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.

Mathematics Dissertation Defense for Marie Meyer.

Mathematics Dissertation Defense for McCabe Olsen.

Speaker: Karthik Chandrasekhar

Title:  Facing up to metrics

Abstract:

Here we study arrangements of geodesic lines on spaces topologically equivalent to the real 2-plane, but having different metrics. We study the regions of all possible of f-vectors. Mainly we study the hyperbolic plane (the upper half plane with a non-Euclidean metric) which reveals a vastly different region of f-vectors.

Nick Early, University of Minnesota.

Canonical bases for permutohedral plates

Abstract:

There is a natural construction according to which the set of all faces of an arrangement of hyperplanes can be made into a vector space, by taking linear combinations of their characteristic functions.  Our space is equipped with a natural basis of characteristic functions of certain polyhedral cones called permutohedral cones passing through the origin, studied as plates by A. Ocneanu, which are labeled by ordered set partitions; these are in duality with faces of the arrangement of reflection hyperplanes xi=xj.  It is interesting to study the affine case as well: recently we conjectured stating that plates situated compatibly with the lattice of integer points in a generalized hypersimplex provide a new combinatorial interpretation of the h*-vector for that generalized hypersimplex.

In this talk, we construct directly a certain canonical basis which is compatible with one or both of two quotients: neglecting characteristic functions of (1) nonpointed cones, and (2) cones of codimension at least 1.  The essential feature here is that subsets of the canonical basis map to bases of the quotients.  As a consequence, we obtain the straightening relations which were originally computed by Ocneanu through the introduction of an auxiliary space of formal linear combinations of layered trees.  Time permitting, we will describe the circumstances which led us to formulate the conjecture for the h*-vector.

For more details, see Discrete CATS website

Speaker:  Carl Lee, University of Kentucky

Title:  The Ingredients of the g-Theorem

Abstract:  I will present some of the historical context of the g-Theorem, which characterizes the numbers of faces of simplicial convex polytopes, and the ingredients of its proof.

For more details, see Discrete CATS website

McCabe Olsen, University of Kentucky.

Title: Ehrhart theory and ordered set partitions

Abstract: Given a lattice polytope P, one of the central questions in Ehrhart theory is to describe the h*-polynomial (or h*-vector) of P, as this encodes and detects certain algebraic and geometric properties of P. Given that the coefficients of the h*-polynomial are nonnegative integers, it is natural (for a combinatorialist) to wishfully think that these coefficients count something; that is, ideally this polynomial encodes some statistic on some combinatorial object. In this talk, we will discuss some conjectures of Nick Early regarding the h*-polynomial of two well-known polytopes, namely dilated unit simplices and hypersimplices, involving a winding number statistic on certain decorated ordered set partitions. We will provide a proof to one of these conjectures and discuss the other.

For more details, see Discrete CATS website

Andrés R. Vindas Meléndez, University of Kentucky

Fixed Subpolytopes of the Permutahedron 

Abstract: 

Motivated by the generalization of Ehrhart theory with group actions, this project makes progress towards obtaining the equivariant Ehrhart theory of the permutahedron. The fixed subpolytopes of the permutahedron are the polytopes that are fixed by acting on the permutahedron by a permutation. We prove some general results about the fixed subpolytopes. In particular, we compute their dimension, show that they are combinatorially equivalent to permutahedra, provide hyperplane and vertex descriptions, and prove that they are zonotopes. Lastly, we obtain a formula for the volume of these fixed subpolytopes, which is a generalization of Richard Stanley's result of the volume for the standard permutahedron. This is joint work with Federico Ardila (San Francisco State) and Anna Schindler (University of Washington).

For more details, see Discrete CATS website

Ben Braun, University of Kentucky.

Ehrhart h* polynomials, unit circle roots, and Ehrhart positivity

Abstract:

We will introduce the basics of Ehrhart theory, then discuss methods for establishing Ehrhart positivity using h* polynomials with roots on the unit circle.  This is based on joint work with Fu Liu.

For more details, see Discrete CATS website

Speaker:  Gabor Hetyei UNC Charlotte

Title:  Partitions of a fixed genus have an algebraic generating function

Abstract:

We show that, for any fixed genus g, the ordinary generating function for the genus g partitions of an n-element set into k blocks is algebraic. The proof involves showing that each such partition may be reduced in a unique way to a primitive partition and that the number of primitive partitions of a given genus is finite. We illustrate our method by finding the generating function for genus 2 partitions, after identifying all genus 2 primitive partitions, using a computer-assisted search. This is joint work with Robert Cori.

For more details, see Discrete CATS website