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Resources and Background Readings

This page contains some links to background reading. It was originally set up as useful reading for the participants in preparation for the conference, but now has a second life as useful reading for researchers looking to learn more about this active, coalescing area of mathematical investigation.

Exercise Sets

At the conference there were two sets of introductory exercises that were passed out.

Warm-up in Type A: noncrossing, nonnesting partitions and associahedra by Vic Reiner (10 pages)
How noncrossing partitions occur in free probability theory by Andu Nica (17 pages)

Non-crossing Partitions

Non-crossing Partitions in Surprising Locations (14 pages). Introductory article on non-crossing partitions, written for a fairly general audience by Jon McCammond. It includes brief references to the ways in which noncrossing paritions arise in the study of braid groups, associahedra, and free probability.
Generalized Catalan numbers, Weyl groups and arrangements of hyperplanes (9 pages). A short article by Christos Athanasiadis establishing several connections involving non-crossing partitions and other combinatorially defined objects.
Lattices in finite real reflection groups (29 pages) The uniform proof of the lattice property for noncrossing partitions and the explicit connection with generalized associahedra announced by Tom Brady at the conference (joint with Colum Watt).

Cluster Algebras and Associahedra

Root Systems and Generalized Associahedra (69 pages). Lecture notes (written by Sergey Fomin and Nathan Reading) for the lectures delivered by Fomin at the 2004 Park City Mathematics Institute workshop on Geometric Combinatorics. Aimed at graduate students in geometric combinatorics.
Enumerative Properties of Generalized Associahedra (15 pages). A recent preprint by Frédéric Chapoton containing numerous conjectures and results.

Free Probability

Free Probability Theory and Non-crossing Partitions (38 pages). Survey paper by Roland Speicher that appeared in the Séminaire Lotharingien de Combinatoire series.

Garside Structures and Braid Groups

An Introduction to Garside Structures (28 pages). A survey article by Jon McCammond, written primarily for this conference.
Groupes de Garside (46 pages - in french). The original article by Patrick Dehornoy defining Garside groups.
The Dual Braid Monoid (42 pages). An early article by David Bessis proving and highlighting the key properties of the dual Garside structures for the Artin groups of finite-type.

ad-Nilpotent ideals of Borel subalgebras

ad-Nilpotent ideals of a Borel subalgebra II (9 pages). An article by Paola Cellini and Paolo Papi where they prove an explicit bijection between ad-Nilpotent ideals and the orbits of Q/(h+1)Q, a set known to have close connections with non-crossing partitions.

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Last Modified on 16/Sep/24