Braid Groups, Clusters and Free Probability

January 10 to January 14, 2005

at the

American Institute of Mathematics, Palo Alto, California

organized by

Jon McCammond, Alexandru Nica, and Victor Reiner

This workshop, sponsored by AIM and the NSF, 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

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and working sessions.

Invited participants include C. Athanasiadis, D. Bessis, T. Brady, N. Brady, F. Chapoton, R. Charney, B. Collins, R. Corran, J. Crisp, P. Dehornoy, S. Fomin, F. Goodman, J. McCammond, J. Meier, A. Nica, N. Reading, V. Reiner, M. Shapiro, P. Sniady, E. Sommers, H. Thomas, M. Wachs, K. Whittlesey, and A. Zelevinsky.

The deadline to apply for support to participate in this workshop has passed.

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