#
Braid Groups, Clusters and Free Probability

January 10 to January 14, 2005
at the

American Institute of Mathematics,
San Jose, California

organized by

Jon McCammond,
Alexandru Nica,
and Victor Reiner

## Original Announcement

This workshop will be devoted to
deciphering the mysterious connections between the following objects:
- Garside monoid structures for Coxeter and braid groups,
and the associated "lattices of non-crossing partitions"
- the cluster algebras of Fomin and Zelevinsky, and
the associated polytopes known as "generalized associahedra"
- ad-nilpotent ideals within Borel subalgebras of
semisimple Lie algebras, or equivalently, subsets of
pairwise incomparable positive roots

In each case, the objects of finite type within these classes obey the
same numerology, but without satisfactory explanation (so far). The
lattices of non-crossing partitions for Weyl groups of types A and B
are also closely connected with Voiculescu's notion of free probability
and the "R-transform".
The main topics for the workshop are

- Garside monoids and non-crossing partitions
- Non-crossing partitions and free probability
- Cluster algebras and generalized associahedra
- Ad-nilpotent ideals in Borel subalgebras

## Material from the workshop

A list of participants.
The workshop schedule.

A report on the workshop activities.

Some
useful
background material.

Problem sets for the talks of
Victor Reiner and Andu Nica.

Papers arising from the workshop: