Friday, February 3 at 2:30 pm in the conference room

Margaret Bayer, University
of Kansas

Polyhedra from a Combinatorial Viewpoint

"A convex polytope is the convex hull of a finite set of points in
Euclidean space. A polytope of dimension d has faces of dimensions 0
through d1. Ordered by inclusion, they form the face lattice of the
polytope. This talk concerns the study of face lattices of convex
ddimensional polytopes. The combinatorial study of polytopes is
important in the analysis of algorithms for linear programming and
computational geometry.
It also has remarkable connections with commutative algebra and
algebraic geometry. The talk will be a survey of key results, current
themes, and open problems."


Friday, March 16 at 2:00 pm in Fretwell 207

Koya
Shimokawa, Saitama University, Japan

Tangle analysis of sitespecific recombination

"Knot theory is applied to studies of sitespecific recombination of DNA.
In this talk I will discuss our recent results on characterization of
topological mechanism of Xer recombination.
"


Friday, April 13 at 1:30 pm in the conference room

Margaret Readdy, University
of Kentucky

Euler flag enumeration of Whitney stratified spaces

"
The flag vector contains all the face incidence data of a polytope,
and in the poset setting, the chain enumerative data. It is a
classical result due to Bayer and Klapper that for face lattices of
polytopes, and more generally, Eulerian graded posets, the flag vector
can be written as a cdindex, a noncommutative polynomial which
removes the generalized DehnSommerville relations, that is, all the
linear redundancies among the flag vector entries discovered by Bayer
and Billera. This result holds for regular CW complexes.
We relax the regularity condition to show the cdindex exists for
nonregular CW complexes by extending the notion of a graded poset to
that of a quasigraded poset. This is a poset endowed with an
orderpreserving rank function and a weighted zeta function. This
allows us to generalize the classical notion of Eulerian, and obtain a
cdindex in the quasigraded poset arena.
Generally speaking, for an arbitrary quasigraded poset the weighted
zeta function is not unique. However, for a manifold having a Whitney
stratification, selecting the weighted zeta function of an interval
using the Euler characteristic gives the extended notion of
Eulerianess geometric meaning.
This is joint work with Richard Ehrenborg (University of Kentucky)
and Mark Goresky (Institute for Advanced Study).
"
