The lecture announcements will be continually updated. The arrangement of the upcoming lectures is as follows:
Lecture XIX - 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.
To Join Zoom Meeting:
Meeting ID：622 9418 2329
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/7C9E63FCCD7F4AD1C2F5D6C36580D771
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.
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.
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.
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.
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.
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
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
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
Transposed Poisson algebras
Time: 2020-10-15 17:00 Beijing time (16:00 Novosibirsk time)
Speaker: Prof. Bai Chengming (Nankai Institute)
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.
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.
Higher, Super, and Quantum
Speaker：Vincent Bouchard (University of Alberta, Саnada)
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.
Exponential Networks and enumerative invariants of local CY
Speaker：Mauricio Romo (Tsinghua University)
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.
A discretization of complex analysis for triangulated surfaces.
Time：2020-12-10 17:00 Beijing time (12:00 Moscow time, 16:00 Novosibirsk time)
Speaker：Ivan Dynnikov ( Steklov Mathematical Institute, Russia)
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.
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.
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
◆ Huijun FAN
◆ A.E. MIRONOV
◆ I.A. TAIMANOV
◆ Youjin ZHANG