Title: Introduction to complex hyperbolicity
Abstract:
In the year 1970, Kobayashi introduced an intrinsic canonical metric for every complex manifold X,
and when this obtained metric is everywhere non-degenerated, X is said to be complex hyperbolic.
He conjectured that, in all dimensions, most manifolds should be complex hyperbolic.
In this talk we shall present the history and some recent developments in this subject,
including the Green-Griffiths-Lang conjecture.
Title: Large Genus Asymptotics for Intersection Numbers and Strata Volumes I
Abstract:
In these talks we describe several universality results concerning the large genus asymptotics for geometric invariants associated with moduli spaces of curves and differentials.
These invariants include intersection numbers between \psi-classes on the moduli space of stable curves, as well as Masur-Veech volumes and Siegel-Veech constants associated with strata of holomorphic and quadratic differentials.
The proofs of these asymptotic results are based on a combination of probabilistic, complex analytic, and combinatorial studies of certain algebraic identities satisfied by these invariants.
Title: Large Genus Asymptotics for Intersection Numbers and Strata Volumes II
Abstract:
In these talks we describe several universality results concerning the large genus asymptotics for geometric invariants associated with moduli spaces of curves and differentials.
These invariants include intersection numbers between \psi-classes on the moduli space of stable curves, as well as Masur-Veech volumes and Siegel-Veech constants associated with strata of holomorphic and quadratic differentials.
The proofs of these asymptotic results are based on a combination of probabilistic, complex analytic, and combinatorial studies of certain algebraic identities satisfied by these invariants.
Title: Quasi-modular and quasi-Jacobi forms of genus two, and topological recursion
Abstract:
I will start by reviewing how natural constructions on the moduli space of genus one
stable curves give rise to differential rings of quasi-modular forms and quasi-Jacobi forms.
Then I will explain the construction of a theory of quasi-modular forms and quasi-Jacobi forms at genus two developed in a joint work
with Yongbin Ruan and Yingchun Zhang. Finally I will discuss the application in topological recursion
and in the open/closed Gromov-Witten theory of a special class of toric Calabi-Yau threefolds whose
mirror curves are of genus one or two. The talk is mainly based on joint works with Bohan Fang, Yongbin Ruan and Yingchun Zhang.
Title: Poisson-Lie dual of semisimple groups and cluster algebras
Abstract:
Let G be a complex connected semisimple group with the standard Poisson-Lie structure and let G^\ast be its dual Poisson-Lie group.
The coordinate ring \mathcal{O}(G^\ast) is a Poisson algebra whose deformation quantization gives rise to the
quantum group \mathcal{U}_q(\mathfrak{g}). We embed \mathcal{O}(\G^\ast) into a larger cluster Poisson algebra together with a Weyl group action.
By applying bases theory of cluster algebras, we obtain a natural linear basis \Theta of \mathcal{O}(\G^\ast) with positive integer structure coefficients.
We show that the basis \Theta, as a set, is invariant under Lusztig's braid group action.
Title: Contractibility of space of stability conditions on the projective plane via global dimension function
Abstract:
The global dimension function \mathrm{gldim} is a continuous function defined on Bridgeland stability manifold,
and it maps a stability condition to a non-negative real number. We compute the global dimension function \mathrm{gldim}
on the principal component \mathrm{Stab}^{\dag}(\mathbb{P}^2) of the space of Bridgeland stability conditions on
projective plane \mathbb{P}^2. It admits 2 as minimum value and the preimage \mathrm{gldim}^{-1}(2) is contained in
the closure \overline{\mathrm{Stab}^{\mathrm{Geo}}(\mathbb{P}^2)} of the subspace consisting of geometric stability conditions.
We show that \mathrm{gldim}^{-1}[2,x) contracts to \mathrm{gldim}^{-1}(2)
for any real number x\geq 2 and that \mathrm{gldim}^{-1}(2) is contractible.
This is a joint work with Yu-Wei Fan, Chunyi Li and Yu Qiu. The preprint is available at arXiv:2001.11984.
