Introduction to braid groups
Braid groups are special types of groups which appear in many areas of pure and applied mathematics. We will mainly focus on the family of "standard" braid groups (Artin braid groups), which are closely connected to the family of symmetric groups. They can be presented by generators & relations but also have more intuitive topological reailizations. There are natural connections for instance with Coxeter groups, reflection groups and Iwahori-Hecke algebras, with semisimple Lie algebras, with knot theory and with mathematical physics. We will explore many of these connections and discuss some important representations. The course is aimed at postgraduate students in mathematics and related disciplines.
讲师
日期
2023年10月18日 至 2024年01月11日
位置
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| 周三 | 13:30 - 15:05 | A3-1a-205 | ZOOM 03 | 242 742 6089 | BIMSA |
| 周四 | 15:20 - 16:55 | A3-1a-205 | ZOOM 02 | 518 868 7656 | BIMSA |
修课要求
Essential prerequisites: undergraduate group theory, linear algebra and introductory topology. Desirable prerequisites: introductory representation theory.
课程大纲
In the course we will discuss the following topics, mainly from the textbook but with some additional material:
-algebraic definition of braid group (standard and Birman-Ko-Lee presentation) and basic properties
-geometrical and topological braids, braid diagrams and Reidemeister moves
-configuration spaces and their fundamental groups
-homotopy sequences of locally trivial fibrations
-pure braid group, normal/combed forms and further algebraic properties
-links and knots, Alexander's theorem and Markov's theorem
-Burau and reduced Burau representation of the braid group and Alexander-Conway polynomials
-faithful braid group action on free groups
-braid group as mapping class group
-Iwahori-Hecke algebra, Temperley-Lieb algebra and Jones-Conway/HOMFLY-PT polynomial
-generalizations: braid monoids and Artin-Tits braid groups of other types
-algebraic definition of braid group (standard and Birman-Ko-Lee presentation) and basic properties
-geometrical and topological braids, braid diagrams and Reidemeister moves
-configuration spaces and their fundamental groups
-homotopy sequences of locally trivial fibrations
-pure braid group, normal/combed forms and further algebraic properties
-links and knots, Alexander's theorem and Markov's theorem
-Burau and reduced Burau representation of the braid group and Alexander-Conway polynomials
-faithful braid group action on free groups
-braid group as mapping class group
-Iwahori-Hecke algebra, Temperley-Lieb algebra and Jones-Conway/HOMFLY-PT polynomial
-generalizations: braid monoids and Artin-Tits braid groups of other types
参考资料
Kassel & Turaev, Braid groups, Graduate Texts in Mathematics 247, Springer.
Birman, J. S., & Brendle, T. E. (2005). Braids: a survey. In Handbook of knot theory (pp. 19-103). Elsevier Science. Available at arxiv/math/0409205.
Birman, J. S., & Brendle, T. E. (2005). Braids: a survey. In Handbook of knot theory (pp. 19-103). Elsevier Science. Available at arxiv/math/0409205.
听众
Graduate
, 博士后
视频公开
公开
笔记公开
公开
语言
英文
讲师介绍
Bart Vlaar于2022年9月以副研究员身份全职入职BIMSA。他的研究兴趣包括代数和表示论，以及它们在数学物理上的应用。他在苏格兰格拉斯哥大学获得博士学位，之后先后在阿姆斯特丹大学、诺丁汉大学、约克大学和苏格兰赫瑞瓦特大学任职位，并访问位于波恩的马斯克博朗克数学研究所。