三维流形及纽结理论高级讲习班

There will be four short courses in this summer school. (1) Topology on 3-manifold : Our course will focus on how to use topological methods to study such problems, including taut foliations, measured laminations, complex of curves and etc. Besides the fundamentals, we will include the following: (a) The existence and property of Heegaard splittings of a three-manifold with large distance. (b) A proof of Waldhausen conjecture on Heegaard splittings of 3-manifolds. Talked by Tao Li and Ruifeng Qiu. (2) Geometry on 3-manifold: After Thurston's revolutionary work, the study of hyperbolic three-manifolds has been very active and of great importance in low-dimensional topology. In this short course, we will talk about the fundamental theory about hyperbolic three-manifolds, including Mostow rigidity, Teichmuller space, and the eight geometries of Thurston. Talked by Hongbin Sun. (3) The virtual Haken conjecture: In 2012, combining works of Bergeron, Kahn, Markovic, Sageev, Wise and himself, Agol proved the virtual Haken conjecture and the virtually fibering conjecture. In this short course, we will introduce their proof. Talked by Yi Liu. (4) Knot theory: In this short course, we will cover the fundamentals in knot theory, including Seifert surfaces, Alexander polynomial, Dehn surgery, Jones polynomial, and etc. Talked by Jiajun Wang and Hao Zheng. In addition, there will be twenty research talks.

本次讲习班包括四门短课。一. 三维流形上的拓扑：我们将讲解如何用叶状结构、曲线复形等方法来研究三维流形上的Heegaard分解。 主讲人李韬、邱瑞锋。二. 三维流形上的几何：自从瑟斯顿的工作之后，双曲流形的研究一直是三维拓扑中的非常重要与活跃的分支。我们将讲解双曲流形的基础知识，如Mostow刚性定理、Teichmuller空间等. 主讲人孙洪宾。 三. 关于Virtual Haken猜想：最近， Agol在Kahn, Markovic, Sageev, Wise等人的工作的基础上， 给出了Virtual Haken猜想和Virtual fibering猜想的证明。我们讲介绍这方面的工作. 主讲人刘毅。四. 纽结理论：我们将讲解纽结理论中的一些基础知识，包括Seifert曲面，Alexander多项式，Jones多项式等等. 主讲人王家军、郑浩。此外，将有二十个前沿报告。

本项目资助由华东师范大学和辽宁师范大学联合举办的“三维流形及纽结理论高级讲习班”，举办地点是辽宁省大连市， 时间是7月14日至8月31日。共有5门短课和20个学术报告，其中短课涵盖三维流形的双曲几何、Virtual Haken猜想、组合理论及纽结不变量理论等。共有20余名专家及40名研究生参加了本次活动。该讲习班已已按计划圆满结束。