Speaker:  Ben Braun, U Kentucky

Speaker:  Margaret Readdy

Title:   Combinatorial identities related to the calculation of the cohomology of Siegel modular varieties

Abstract:

In the computation of the intersection cohomology of Shimura varieties, or of the L² cohomology of equal rank locally symmetric spaces, combinatorial identities involving averaged discrete series characters of real reductive groups play a large technical role. These identities can become very complicated and are not always well-understood. We propose a geometric approach to these identities in the case of Siegel modular varieties using the combinatorial properties of the Coxeter complex of the symmetric group

Joint work with Richard Ehrenborg and Sophie Morel.

Discrete CATS Seminar

Speaker:  Richard Ehrenborg

Title:         Uniform flag triangulations of the Legendre polytope

                (or how I spent my summer holiday)

Abstract:

The Legendre polytope, also known as the full root polytope of type A,

is the convex hull of all pairwise differences of the basis vectors.

We describe all flag triangulations of this polytope that are uniform,

that is, the edges are described in terms of the relative order of the

indices of the four basis vectors involved.  We obtain three classes

of triangulations: the lex class, the revlex class and the Simion

class. We also do a refined enumeration of faces of these

triangulations that keeps track of the number of forward and backward

arrows, and surprisingly the enumeration result only depends on which

class the triangulation belongs to.



Joint work with Gabor Hetyei and Margaret Readdy.

Masters Examination

Kyle Franz

Masters Exam

Jessica Doering

Alex Happ

PhD Dissertation Defense

Title:  A combinatorial miscellany: antipodes, parking cars, and descent set powers

PhD Advisor:  Richard Ehrenborg

Speaker:  Martha Yip, University of Kentucky

Title: The volume of the caracol polytope

Speaker:  Aida Maraj, University of Kentucky

Title: Hierarchical models and their toric ideals 

Mathematics Qualifying Exam for Julie Vega.

Title: Chromatic Numbers, Hom Complexes, and Topological Obstructions (or a great way to organize your favorite maps)

Abstract: Imagine a pleasant graph, but in color. Now let’s make it a little more exciting by coloring vertices such that no two adjacent vertices have the same color. The smallest number for which you can do this is called the “chromatic number.” In ’78 Lov`asz used the connectivity of the neighborhood complex to find lower bounds for the chromatic number. Later, he generalized this idea to the Hom Complex, Hom(G, H), which encodes graph homomorphisms between G and H and their relationship. In this talk, following an argument by Babson and Kozlov, we will look at Hom complexes and use Stiefel Whitney classes to find lower bounds on the chromatic number of a graph.