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 1516, 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
SiQi 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 mah@tsinghua.edu.cn
Schedule
January 15,2024 
A404,Science Building, Tsinghua 
Time 
Schedule 
8:208:40 
Registration 
8:408:50 
Opening Remark 
Time 
Speaker 
Title 
Chair 
8:509:35 
Iskander Taimanov (Sobolev Institute of Mathematics) 
Bloch functions on nonsimplyconnected manifolds 
Youjin Zhang (Tsinghua U.) 
9:3510:10 Group photo and Tea break 
10:1010:55 
Di Yang (USTC) 
Large genus asymptotics of psiclass intersection numbers 
Andrey Mironov (Sobolev Institute of Mathematics) 
11:0011:45 
Huagui Duan (Nankai U.) 
Maslovtype index theory and closed orbits 
Lunch 
Time 
Speaker 
Title 
Chair 
14:0014:45 
Andrey Mironov (Sobolev Institute of Mathematics) 
Rank one commuting ordinary differential operators as a limit of commuting difference operators 
SiQi Liu (Tsinghua U.) 
14:5015:35 
Xiaomeng Xu (Peking U.) 
Quantum Stokes matrices at arbitrary order poles 
15:3515:55 Tea break 
15:5516:40 
Chaozhong Wu (Sun Yatsen U.) 
A KPmKP hierarchy via pseudodifferential operators with two derivations 
Nataliya Daurtseva (Sobolev Institute of Mathematics.) 
16:4517:30 
Xin Wang (Shandong U.) 
Universal structures in Hodge integrals 
Dinner 

Schedule
January 16,2024 A404,Science Building, Tsinghua 

Time 
Speaker 
Title 
Chair 
8:309:15 
Huijun Fan (Peking U.) 
On the Geometry of LandauGinzburg Model 
Iskander Taimanov (Sobolev Institute of Mathematics) 
9:2010:05 
Chenglang Yang (AMSS) 
BKP hierarchy, a formula for connected npoint functions, and applications 
10:0510:25 Tea break 
10:2511:10 
Aleksandr Buryak (Higher School of Economics) 
On integrable systems corresponding to partial cohomological field theories 
Dafeng Zuo (USTC) 
11:1512:00 
Chunhui Zhou (USTC) 
On a kind of generalized Frobenius manifolds and the corresponding integrable systems 
Lunch 
Time 
Speaker 
Title 
Chair 
14:0014:45 
Nataliya A. Daurtseva (Sobolev Institute of Mathematics) 
About almost Hermitian structures that can be realized on S^6 
Hui Ma (Tsinghua U.) 
14:4515:05 Tea break 
15:0515:50 
Chao Qian (Beijing Institute of Technology) 
Symmetry of hypersurfaces with symmetric boundary 
Jian Zhou (Tsinghua U.) 
15:5516:40 
Xiaobo Liu (Peking U.) 
Isoparametric Submanifolds and Mean Curvature Flow 
16:5017:00 
Closing Remark 
Dinner 
Titles and Abstracts
Day 1
Bloch functions on nonsimplyconnected manifolds
Iskander Taimanov (Sobolev Institute of Mathematics, Russia)
Abstract：We would like to speak on an extension of the FloquetBloch theory to nonsimplyconnected manifolds and its relation to the Aharonov Bohm type effects.
Large genus asymptotics of psiclass 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.
Maslovtype 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 Maslovtype 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 onepoint 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 qStokes 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 qStokes matrices as a quantization of the irregular RiemannHilbert map at pole of order k.
A KPmKP hierarchy via
pseudodifferential operators with two derivations
Chaozhong Wu (Sun Yatsen University)
Abstract：By using pseudodifferential operators containing two derivations, we extend the KP hierarchy to a certain KPmKP hierarchy. For the KPmKP hierarchy, we obtain its B\"{a}cklund transformations, bilinear equations of BakerAkhiezer 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 GromovWitten 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 LandauGinzburg 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 CalabiYau (CY) manifold, we can define GromovWitten 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 npoint functions,
and applications
Chenglang Yang (Academy of Mathematics and Systems Sciences, CAS)
Abstract：In this talk，I will introduce a formula calculating connected npoint functions of taufunctions of the BKP hierarchy via corresponding affine coordinates， and its applications to KontsevichWitten and BrezinGrossWitten taufunctions. Some preliminaries about the BKP hierarchy and KW taufunction 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 LiuRuanZhang 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 DubrovinZhang 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 AblowitzLadik 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 6dimensional sphere. Usually, an almost Hermitian 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 quasiKähler and semiKähler 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 Helfrichtype 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 nonnegative 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 ChuuLian Terng and Marco Radeschi.
附：清华大学校园地图 (Tsinghua University Campus Map)
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