Sep 4 - Sep 7, 2023. Hosted by Morningside Center of Mathematics, CAS. The summur school will have in-person sessions as well as online talks, both can be accessed virtually via Zoom. We will be able to cover accomodations for some participants. Please register for further notices. If you need to apply for assistance with accomodation, please register before Aug 20th.
High dimensional convex hypersurfaces and their contact homologies
Cluster algebras and 4D symplectic topology
Geometry of holomorphic isometric embeddings between bounded symmetric domains and applications
勒让德纽结及其不变量
G2-manifolds and instantons
Wrapped Floer theory for Lagrangian surfaces
Character Varieties of Tangles and Singular Instanton Homology
Decompositions of symplectic manifolds
Construction of mixed polarization on K\"ahler manifolds with T-symmetry
Recently there's been a number of breakthroughs in dim>3 contact topology around convex hypersurfaces (mostly from a geometric topology point of view). The course will give a friendly way to think about convex hypersurfaces and explain how to compute their SFT invariants. It will cover some homological algebra, explicit examples, and get into analytical details in later lectures. I'll assume familiarity with contact manifolds and their fillings as well as the basics of SFT (holomorphic curves and DGAs).
I will introduce the recent developments connecting the study of Lagrangian fillings of Legendrian links with the theory of cluster algebras. Each step will be motivated with results and examples, building from the ground up. In particular, I will explain the main steps of the construction of cluster algebras on braid varieties using symplectic topology and discuss several related results and applications. Among them, the key ingredients for the proof of surjectivity of the filling-to-cluster map will be presented and there will be a discussion on the Donaldson-Thomas transformation from the viewpoint of Legendrian links.
In the first lecture, we first discuss some fundamental properties of bounded symmetric domains and their Bergman metrics. I will also explain some results of holomorphic isometric embeddings between bounded symmetric domains with respect to the Bergman metrics up to scalar constants.
In the second lecture, I will continue to talk about more results of holomorphic isometric embeddings between bounded symmetric domains. After that, I will explain some results about the asymptotic behavior of local holomorphic curves exiting the boundary of a bounded symmetric domain and their applications based on my joint work with Ngaiming Mok. One of the applications is related to the hyperbolic Ax-Lindemann-Weierstrass conjecture that has been solved by Klingler-Ullmo-Yafaev (2016) for pure Shimura varieties.
勒让德纽结是三维空间中满足一定微分方程的环路。它链接了切触拓扑和低维拓扑，是拓扑中有趣的研究对象。不变量是用来区分纽结的代数工具。我们将介绍如何使用圆盘的组合计数来定义勒让德纽结的不变量，并解释其背后的原理。
In dimension seven, manifolds with special holonomy G2 play an important role in both mathematics and physics. Aside from certain construction of G2-metrics, properties of the moduli space of G2-metrics are barelyknown. In the first half of the lecture series, I will discuss basics in G2-manifolds from scratch, including geometric and topological properties. In the second half, I will switch to the gauge theory side, following the proposal of Donaldson–Thomas, to discuss G2-instantons and the corresponding moduli space. The lectures are meant to be introductory and accessible to anyone with basic knowledge in Riemannian geometry
We will introduce wrapped Floer theory and will show its application on proving rigidity of emmbedded exact Lagrangian fillings. We will also extend to the theory to immersed Lagrangian fillings to obtain lower boundes of double points of immersed exact Lagrangian fillings.
Hedden, Herald, and Kirk initiated a program to study knots through the perturbed traceless SU(2) character varieties of associated tangles. One of the primary motivations for this strategy is its connection to Kronheimer and Mrowka's singular instanton homology. I will present my results about how these character varieties interact with the tangle sum operation defined by Conway. I will explain how these results lead to a better understanding of the relationship between these character varieties and singular instanton homology.
We will talk about different ways to split up a symplectic manifold into simple pieces that fit together in controlled ways. For each of these ways, we will see how to encode the symplectic topology of the manifold diagrammatically.
Geometric quantization on symplectic manifolds plays an important role in representation theory and mathematical physics, deeply relating to symplectic geometry and differential geometry. In this talk, we will focus on the construction of mixed polarization on the K\"ahler manifold using T-symmetry. This is a joint work with Naichung Conan Leung.
Video 1, Video 2, Video 3, Video 4, Note 1, Note 2, Note 3, Note 4.