UBC Mathematics Colloquium
North Carolina State University, USA.

##

## Combinatorics and topology of stratified spaces

## Fri., March. 19, 2010, 3:00pm, MATX 1100

Abstract:

Anders Bj\"orner characterized which finite, graded
partially ordered
sets (posets) are closure posets of finite, regular CW
complexes, and
he also observed that a finite, regular CW complex is homeomorphic
to the order complex of its closure poset. One might therefore
hope to
use combinatorics to determine topological structure of stratified
spaces by studying their closure posets; however, it is possible
for
two different CW complexes with very different topological
structure
to have the same closure poset if one of them is not regular. I
will
talk about a new criterion for determining whether a finite CW
complex
is regular (with respect to a choice of characteristic functions);
this
will involve a mixture of combinatorics and topology. Along the
way, I
will review the notions from topology and combinatorics we
will need.
Finally I will discuss an application: the proof of a conjecture
of
Fomin and Shapiro, a special case of which says that the Schubert
cell
decomposition of the totally nonnegative part of the space of
upper
triangular matrices with 1's on the diagonal is a regular CW
complex
homeomorphic to a ball.