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.
T. Gobet and G. Williamson.
New room and schedule ! Thursday, 3 - 5 pm,
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.
7 (13.09.18): Linear representations III: The
Lawrence-Krammer-Bigelow representation of the classical braid
- 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.
15 (15.10.18): Talk of Tony Licata.