Beijing Geometry and Physics Colloquium

# Workshop on Integrable Systems and Gromov-Witten Invariants

Nov 18 - 20, 2016. Room A404, Department of Mathematical Sciences, Tsinghua University, P. R. China.

## Schedule

Time Nov 18 Nov 19 Nov 20
8:30 ~ 9:15 I. Taimanov Anmin Li Lixin Tian
9:20 ~ 10:05 A. Mironov Li Sheng P. Lorenzoni
10:05 ~ 10:25 Tea break
10:25 ~ 11:10 Huijun Fan Zixiang Zhou Bohan Fang
11:15 ~ 12:00 Dajun Zhang Shuai Guo Huazhong Ke
12:00 ~ 14:00 Lunch break
14:00 ~ 14:45 A. Zheglov Changzheng Qu
14:50 ~ 15:35 Hailong Her Zhengyu Zong
15:35 ~ 15:55 Tea break
15:55 ~ 16:40 Senyue Lou Engui Fan
16:45 ~ 17:30 M. Pavlov Shanzhong Sun

## Details

### Nov 18 (Friday)

8:15 ~ 8:30, Opening

8:30 ~ 9:15, Iskander A. Taimanov (Sobolev Institute of Mathematics)

Title: Frobenius Manifolds via Singular Spectral Curves

Abstract: We demonstrate how to construct Frobenius manifolds by using the finite gap integration methods applied to singular spectral curves. The talk is baed on the joint work with A.E. Mironov.

9:20 ~ 10:05, Andrey E. Mironov (Sobolev Institute of Mathematics)

Title: Integrable Magnetic Geodesic Flows on 2-torus and the Systems of Hydrodynamic Type

Abstract: The only one example has been known of magnetic geodesic flow on the 2-torus which has a polynomial in momenta integral independent of the Hamiltonian. In this example the integral is linear in momenta and corresponds to a one parametric group preserving the Lagrangian function of the magnetic flow. We consider the problem of integrability on one energy level. This problem can be reduced to a remarkable Semi-hamiltonian system of quasilinear PDEs and to the question of existence of smooth periodic solutions for this system. Our main result states that the pair of Liouville metric with zero magnetic field on the 2-torus can be analytically deformed to a Riemannian metric with small magnetic field so that the magnetic geodesic flow on an energy level is integrable by means of a quadratic in momenta integral. Thus our construction gives a new example of smooth periodic solution to the Semi-hamiltonian quasilinear system of PDEs. The talk is based on the joint paper with Misha Bialy (Tel-Aviv) and Sergey Agapov (Novosibirsk).

10:05 ~ 10:25, Tea Break

10:25 ~ 11:10, Huijun Fan (Peking University)

Title: Fukaya Category of Landau-Ginzburg Model

Abstract: In this report, I will give a rigorous mathematical construction of the Fukaya category of Laudau-Ginzburg model, which were described briefly by recent work of Gaiotto-Moore-Witten and Kapranov-Kontsevich-Soibelman on the algebra of the infraed. This is a moduli problem about Witten equation with Lefschetz boundary condition. This is a joint work with Dingyu Yang and Wenfeng Jiang.

11:15 ~ 12:00, Dajun Zhang (Shanghai University)

Title: On Elliptic Aspects of Discrete Integrable Systems

Abstract: There are two ways elliptic curves can play a role in integrable systems: either as elliptic type solutions (i.e. solutions expressible in terms of elliptic functions), or as elliptic deformation of the equations themselves. In either way, the study of the elliptic case is often richer that the rational and trigonometric/hyperbolic case, and reveals many new features of the models in question. In this talk, first, I will give a brief introduction on discrete integrable systems (DIS), including the concept of multidimensional consistency, Adler-Bobenko-Suris’ classification of quadrilateral lattice equations, etc. Then, I will introduce a method used in DIS, Cauchy matrix approach, which is based on the Sylvester equation and discrete dispersion relations. Next, we show that the Cauchy matrix approach works for the study of some elliptic Integrable systems, i.e. some equations in these systems are formulated with an elliptic curve. Finally, we show a kind of soliton solutions expressed via Weirestrass $\sigma$ function.

12:00 ~ 14:00, Lunch Break

14:00 ~ 14:45, Alexander Zheglov (Lomonosov Moscow State University)

Title: On Commuting Ordinary Differential Operators of Rank Two With Polynomial Coefficients

Abstract: I'll talk about several results on commuting ordinary differential operators of rank two with polynomial coefficients, obtained in joint works together with A.E. Mironov and with I. Burban. These results are related with the following Berest conjecture. Let $A_1$ be the first Weyl algebra $\mathbb{C}[x][\partial_x]$. Consider a generic polynomial equation in two variables $f(X,Y)=0$ that has a solution in $A_1$. Each such solution is a pair of commuting ordinary differential operators in $A_1$. A conjecture proposed by Yu. Berest says that the orbit space of the group action of $Aut(A_1)$ on the set of solutions of this equation is infinite if the genus of the corresponding spectral curve is 1, and is finite otherwise. This conjecture has an intimate connection with the famous Dixmier conjecture for $A_1$.

