Workshop on Gromov-Witten Theory and Integrable Hierarchies

Nov 16 - 19, 2017. School of Mathematical Sciences, University of Science and Technology of China.


Time Nov 17 Nov 18 Nov 19
8:30 ~ 9:15 Iskander Taimanov Jian Zhou Bohan Fang
9:20 ~ 10:05 Si Li Andrey Mironov Minxin Huang
10:05 ~ 10:25 Tea break
10:25 ~ 11:10 Senyue Lou Baosen Wu Maxim Pavlov
11:15 ~ 12:00 Li Sheng Zhengyu Zong
12:00 ~ 14:00 Lunch break
14:00 ~ 14:45 Aleksandr Smirnov Xiaochuan Liu
14:50 ~ 15:35 Xiangke Chang Chao-Zhong Wu
15:35 ~ 15:55 Tea break
15:55 ~ 16:40 Alexander Zheglov Xiaojun Chen

Titles and Abstracts

Speaker: Xiangke Chang (Chinese Academy of Sciences)

Title: Several peakon and Toda lattices

In this talk, I'll introduce some recent results on peakon, Toda lattices, and orthogonal polynomials.

Speaker: Xiaojun Chen (Sichuan University)

Title: Calabi-Yau algebras and noncommutative symplectic geometry

In 2012, Pantev, Toën, Vaquié and Vezzosi introduced the notion of shifted symplectic structures for derived stacks. This not only generalizes the classical symplectic geometry to a much broader context, but also reveals many new features on a lot of geometric spaces, especially on the various moduli spaces that mathematicians are now studying. As remarked by the authors, the shifted symplectic structure, if it exists, always comes from the Poincare duality of the corresponding source spaces; they also outlined how to generalize the shifted symplectic structure to non-commutative spaces, such as Calabi-Yau categories.

In this talk, we focus on a special class of non-commutative Calabi-Yau spaces, namely Calabi-Yau algebras, and show that for a Koszul Calabi-Yau algebra, there is a noncommutative shifted symplectic structure, which is also called the shifted bi-symplectic structure, on the cobar construction of its Koszul dual coalgebra adjoining a co-unit. Such shifted bi-symplectic structure naturally induces a shifted symplectic structure on the corresponding differential graded (DG) representation schemes, and also induces a shifted Poisson structure on the derived representation schemes of the Calabi-Yau algebra. Joint work with F. Eshmatov.

Speaker: Bohan Fang (Peking University)

Title: Oscillatory integrals on T-dual branes and Gamma II for toric Fano varieties

I will discuss the integrals on the T-dual branes in the Landau-Ginzburg mirror of toric varieties. The results are genus 0 Gromov-Witten descendant potentials. In particular, based on this result and with some input from homological mirror symmetry, I will sketch a proof of Gamma II conjecture, which concerns such potential's asymptotic behavior.

Speaker: Minxin Huang (University of Science and Technology of China)

Title: Refined invariants for Calabi-Yau three-folds and minimal 6d superconformal theories

6d superconformal theories are constructed by the building blocks of a finite set of minimal theories. We develop method to compute the elliptic genus of tensionless strings in these minimal theories, using Weyl invariant Jacobi forms of simple Lie algebra. The elliptic genus is related to topological string partition functions and Gromov-Witten invariants of the corresponding Calabi-Yau three-folds.

Speaker: Si Li (Tsinghua University)

Title: Topological string field theory and integrable hierarchy

We discuss the physics origin of a two-parameter quantum integrable hierarchies associated to Calabi-Yau geometries as a natural quantum gauge symmetry in B-twisted topological string field theory.

Speaker: Xiaochuan Liu (Xi'an Jiaotong University)

Title: Stability of Peakons

The stability of solitary waves (or solitons) is one of the fundamental qualitative properties of the solutions of nonlinear waves equations. In this talk, I will review some progress on the constructions of the orbital stability of the peaked solitary waves (or peakons) by a direct method of conservation laws; present some recent results for the higher-order Camassa-Holm-type dispersive equations; and investigate the effect on the issue of stability of higher-order nonlinearity.

Speaker: Senyue Lou (Ningbo University)

Title: Four nonequivalent integrable Alice-Bob KdV systems

Using parity and time reversal symmetries, some nonequivalent integrable nonlocal KdV (Alice-Bob KdV) systems are obtained from the well-known systems such as the Hirota-Satsuma system, the usual KdV equation with source, coupled KdV systems and the dispersive long wave equations. Same idea can be used to find other types of nonlinear nonlocal integral systems. For instance, the AB-Burgers, AB-NLS type, AB-DNLS systems can be obtained from the Kaup-Broer system, the Boussinesq system and the third CLL systems respectively.

Speaker: Andrey Mironov (Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences)

Title: Billiard and Algebraic Birkhoff conjecture

We introduce a new dynamical system which we call Angular billiard. It acts on the exterior points of a convex curve in Euclidean plane. In a neighborhood of the boundary curve this system turns out to be dual to the Birkhoff billiard. Using this system we get new results on algebraic Birkhoff conjecture on integrable billiards. The talk is based on the joint paper with M. Bialy (Tel Aviv).

