David Bessis

Geometry and combinatorics of braid groups

This course is an introduction to the theory of braid groups associated to finite reflections groups. Reflection groups are elementary geometric objects which play central roles in many classifications: regular polytopes, Lie groups, algebraic groups and finite simple groups... To each finite reflection group, one may associate a braid group; the intuitive braid group on $n$ strings is a particular example, associated with the symmetric group.

The combinatorial theory of Coxeter systems is a useful tool to study reflection groups. The aim of this course is to explain how related combinatorial constructions can be used to study subtle aspects of braid groups. We plan to cover the following material, or more likely some fraction of it:

basic properties of Coxeter systems;
Brieskorn's theorem about presentations of braid groups;
Garside monoids, word and conjugacy problem;
Deligne's theorem on the topology of complements of complexified hyperplane arrangements;
cohomology of braid groups;
Salvetti's complex and cohomology of reflection groups;
Deligne's theorem about coherence for actions of braid groups on categories;
Bestvina's approach to ``hyperbolic aspects'' of braid groups.

As it will appear, these seemingly different subjects are closely related and involve similar combinatorics. The course will be essentially self-contained, the only prerequisites being basic group theory and basic algebraic topology.