Speaker:  Alex Chandler, NC State

Title:  Thin posets and homology theories

Abstract:

Inspired by Bar-Natan's description of Khovanov homology, we discuss

thin posets and their capacity to support homology and cohomology

theories which categorify rank-statistic generating

functions. Additionally, we present two main applications. The first,

a categorification of certain generalized Vandermonde determinants

gotten from the Bruhat order on the symmetric group by applying a

special TQFT to smoothings of torus link diagrams. The second is a

broken circuit model for chromatic homology, categorifying Whitney's

broken circuit theorem for the chromatic polynomial of graphs.

www.math.uky.edu/~readdy/Seminar

Speaker:  Matthias Köppe, UC Davis

Title:  Normalized antics: Polyhedral computation in an irrational age

Classic polyhedra such as the dodecahedron have irrational coordinates. How do we compute with them if we need exact answers? In this hands-on lecture using the SageMath system, intended to be accessible to advanced undergraduate students, I explain how to compute with polyhedral representations and with real embedded algebraic number fields. I then report on an ongoing project with W. Bruns, V. Delecroix, and S. Gutsche to obtain an efficient implementation in Normaliz, integrated with SageMath, involving the existing software libraries ANTIC, arb, and FLINT.

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