Speaker: Maxim Pavlov (Novosibirsk State University)

Title: Integrability of Exceptional Hydrodynamic Type Systems

We consider integrable (by the Extended Hodograph Method) multi-component hydrodynamic type systems, which possess just one multiple root of characteristic polynomials of velocity matrices.

Speaker: Li Sheng (Sichuan University)

Title: Virtual Neighborhood Technique and Related Analytic Problems


Speaker: Aleksandr Smirnov (St. Petersburg State University of Aerospace Instrumentation)

Title: Spectral curves for the rogue waves

In the present work, we calculate spectral curves for quasi-rational solutions of equations from AKNS hierarchy. Also, we found spectral curves for the simplest trigonometric, hyperbolic and elliptic solutions.

Speaker: Iskander Taimanov (Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences)

Title: The Moutard transformation of Dirac operators and conformal geometry of surfaces in the four-space


Speaker: Baosen Wu (Tsinghua University)

Title: Donaldson-Thomas invariant and its degeneration

Donaldson-Thomas invariants of Calabi-Yau threefolds are holomorphic analogue of Casson invariants of three-dimensional real manifolds. They are also closely related to Gromov-Witten invariants in various ways. I will explain briefly this analogue, the Gromov-Witten/Donaldson-Thomas correspondence. Then I will focus on the degeneration method applying to moduli problem, especially Hilbert schemes. As application, we have the degeneration formula of Donaldson-Thomas invariants. This is joint work with Jun Li.

Speaker: Chao-Zhong Wu (Sun Yat-Sen University)

Title: On string equation and topological solution of Drinfeld-Sokolov hierarchies

We give a general method to compute the expansion of topological tau functions for Drinfeld-Sokolov hierarchies associated to an arbitrary untwisted affine Kac-Moody algebra. Our method consists of two main steps: first these tau functions are expressed as (formal) Fredholm determinants of the type appearing in the Borodin-Okounkov formula, then the kernels for these determinants are found using a reduced form of the string equation. This is a collaboration with Mattia Cafasso.

Speaker: Alexander Zheglov (Moscow State University)

Title: Quantum integrable systems and torsion free sheaves on algebraic varieties

I will talk about the correspondence between quantum integrable systems and algebro-geometric spectral data. The most important ingredient of these data are spectral sheaves --- torsion free sheaves of special kind on algebraic varieties. I will also discuss how explicit is this correspondence, in particular, could we produce integrable systems starting from spectral data, even if the spectral variety has dimension greater than one.

Speaker: Jian Zhou (Tsinghua University)

Title: Kepler problem and AdS/CFT correspondence

Kepler system is one of the first integrable systems. Modern study of the Kepler problem reveals some rich connections with other mathematical objects, for example hidden symmetry of the Kepler system and minimal coadjoint orbits, relationships with Penrose twistor theory and Howe duality, etc. There are also generalization of the Kepler problems, which are related to Jordan algebras and bounded symmetric domains. We will introduce a connection of the Kepler problem with the AdS/CFT correspondence. This leads us to hidden supersymmetry that unites all aspects of the Kepler problem.

Speaker: Zhengyu Zong (Tsinghua University)

Title: Gromov-Witten/Donaldson-Thomas correspondence for local gerby curves

In 2004 and 2005, the work of Bryan-Pandharipande and Okounkov-Pandharipande build the Gromov-Witten/Donaldson-Thomas correspondence for local curves. In this talk, I will discuss an ongoing project on the Gromov-Witten/Donaldson-Thomas correspondence for local gerby curves, which is an orbifold generalization of the above work. More concretely, we will consider a $\mathbb{Z}_{n+1}$-gerbe $C$ over a smooth genus $g$ curve. Let $L_1$ and $L_2$ be two orbifold line bundles over $C$ such that the isotropy group $\mathbb{Z}_{n+1}$ acts on them via fiberwisely multiplication by $\exp\left(\frac{2\pi i}{n + 1}\right)$ and $\exp\left(-\frac{2\pi i}{n + 1}\right)$ respectively. We will study the Gromov-Witten/Donaldson-Thomas correspondence for the total space of $X := L_1\oplus L_2$. The strategy of the proof is to apply degeneration formula to $X$ to reduce the problem to the case of cap, cylinder, and pair of pants. This work is joint with Zijun Zhou.


Scientific Committee

Youjin Zhang

Jian Zhou

Organizing Committee

Dafeng Zuo

Hui Ma

Si-Qi Liu

Youjin Zhang

Jian Zhou

Invited Speakers

Xiangke Chang

Xiaojun Chen

Bohan Fang

Minxin Huang

Si Li

Xiaochuan Liu

Senyue Lou

Andrey Mironov

Maxim Pavlov

Li Sheng

Aleksandr Smirnov

Iskander Taimanov

Baosen Wu

Chao-Zhong Wu

Alexander Zheglov

Jian Zhou

Zhengyu Zong


Dafeng Zuo
Kelei Tian


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