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# 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.