14:50 ~ 15:35, Hailong Her (Nanjing Normal Universtiy)

Title: Algebra Structures from Fukaya Categories and Linearized Contact Homology

Abstract: In this talk, we will discuss the Poisson structure on Fukaya categories and Lie bialgebra structures appearing on both the cyclic cohomology of Fukaya category and the linearized contact homology of exact symplectic manifolds with contact type boundary. These are joint works with X. Chen, S. Sun and X. Yang. We will further discuss the analytic difficulty of establishing a relating map between the cyclic cohomology and the linearized contact homology, a method based on virtual neighborhood techniques will be sketched.

15:35 ~ 15:55, Tea Break

15:55 ~ 16:40, Senyue Lou (Ningbo University)

Title: TBA

Abstract: TBA

16:45 ~ 17:30, Maxim Pavlov (Novosibirsk State University)

Title: WDVV Associativity Equations as a High Frequency Limit

Abstract: In this talk we consider a new dispersive integrable systemof Camassa--Holm type, which possesses two distinguish limits: Long wave (dispersionless) limit to diagonalisable quasilinear system of first order and short wave (high frequency) limit to nondiagonalisable quasilinear system of first order known as the WDVV associativity equations. Also we present a bi-Hamiltonian structure of this Intermediate system.

### Nov 19 (Saturday)

8:30 ~ 9:15, Anmin Li (Sichuan University)

Title: Virtual Neighborhood Technique for Holomorphic Curve Moduli Spaces

Abstract: Recently, there is a great deal of interest to re-visit the approaches for Gromov-Witten invariants for general symplectic manifolds . The main complication is that the moduli space has various lower strata. How to deal with these lower strata is one of main issues discussed recently. We show that the Gromov-Witten invariants can be defined as an integral over top strata of virtual neighborhood. Therefore, all the complication near lower strata of the moduli space can be avoided entirely.

9:20 ~ 10:05, Li Sheng (Sichuan University)

Title: The Exponential Decay of Gluing Maps for J-Holomorphic map Moduli Spaces

Abstract: We prove the exponential decay of the derivative of the gluing maps with respect to the gluing parameter.

10:05 ~ 10:25, Tea Break

10:25 ~ 11:10, Zixiang Zhou (Fudan University)

Title: Darboux transformations and global solutions for nonlocal Davey-Stewartson I equation

Abstract: For the nonlocal Davey-Stewartson I equation, the Darboux transformation is considered and explicit expressions of the solutions are obtained. Like the nonlocal equations in $1+1$ dimensions, many solutions may have singularities. However, by suitable choice of parameters in the solutions of the Lax pair, it is proved that the solutions obtained from seed solutions which are zero and an exponential function of $t$ respectively, by a Darboux transformation of degree $n$ are global solutions of the nonlocal Davey-Stewartson I equation. The derived solutions are soliton solutions when the seed solution is zero, in the sense that they are bounded and have $n$ peaks, and "line dark soliton" when the seed solution is an exponential function of $t$, in the sense that they are bounded and their norms change fast along some straight lines.

11:15 ~ 12:00, Shuai Guo (Peking University)

Title: Elliptic Global Mirror Symmetry for the Quintic Threefold

Abstract: In this talk, we mainly introduce two results to establish the elliptic global mirror symmetry. First, we compute an explicit formula for the genus-one FJRW invariants of the quintic, verifying the genus-one mirror theorem. Next, we verify the genus-one Landau-Ginzburg/Calabi-Yau correspondence. The first result is proved by two steps: a comparison result and a computation of the twisted invariants. The second result is proved by the graph sum formula. This is a joint work with Dustin Ross.

12:00 ~ 14:00, Lunch Break

14:00 ~ 14:45, Changzheng Qu (Ningbo University)

Title: Liouville Correspondences between Some Integrable Hierarchies

Abstract: In this talk, the Liouville correspondence between the integrable modified KdV hierarchy and its dual integrable modified Camassa-Holm hierarchy is found. It is shown that the Liouville transformation connects their Hamiltonian conservation laws. Furthermore, a novel transformation mapping the modified Camassa-Holm equation to the Camassa-Holm equation is discovered. In addition, the Liouville correspondence between the Degasperis-Procesi hierarchy and Kaup-Kupershmidt hierarchy is also established.

