator algebras, in particular W-algebras. I will comment (and speculate) on the enumerative interpretation of these new constructions in terms of intersection numbers on various moduli spaces. If time permits, I may also briefly explain how these higher and super quantum Airy structures further expand the definition of the Eynard-Orantin topological recursion.
Lecture XVI
Video:https://disk.pku.edu.cn:443/link/3E5C254A4C00FAE8BAB460A757F08C66
Video Until:2025-01-01
Exponential Networks and enumerative invariants of local CY
Speaker：Mauricio Romo (Tsinghua University)
Time：2020-11-26 17:00
Abstract: Exponential networks (EN) are a variant of the spectral networks of Gaiotto-Moore-Neitzke, for the case of logarithmic differentials and they naturally lead to Donaldson-Thomas type invariants of local CY 3-folds. I will define ENs and subsequently describe how to get the invariants, illustrated by some examples. If time permits I will show some recent development for cases with compact 4-cycles.
Lecture XVII
Video:https://disk.pku.edu.cn:443/link/79BFC86B8E4BF6B39A0F7342732A65F9
Video Until:2025-01-01
A discretization of complex analysis for triangulated surfaces.
Speaker：Ivan Dynnikov ( Steklov Mathematical Institute, Russia)
Time：2020-12-10 17:00 Beijing time (12:00 Moscow time, 16:00 Novosibirsk time)
Abstract:
I will overview results on a particularly simple discrete version of the notion of a holomorphic function. It was suggested by S.P.Novikov and myself in 2002 and stemmed from the idea that the discrete analogue of the Cauchy--Riemann operator must be a first order difference operator. This is most naturally defined on a triangular lattice or a triangulated surface admitting a checkerboard coloring.
Lecture XVIII
Multiple Orthogonal Polynomials with respect to Hermite weights: applications and asymptotics
Speaker: A.I. Aptekarev, (Keldysh Institute of Applied Mathematics RAS),
Joint work with S. Yu. Dobrokhotov, A.V. Tsvetkova (Ishlinsky Institute for Problems in Mechanics RAS) and D. N. Tulyakov (Keldysh Institute of Applied Mathematics RAS)
Time: 2021-01-21 17:00
Abstract: We start with the definition of the Hermite multiple orthogonal polynomials by means of orthogonality relations. Then we present several recent applications, such as eigenvalues distribution of random matrices ensembles with external field and Brownian bridges. The main goal of the talk will be the asymptotics of this polynomial sequence when the degree of the polynomial is growing in the scale corresponding to its variable (so called Plancherel – Rotach type asymptotics). The starting point for our asymptotical analysis is the recurrence relations for multiple orthogonal polynomials. We will present an approach based on the construction of decompositions of bases of homogeneous difference equations. Another approach, based on the semiclassical asymptotics in the case of complex-valued phases will be presented in S. Yu. Dobrokhotov’s talk.
Lecture XIX
Real-valued semiclassical approximation for the asymptotics with complex-valued phases of the Hermitian type orthogonal polynomials
S. Yu. Dobrokhotov, A.V. Tsvetkova (Ishlinsky Institute for Problems in Mechanics RAS)
based on joint work with A.I. Aptekarev, D. N. Tulyakov (Keldysh Institute of Applied Mathematics RAS)
Time: 2021-02-04 17:00 Beijing time, 16:00 Novosibirsk time
Lecture XX - February 18th，2021
Fukaya category for Landau-Ginzburg orbifolds and Berglund-H\"ubsch homological mirror symmetry for curve singularities.
Speaker: Prof. Cheol-Hyun Cho（Seoul National University）
Time: 2021-02-18 17:00 Beijing time, 16:00 Novosibirsk time
Abstract: For a weighted homogeneous polynomial and a choice of a diagonal symmetry group, we define a new Fukaya category based on wrapped Fukaya category of its Milnor fiber together with monodromy information. It is analogous to the variation operator in singularity theory. As an application, we formulate a complete version of Berglund-H\"ubsch homological mirror symmetry and prove it for two variable cases. This is a joint work with Dongwook Choa and Wonbo Jung.
Lecture XXI - March 18th，2021
Virasoro constraints for Drinfeld-Sokolov hierarchies and equations of Painlevé type
Speaker: Prof. Wu Chaozhong（Sun Yat-Sen University）
Time: 2021-03-18 17:00 Beijing time, 16:00 Novosibirsk time
Abstract: By imposing Virasoro constraints to Drinfeld-Sokolov hierarchies, we obtain their solutions of Witten-Kontsevich and of Brezin-Gross-Witten types, and those characterized by certain ordinary differential equations of Painlevé type. We also show the existence of affine Weyl group actions on solutions of such Painlevé-type equations, which generalizes the theory of Noumi and Yamada on affine Weyl group symmetries of the Painlevé-type equations. This work is joint with Si-Qi Liu and Youjin Zhang.
Lecture XXII - April 15th，2021
Homological mirror symmetry for chain type polynomials
Speaker: Umut Varolgunes
Time: 2021-04-15 12:00 Beijing time, 11:00 Novosibirsk time
Abstract: I will start by explaining Takahashi's homological mirror symmetry (HMS) conjecture regarding invertible polynomials, which is an open string interpretation of Berglund-Hubsch-Henningson mirror symmetry. In joint work with A. Polishchuk, we resolve this HMS conjecture in the chain type case up to rigorous proofs of general statements about Fukaya-Seidel categories. Our proof goes by showing that the categories in both sides are obtained from the category Vect(k) by applying a recursion. I will explain this recursion categorically and sketch the argument for why it is satisfied on the A-side assuming the aforementioned foundational results. If time permits, I will also mention what goes into the proof in the B-side.
Lecture XXIII - April 29th，2021
Spinorial description of G_2 and SU(3)-manifolds
Speaker: I. Agricola (Marburg, Germany)
Time: 2021-04-29 17:00 Beijing time, 16:00 Novosibirsk time, 12:00 Moscow time
Video:
https://cloud.mail.ru/stock/3XmMgG8dduEKd55qxfS2fCDy
or
https://disk.pku.edu.cn:443/link/69A1A095EFD2E28809739E9664B9B6B9
Video Until: 2025-01-012026-06-30 23:59
Lecture XXIV - May 20th，2021
Virasoro conjecture for FJRW theory
Speaker: Dr. He Weiqiang (Sun Yat-sen University)
Time: 2021-05-20 17:00 Beijing time, 16:00 Novosibirsk time, 12:00 Moscow time
Abstract:
Virasoro conjecture is one of the most fascinating conjecture in Gromov-Witten theory, which is introduced by Eguchi-Hori-Xiong. It state that the Gromov-Witten potential Z is a solution of a sequence of nonlinear differential equation: L_k(Z)=0, k>=-1. And L_k satisfies the following Virasoro relation [L_m, L_n]=(m-n)L_{m+n}
In this talk, I will give a survey on Virasoro conjecture. I will also talk about the explicit form of Virasoro constraints on FJRW theory and prove it in some simple case, base on the joint work with Yefeng Shen.
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Organization Committee
◆ Huijun FAN
◆ A.E. MIRONOV
◆ I.A. TAIMANOV
◆ Youjin ZHANG
