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

Title: Graph constructions and chromatic numbers

Abstract: We will survey various graph constructions related to proper k-colorability. Possible topics for discussion include constructions of k-chromatic graphs due to Hajos, Ore, and Urquhart, constructions of graphs with high girth and high chromatic number due to Alon, Kostochka, Reiniger, West, and Zhu, and constructions of k-critical graphs with minimal possible number of edges due to Kostochka and Yancey.

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

Abstract: A number of flow polytopes have volumes that are products of nice combinatorial quantities, but perhaps surprisingly, there are no known combinatorial proofs of these formulas.  In this talk, I will give a combinatorial interpretation of the Lidskii volume formula of Baldoni and Vergne, and use this to prove that the volume of the caracol polytope is the product of a Catalan number and the number of parking functions of length n.  This is joint work with Benedetti, Gonzalez D'Leon, Hanusa, Harris, Khare and Morales.

Speaker:  Aida Maraj, University of Kentucky

Title: Hierarchical models and their toric ideals 

Abstract: 

This talk will be about Hierarchical Models in Algebraic Statistics. I will present an example where these models apply and introduce their corresponding toric ideals. A formula for calculating Krull dimension will be given. To describe generating sets of these ideals one can use a symmetric group action. Using this tool, we will describe generating sets for some classes of these ideals as decomposable and non-reducible Models.  This is joint work with Uwe Nagel.