Title: Gamma猜想简介
Abstract:
Gamma猜想是由Galkin，Golyshev以及Iritani提出的关于Fano流形量子上同调的一系列猜想，
包括猜想O以及Gamma猜想I和II。这些猜想关注的是Fano流形量子上同调的Dubrovin联络的平坦截面在非正则奇点附近的渐近行为，
期望可以用Fano流形的导出范畴通过量子上同调的整结构（integral structure）来描述这些渐近行为。
我们将会讲解相关概念（量子上同调，Dubrovin联络，整结构，导出范畴……），介绍这一系列猜想。最近我与胡晓文合作，
对于二次超曲面证明了Gamma猜想II。如果时间允许，我们也会谈及这项工作。
Title: Courant代数胚的范畴化
Abstract:
Courant代数胚是刘张炬-Weinstein-徐平在研究李双代数胚的和空间上的结构时引入的，
在广义复几何与拓扑场论等领域中有重要应用。我们定义了CLWX 2-代数胚，它是Courant代数胚的范畴化，
建立了CLWX 2-代数胚与李3-代数、度为3的辛NQ流形以及李2-代数胚的高阶Pontryagin示性类等结构之间的联系。
Title: On the q-mKP hierarchy
Abstract: In the talk, we will show to you some results on the Sato theory of q-mKP hierarchy. Sato theory, developed by Mikio Sato and his school in Kyoto, contains many interesting mathematical structures and in particular provides
the construction of hierarchies of soliton equations, written in terms of tau-functions depending on infinite number of independent variables. In this talk, the tau function and additional symmetries of q-mKP hierarchy are introduced.
Title: Davey-Stewartson系统以及相关问题
Abstract:
Davey-Stewartson系统是重要的（2+1）维可积模型。
在这次报告中，我主要介绍在经典的Davey-Stewartson系统研究基础上所做的一些工作，
包括新定义的Davey-Stewartson可积系列，无色散Davey-Stewartson系统以及相关的无色散可积系列.
Title: Open KdV方程簇的一类新的tau函数
Abstract: Open KdV方程簇来源于对带边黎曼面模空间相交理论的研究，它刻画了靶空间为单点的情形下的Open Gromov-Witten不变量与Witten-Kontsevich配分函数之间的联系。在近期的工作中，我们从KdV方程簇的一个一般的tau函数出发，
建立其与Open KdV方程簇的tau函数之间的联系。作为应用，我们给出Open Gromov-Witten不变量的一个有效计算方法；同时，我们从KdV方程簇的广义BGW tau函数出发构造Open KdV方程簇的一类新的tau函数，并给出其显式公式。
Title: Borel-Laplace multi-transform, and integral representations of solutions of qDEs
Abstract: The quantum differential equation (qDE) is a rich object attached to a smooth projective variety X. It is an ordinary differential equation
in the complex domain which encodes information of the enumerative geometry of X, more precisely its Gromov-Witten theory. Furthermore, the monodromy of
its solutions conjecturally rules also the topology and complex geometry of X. In this talk I will introduce some analytic integral multitransforms of Borel-Laplace type,
and I will use them to obtain Mellin-Barnes integral representations of solutions of qDEs. Based on arXiv:2005.08262.
Title: Curvature flows of hypersurfaces and geometric inequalities
Abstract: Curvature flows of hypersurfaces are characterized by a family of hypersurfaces evolving in an ambient manifold with velocity determined by their extrinsic curvatures.
The equations that arise are nonlinear parabolic differential equations. The curvature flows of hypersurfaces have many applications including the proof of some sharp geometric inequalities.
In this talk, I will describe some recent work on this topic, with focus on the applications of curvature flows in the proof of isoperimetric type inequalities in Euclidean space and in hyperbolic space.
Title: 3维引力:物理，几何与算术
Abstract: 3维引力是由E.Witten建议研究的低维引力模型，它具有很多迥异于高维引力的特殊性质。在这个报告中，我们介绍经典3维引力模型的物理内容及其各种推广，
由Ads/CFT对应建立相应量子理论的方法与问题，我们也将看到一些重要几何与算术问题在其中的自然出现。
Title: Motivicity vs Periodicity
Abstract: In this talk I shall explain what the two notions in the title mean and why they should be important in algebraic geometry.
Title: Exact solution to a 2-species exclusion process and its asymptotics (双粒子排他过程的精确解以及它的渐近性)
Abstract: Exclusion process has been the default model for transportation phenomenon. One fundamental issue is to solve the master equation analytically,
which, in principle, gives all the correlation function of the system. In this talk, we will focus on an exclusion process with 2 species particles: the AHR (Arndt-Heinzl-Rittenberg) model.
We will give a full derivation of its Green's function as well as its joint current distributions. We will also study its long time behaviour with step type initial conditions.
This is a joint work with Jan de Gier, Iori Hiki, Tomohiro Sasamoto and Masato Usui. （排他过程是研究传输现象的经典范例模型，其中最本质的问题之一就是给出其格林函数，
从而得到其current distribution。在本报告中，我们考虑双粒子模型AHR model，
我们将给出其格林函数以及 joint curren tdistributions的exact formula。同时在跃迁类型的初始条件下，我们会给出其渐近分析，
从而得到其在时间趋向无穷时的current distribution的极限。）