Matthew Dyer (University of Notre Dame) Friday 28 April, 12-1pm, Place: Carslaw 375 Title: Bruhat order, weak order and closure operators on root systems Abstract: Reflection orders of Coxeter groups are certain total orders of the reflections (or positive roots). They are analogues for infinite Coxeter groups of reduced expressions of longest elements of finite Coxeter groups. Their initial sections parameterise (actually or conjecturally, depending on context) natural twists of certain algebraic, geometric, combinatorial and representation theoretic objects attached to Coxeter groups, such as Bruhat order. Many long standing conjectures concerning initial sections may be viewed as natural extensions, to a completion by "infinitely long elements", of trivial or well known properties of weak order of Coxeter groups. This talk will survey recent results, some due to Francois Viard and Weijia Wang, arising from attempts to refine, test and prove (at least in tractable special cases) these conjectures.