Artin--Tits groups are certain groups defined either algebraically as ``lifts'' (throwing away s^2 = 1) of Coxeter groups, or topologically as fundamental groups of complexified hyperplane complements associated Coxeter systems. Unlike Coxeter groups, however, many their group theoretical problems remain open; one prominent conjecture is the K(pi,1) conjecture, which states that the complexified hyperplane complement is indeed the classifying space of the Artin--Tits group.
In a (rather distant) world, Artin--Tits groups show up naturally as symmetries of triangulated categories. These categories each have an associated complex manifold known as the space of Bridgeland stability conditions, of which the corresponding Artin—Tits group also acts. These spaces are (known for simply-laced cases, conjecturally for all) coverings of hyperplane complements, and for the finite ADE cases (i.e. spherical types), it was shown -- independently from Deligne's result -- that the associated spaces of stability conditions are indeed contractible, reproving the K(pi,1) conjecture for these cases (universality comes for free!).
The first part of this talk is aimed at discussing the general picture above. Starting with the Artin--Tits groups of type A (i.e. classical braid groups), I will showcase certain analogy between Teichmuller theory for surfaces (disk with punctures in the type A case) and the theory of Bridgeland stability conditions for triangulated categories.
The second part of this talk is aimed at posing problems arising from this perspective. While these technically involve the study of certain submanifolds of the spaces of stability conditions (obtained as "fixed points"), I will mainly present problems that can be understood locally, one of which boils down to the following: when are "nice" subspaces of hyperplane complements associated to Coxeter systems again Coxeter?
No prior knowledge of (triangulated) categories will be needed for both talks.
