The purpose of the workshop is to foster discussions and collaborations among participants, with a focus on exchanging ideas and experiences related to Geometry and Integrable Systems. The workshop is scheduled to take place on January 15-16, 2024, at the Department of Mathematical Sciences, Tsinghua University. 

 

Scientific Committee

Andrey Mironov  Iskander Taimanov (Sobolev Institute of Mathematics, Russia)

Youjin Zhang     Jian Zhou   (Tsinghua University) 

 

Organizing Committee

Si-Qi Liu    Hui Ma   (Tsinghua University)

 

Sponsor

Department of Mathematical Sciences, Tsinghua University

 

Support fundings

中俄合作项目“几何与数学物理中的量子不变量研究（NSFC 12061131014）

 

Conference location

A404 Science Building, Tsinghua University  清华大学理科楼A404

 

Contact information

陈新湃 chenxinpai@tsinghua.edu.cn

马辉 ma-h@tsinghua.edu.cn

 

Schedule

 

 

Titles and Abstracts

Day 1

 

Bloch functions on non-simply-connected manifolds

Iskander Taimanov (Sobolev Institute of Mathematics, Russia)

Abstract：We would like to speak on an extension of the Floquet-Bloch theory to non-simply-connected manifolds and its relation to the Aharonov Bohm type effects.

 

Large genus asymptotics of psi-class intersection numbers

Di Yang (University of Science and Technology of China)

Abstract：We give a new proof of the DGZZ conjecture. If time permits, we will present some new progress. The work is joint with Jindong Guo.

 

Maslov-type index theory and closed orbits

Huagui Duan (Nankai University)

Abstract：In this talk, I will introduce two kinds of closed orbit problems, i.e., closed geodesics on manifolds and closed orbits on hypersurfaces with the fixed energy. Then I will introduce some recent progress in this field, and simply explain how to deal with these problems by using the Maslov-type index theory.

 

Rank one commuting ordinary differential operators

as a limit of commuting difference operators

Andrey Mironov (Sobolev Institute of Mathematics, Russia)

Abstract：We study a connection between commuting ordinary differential operators and commuting difference operators. We show that rank one commuting ordinary differential operators can be extended to a family of one-point commuting difference operators depending on a small parameter. The spectral data for the difference operators are obtained by a natural extension of the classical Krichever's spectral data for commuting differential operators of rank one.

Quantum Stokes matrices at arbitrary order poles

Xiaomeng Xu (Peking University)

Abstract：This talk first introduces the quantum Stokes matrices at a pole of order k. It then proves that the q-Stokes matrices naturally satisfy certain degree k algebraic relations. For the case k=2, these algebraic relations recover the defining relation of quantum groups. In the end, it explains q-Stokes matrices as a quantization of the irregular Riemann-Hilbert map at pole of order k.

 

A KP-mKP hierarchy via

pseudo-differential operators with two derivations

Chaozhong Wu (Sun Yat-sen University)

Abstract：By using pseudo-differential operators containing two derivations, we extend the KP hierarchy to a certain KP-mKP hierarchy. For the KP-mKP hierarchy, we obtain its B\"{a}cklund transformations, bilinear equations of Baker-Akhiezer functions and Hirota equations of tau functions. Moreover, we show that this hierarchy is equivalent to a subhierarchy of the dispersive Whitham hierarchy associated to the Riemann sphere with its infinity point and one movable point marked. This work is joint with Lumin Geng and Jianxun Hu.

 

Universal structures in Hodge integrals

Xin Wang (Shandong University)

Abstract：In this talk, we will discuss two universal structures in Hodge integrals. One is Virasoro constraints for Hodge integrals of any target varieties. Another one is partial differential equations for higher genus Gromov-Witten invariants from Hodge integrals of any target varieties. This talk is partially based on joint work with Felix Janda.

 

 

 

 

Day 2

 

On the Geometry of Landau-Ginzburg Model

Huijun Fan(Peking University)

Abstract: An LG model (M, f) is given by a noncompact complex manifold M and the holomorphic function f defined on it, which is an important model in string theory. Because of the mirror symmetry conjecture, the research on the geometric structure and quantization theory of LG model has attracted more and more attention. Given a Calabi-Yau (CY) manifold, we can define Gromov-Witten theory (A theory) on it, and also study the variation of Hodge structure on its mirror manifold (B theory). Accordingly, LG model includes A theory - FJRW theory and Hodge structure variational theory.

