On Groups G_{n}^{k} and \Gamma_{n}^{k}: a study of Dynamics, Manifolds, and Inva

任课教师 Speaker:Vasily Manturov
时间 Time: 每周一15:20-16:55,2019-7.22 ~ 9-30
地点 Venue:清华大学近春园西楼第三会议室

课程描述 Description

In 2015, I defined the notion of groups G_{n}^{k} and formulated the principle: if dynamical systems describe a motion of n particles with a nice codimension 1 property then they have topological invariants valued in groups G_{n}^{k}. I will treat both topological properties (which topological spaces can be studied in this way) and algebraic(word problem, conjugacy problem, relation to braid groups, Coxeter groups and other groups).
In the second part of the course, I will define similar groups \Gamma_{n}^{k} which origninate from triangulations of manifolds and define the manifold of triangulations for each smooth manifold. I will also define the braid group for arbitrary manifold and study its properties.


Lecture 1. General Theory of groups G_{n}^{k}. Braid groups and G_{n}^{3},G_{n}^{4}第一课 G_{n}^{k}群论。辫子群,G_{n}^{3},G_{n}^{4}
Lecture 2. Indices. Examples and Calculations. Secondary indices. Brunnian braids
指数。李子和计算。二次的指数。Brunn 的辫子
Lecture 3. Rewriting algorithm. G_{n}^{k} groups and Coxeter groups.
Word problem in G_{n}^{2}: two solutions. The diamond lemma.
重写算法。G_{n}^{k}群和Coxeter 的裙。
自问题在G_{n}^{2}: 两个解法.
Lecture 4. Higher groups G_{n}^{k} and restricted configuration spaces.
Realisation of G_{k+1}^{k}.
高等G_{n}^{k}群限制的构形空间. G_{k+1}^{k}的实现
Lecture 5. Two generalisations of G_{n}^{3}
G_{n}^{3}群的两个普遍化
Lecture 6. Projective duality. Curves on surfaces. Moduli spaces.
Complex of G_{n}^{k}
对偶性。线在两位的流行。莫空间。 G_{n}^{k} 的复形
Lecture 7. Small cancellation. The word problem solution for G_{k+1}^{k}
小的约分 自问题解对G_{k+1}^{k}
Lecture 8. \Gamma_{n}^{4} and the pentagon relation.
\Gamma_{n}^{4}群和五角星的关系
Lecture 9. The connection between G_{n}^{k} and \Gamma_{n}^{k}
G_{n}^{k}和\Gamma_{n}^{k}
Lecture 10. Braid groups for arbitrary manifolds.
高维的流行的鞭子群
Lecture 11. Unsolved Problems
不解定的问题

预备知识 Prerequisites

Basics of algebra, basics of topology, possibly, some knot theory

参考资料 References