Braid groups

Abstract: The theory of braid groups occupies a beautiful place in modern mathematics. The theory began with Artin in an attempt to understand knots; nowadays braids play an important role in topology, representation theory and symplectic geometry. This course will be an introduction to the theory with an emphasis on current research directions. We will begin with the basic theory of Coxeter groups and classical topological approaches to the type A braid group. We will then cover the Burau and Krammer representations, and discuss central problems including linearity, the word problem, the centre and the description of classifying spaces. We will then discuss the recent theory of categorical actions of braid groups, discussing in detail the work of Khovanov-Seidel and Brav-Thomas.

Lecturers: T. Gobet and G. Williamson.

! New room and schedule ! Thursday, 3 - 5 pm, Carslaw 830.

Extended abstract and bibliography

• Lecture 1 (02.08.18): Coxeter groups I: Coxeter groups (Definition, exchange lemma, word problem). Solution 1 Solution 2.
  
• Lecture 2 (09.08.18): Coxeter groups II: Finite reflection groups, root systems, Tits representation of an arbitrary Coxeter group. Sol. to ex. 6
  
• Lecture 3 (16.08.18): Braid groups I: Classical braid groups, topologically (Fundamental groups of configuration spaces and mapping class group of the punctured disk, action on the free group, Artin's solution to the word problem).
  
• Lecture 4 (23.08.18): Braid groups II: Braid groups, algebraically (Artin-Tits groups attached to arbitrary Coxeter groups, Conjectures, Garside structures).
  
• Lecture 5 (30.08.18): Linear representations I: Homological representations of mapping class groups. Solution to an exercise.
  
• Lecture 6 (06.09.18): Linear representations II: The Burau representation of the classical braid group, unfaithfulness ; Hecke algebras.
  
• Lecture 7 (13.09.18): Linear representations III: The Lawrence-Krammer-Bigelow representation of the classical braid group, faithfulness.
• Lecture 8 (20.09.18): Categorical actions I: Actions of groups on categories.
  
• Lecture 9 (04.10.18): Categorical actions II: Homotopy categories, zig-zag algebras.
  
• Lecture 10 (11.10.18): Categorical actions III: Khovanov and Seidel's faithful categorical action.
  
• Lecture 11 (18.10.18): Categorical actions IV: Weight structures and t-structures on triangulated categories. The canonical t-structure induced by the grading on the bounded homotopy category of graded projective modules over a type ADE zigzag algebra.
  
• Lecture 12 (25.10.18): Categorical actions V: Brav and Thomas' proof of faithfulness of the categorical action of a braid group of type ADE.
  
• Lecture 13 (01.11.18): Affine Weyl groups; loop presentations of affine braid groups.
• Lecture 14 (08.11.18): Talk of Bob Howlett on the automatic structure of Coxeter groups.
• Lecture 15 (15.10.18): Talk of Tony Licata.