This report starts with some examples, gives the geometric and topological information contained by a LG model, and derives the relevant Witten equation (nonlinear) and Schrodinger equation (linear). The study of the solution space of these two sets of equations will lead to different quantization theories. Secondly, we give our recent correspondence theorem of Hodge structures between LG model and CY manifold. Finally, we will discuss some relevant issues.

 

BKP hierarchy, a formula for connected n-point functions, 

and applications

Chenglang Yang (Academy of Mathematics and Systems Sciences, CAS)

Abstract：In this talk，I will introduce a formula calculating connected n-point functions of tau-functions of the BKP hierarchy via corresponding affine coordinates， and its applications to Kontsevich-Witten and Brezin-Gross-Witten tau-functions. Some preliminaries about the BKP hierarchy and KW tau-function will also be included. This talk is mainly based on joint works with Professors Xiaobo Liu，Zhiyuan Wang.

 

On integrable systems corresponding to partial cohomological field theories

Aleksandr Buryak

(Higher School of Economics, Russia)

Abstract：The notion of a partial cohomological field theory (CohFT) was introduced by Liu-Ruan-Zhang in 2015. This is the same as a CohFT except that we don't require the gluing loop axiom. I would like to discuss the Dubrovin-Zhang and the DR hierarchies associated to partial CohFTs. In particular, I would like to show the computations giving an evidence that in the case of a homogeneous semisimple partial CohFT both hierarchies are bihamiltonian, and the central invariants of the corresponding pair of Poisson brackets are constants. Moreover, varying the higher genus part of the partial CohFT, one can make this central invariant equal to an arbitrary given collection of constants.

 

 

 

On a kind of generalized Frobenius manifolds

and the corresponding integrable systems

Chunhui Zhou (University of Science and Technology of China)

Abstract：In this talk, we will discuss a kind of integrable systems whose associated Frobenius manifolds have no Euler vector fields. They can be viewed as the almost dualities of the Frobenius manifolds associated with extended Toda hierarchy and Ablowitz-Ladik hierarchy. We will also show how to use this result and the GW invariants of CP^1 to calculate the equivariant GW invariants of resolved conifold in two particular cases.

 

About almost Hermitian structures that can be realized on $S^6$

Nataliya A. Daurtseva (Sobolev Institute of Mathematics, Russia)

Abstract：I plan to talk about different almost Hermitian structures on the 6-dimensional sphere. Usually, an almost H-ermitian structure induced by embedding of S^6 in R^7 is considered. This structure has a number of important properties, it is invariant on the homogeneous space S^6=G_2/SU(3) and is nearly Kähler. Unfortunately, there are no other invariant structures on this space. However, it is possible to define the cohomogeneity one action of group SU(2)xSU(2) on S^6. The set of cohomogeneity one structures is much wider than the set of homogeneous ones. This allows us to construct examples of other almost Hermitian structures on S^6, in particular quasi-Kähler and semi-Kähler ones.

 

 

Symmetry of hypersurfaces with symmetric boundary

Chao Qian (Beijing Institute of Technology)

Abstract: The main motivation of this talk is to study which kind of boundary symmetry can pass to the interior of certain interesting hypersurface M^n in R^{n+1}. We will talk about recent progress on this problem for minimal hypersurfaces, hypersurfaces with constant mean curvature，hypersurfaces with constant higher order mean curvature and Helfrich-type hypersurfaces in R^{n+1}. This talk is based on joint work with Hui Ma, Jing Wu and Yongsheng Zhang.

 

Isoparametric Submanifolds and Mean Curvature Flow

Xiaobo Liu (Peking University)

Abstract： Ancient solutions are important in studying singularities of mean curvature flows (MCF). So far most rigidity results about ancient solutions are modeled on shrinking spheres or spherical caps. In this talk, I will describe the behavior of MCF for a class of submanifolds, called isoparametric submanifolds, which have more complicated topological type. We can show that all such solutions are in fact ancient solutions, i.e. they exist for all time which goes to negative infinity. Similar results also hold for MCF of regular leaves of polar foliations in simply connected symmetric spaces with non-negative curvature. I will also describe our conjectures proposed together with Terng on rigidity of ancient solutions to MCF for hypersurfaces in spheres. These conjectures are closely related to Chern’s conjecture for minimal hypersurfaces in spheres. This talk is based on joint works with Chuu-Lian Terng and Marco Radeschi.

 

 

  

附：清华大学校园地图 (Tsinghua University Campus Map)