三维流形及其相关领域专题讲习班

There will be four course in this project,which are: 1. Homotpy aspects of geometric groups, we teach mainly simplicial homotopy, homotopy aspects of some braid group and mapping group, knot group and link group, Lie algebra of braid Vassilier invariant and topological structure. 2. Some combination methods of three-dimension manifold, we teach mainly Heegaard splitting and its applications, this course will introduce Heegaard distance from the definition and properties of curve complex, and slope conjecture. 3. The applications of topology in biology, we introduce prsistent homology forextracting molecular topological fingerprints (MTFs) based on the persistence of molecular topological invariants. MTFs are utilized for protein characterization, identification, and classification. 4. At present kont theory is a wide area of mathematics having deep relations with from front lines of topology, geometry, algebra, combinatorics,theoretical physics and DNA. In the class, we will discuss basic ideas and classical theorem. We will construct various invariants and use them to distinguish knots . We will comlete the courses with modern ideas ang results.

本项目将有四个短课程，它们是：1、几何群的同伦属性，主要讲授单纯同伦论、各种类型的辫群与映射类群的同伦属性、纽结群与链环群的同伦属性、辫群的李代数与Vassiliev不变量和子图的拓扑结构等；2、三维流形的一些组合方法，主要讲授Heegaard分解理论及其应用，该课程将从曲线复形的定义、性质出发，引入Heegaard分解距离，同时讲授斜率猜想等；3、拓扑学在生物学中的应用，根据拓扑分子不变量的持久性，介绍持续同调性，主要为了提取分子拓扑指纹图(MTFs)，将MTFs应用于蛋白质的刻画、鉴定和分类；4、讲授结理论与拓扑学、几何学、代数、组合学、理论物理学和DNA等前沿学科的内在联系和应用，同时将讨论基本思想和经典定理。将构造各种不变量，并利用它们来区分纽结。我们将用现代的数学思想和结果完成这些课程。