14:50 ~ 15:35, Zhengyu Zong (Tsinghua University)

Title: Gromov-Witten/Donaldson-Thomas Correspondence for $A_n$ Orbifold and Crepant Resolution Conjecture

Abstract: In this talk, I will discuss the Gromov-Witten/Donaldson-Thomas correspondence for $\mathbb{P}^1\times [\mathbb{C}^2/\mathbb{Z}_{n+1}]$ relative to arbitrary number of fiber divisors. The smooth counterpart of this story is the Gromov-Witten/Donaldson-Thomas correspondence for $\mathbb{P}^1\times \mbox{($A_n$-resolution)}$ which was proved by Maulik in 2008. I will also talk about the Crepant Resolution Conjecture for the relative theory of $\mathbb{P}^1\times [\mathbb{C}^2 /\mathbb{Z}_{n+1}]$. In the end, I will discuss the future application of this result to the Gromov-Witten/Donaldson-Thomas correspondence and Crepant Resolution Conjecture for local gerby curves with $A_n$ singularities. This work is joint with Zijun Zhou.

15:35 ~ 15:55, Tea Break

15:55 ~ 16:40, Engui Fan (Fudan University)

Title: Integrable System, Orthogonal Polynomials, Random Matrix Theory, and Riemann-Hilbert Problem

Abstract: In this talk, we provide the links of Riemann-Hilbert Problem with integrable system, orthogonal polynomials and random matrices. So that Riemann-Hilbert approach can be applied in these three research fields.

16:45 ~ 17:30, Shanzhong Sun (Capital Normal Universtiy)

Title: Moyal quantization of the cyclic cohomology of Fukaya categories

Abstract: In our attempts to understand the higher genus version of Fukaya categories of symplectic manifolds, we develop the Moyal quantization of the Lie bialgebra structure on the cyclic cohomology of the Fukaya category. I will report our joint work in progress with X. Chen, Y. Chen and F. Eshmatov in this direction.

### Nov 20 (Sunday)

8:30 ~ 9:15, Lixin Tian (Nanjing Normal University)

Title: TBA

Abstract: TBA

9:20 ~ 10:05, Paolo Lorenzoni (University of Milan)

Title: Bi-flat F-manifolds, Painlevé transcendents and complex reflection groups

Abstract: We study F-manifolds equipped with a pair of flat connections (and a pair of F-products), that are required to be compatible in a suitable sense. In the first part of the talk I will discuss some examples related to complex reflection groups. In the second part of the talk I will show that bi-flat F-manifolds in dimension three are locally parameterized by solutions of the full Painlevé IV, V, and VI equations, according to the Jordan normal form of the operator of multiplication by the Euler vector field. Based on joint works with Alessandro Arsie.

10:05 ~ 10:25, Tea Break

10:25 ~ 11:10, Bohan Fang (Peking University)

Title: The conifold transition of a torus knot and open invariants

Abstract: The conifold transition of the conormal bundle of a torus knot in the cotangent bundle of a 3-sphere is a non-compact Lagrangian submanifold in the resolved conifold. I will review the open Gromov-Witten invariants w.r.t. this Lagrangian defined via localization by Diaconescu-Shende-Vafa, and will introduce another definition by relative Gromov-Witten invariants. Then I will describe an effective algorithm to compute these invariants using the Eynard-Orantin recursion. This talk is based on the joint work with Zhengyu Zong.

11:15 ~ 12:00, Huazhong Ke (Sun Yat-sen University)

Title: Semisimplicity of Quantum Cohomology under Curve Blow-Up (work in progress)

Abstract: A question of Ruan asks whether semisimplicity of quantum cohomology is a birational invariant. In dimension two, we show that a projective (resp. symplectic) surface has semisimple quantum cohohomology if and only if it is rational (resp. symplectically rational). The proof is based on a result of Bayer and Manin that semisimplicity of quantum cohomology is invariant under point blow-up. To study Ruan’s question in dimension three, we take a first step to show that the blow-up of $\mathbb{P}^3$ along a rational curve (of any degree) has semisimple quantum cohomology.

Scientific Committee

Youjin Zhang

Jian Zhou

Organizing Committee

Si-Qi Liu

Hui Ma

Youjin Zhang

Jian Zhou

Invited Speakers

Engui Fan

Huijun Fan

Bohan Fang

Shuai Guo

Hailong Her

Huazhong Ke

Anmin Li

Paolo Lorenzoni

Senyue Lou

Maxim Pavlov

Changzheng Qu

Andrey E. Mironov

Li Sheng

Shanzhong Sun

Iskander A. Taimanov

Lixin Tian

Alexander Zheglov

Dajun Zhang

Zixiang Zhou

Zhengyu Zong

Contact

Xinpai Chen
Phone: (+86) 010-62772984
Email: xpchen@math.tsinghua.edu.cn

Accommodation

Jiasuo Guesthouse
Phone: (+86) 010-62783166
Zhongguanxinyuan Hotel
Phone: (+86) 010-62752288

Map