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.
Lecturer
Date
18th October, 2023 ~ 11th January, 2024
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Wednesday | 13:30 - 15:05 | A3-1a-205 | ZOOM 03 | 242 742 6089 | BIMSA |
| Thursday | 15:20 - 16:55 | A3-1a-205 | ZOOM 02 | 518 868 7656 | BIMSA |
Prerequisite
Essential prerequisites: undergraduate group theory, linear algebra and introductory topology. Desirable prerequisites: introductory representation theory.
Syllabus
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
Reference
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.
Audience
Graduate
, Postdoc
Video Public
Yes
Notes Public
Yes
Language
English
Lecturer Intro
Dr. Bart Vlaar has joined BIMSA in September 2022 as an Associate Professor. His research interests are in algebra and representation theory and applications in mathematical physics. He obtained a PhD in Mathematics from the University of Glasgow. Previously, he has held positions in Amsterdam, Nottingham, York and Heriot-Watt University. Before coming to BIMSA he visited the Max Planck Institute of Mathematics in Bonn.