|Date: Friday, November 01, 2013
Location: 3866 East Hall (4:10 PM to 5:00 PM)
Title: Regular cell complexes in total positivity
Abstract: Sergey Fomin and Michael Shapiro studied stratified spaces arising naturally from representation theory and having the Bruhat orders as their closure posets, and conjectured that these were regular CW complexes homeomorphic to balls. Specifically, they considered the stratification into Bruhat cells of the link of the identity in the totally nonnegative real part of the space of upper triangular matrices with 1's on the diagonal (and of more general unipotent radicals). I will discuss some ingredients that went into the proof of their conjecture and which may also be useful for studying other maps on polytopes (such as Postnikov's measurement map). I will also describe a second, more traditionally topological approach to the Fomin-Shapiro Conjecture based on the CE approximation theorem and local contractibility of the space of homeomorphisms from a sphere to itself. Along the way, I will briefly explain how subword complexes encapsulate the combinatorics of the fibers of these maps from polytopes and how reduced and nonreduced words govern flows to a base point within the fibers. As time permits, I will discuss ways in which this picture could perhaps be generalized to other totally nonnegative varieties.
Speaker: Patricia Hersh
Event Organizer: Sergey Fomin