Discrete CATS Seminar - Fall 2015
Mondays 2:00-3:00pm in POT 745

November 23

Liam Solus (Dissertation Defense), University of Kentucky

Polyhedral Problems in Combinatorial Convex Geometry
Polyhedra play a special role in combinatorial convex geometry in the sense that they are both convex sets and combinatorial objects. As such, a polyhedron can act as either the convex set of interest or the combinatorial object describing properties of another convex set. We will examine two instances of polyhedra in combinatorial convex geometry, one exhibiting each of these two roles. The first instance arises in the context of Ehrhart theory, and the polyhedra are the central objects of study. We will examine the Ehrhart h*-polynomials of a family of lattice polytopes called the r-stable (n,k)-hypersimplices, providing some combinatorial interpretations of their coefficients as well as some results on unimodality of these polynomials. The second instance arises in algebraic statistics, and the polyhedra act as a conduit through which we study a nonpolyhedral problem. For a graph G, we study the extremal ranks of the closure of the cone of concentration matrices of G via the facet-normals of the cut polytope of G. Along the way, we will discover that real-rooted polynomials are lurking in the background of all of these problems.

November 16

Thomas Barron, University of Kentucky

n-semigroups and covering semigroups
A semigroup is a set with an associative binary operation. In this talk we give an introduction to semigroups and n-semigroups (the higher-arity analog of semigroups), leading to the notion of covering semigroups, which are binary semigroups containing the given n-semigroup in a certain sense. We end with some notes on enumerating finite semigroups.

November 09

Jose Samper, University of Washington

Relaxations of the matroid axioms
Motivated by a question of Duval and Reiner about eigenvalues of combinatorial Laplacians, we develop various generalisations of (ordered) matroid theory to wider classes of simplicial complexes. In addition to all independence complexes of matroids, each such class contains all pure shifted simplicial complexes, and it retains a little piece of matroidal structure. To achieve this, we relax many cryptomorphic definitions of a matroid. In contrast to the matroid setting, these relaxations are independent of each other, i.e., they produce different extensions. Imposing various combinations of these new axioms allows us to provide analogues of many classical matroid structures and properties. Examples of such properties include the Tutte polynomial, lexicographic shellability of the complex, the existence of a meaningful nbc-complex and its shellability, the Billera-Jia-Reiner quasisymmetric function, and many others. We then discuss the h-vectors of complexes that satisfy our relaxed version of the exchange axiom, extend Stanley's pure O-sequence conjecture about the h-vector of a matroid, solve this conjecture for the special case of shifted complexes, and speculate a bit about the general case. Based on joint works with Jeremy Martin, Ernest Chong and Steven Klee.

November 02

Joshua Hallam, Wake Forest University

Switching the Whitney Numbers of a Poset
For a ranked poset, P, let w_k(P) be the kth Whitney number of the first kind and let W_k(P) be the kth Whitney number of the second kind. In this talk we will discuss if given a ranked poset, P, there is another ranked poset, Q, such that |w_k(P)|=W_k(Q) and |w_k(Q)|=W_k(P) for all k. In particular, we will discuss how to use certain edge labelings to construct such posets. This is joint work with Rafael Gonzalez d'Leon.

October 26

McCabe Olsen (Qualifying Exam), University of Kentucky

A Unified Approach to the Euler-Mahonian Identity
Originally due to Carlitz in 1975, the Euler-Mahonian identity is a q-analogue of the well known Euler polynomial identity. This identity has been proven using many diverse methods. In this talk, we will discuss two proofs of the identity. The first proof due to Beck and Braun uses polyhedral geometry. The second proof due to Adin, Brenti, and Roichman uses a descent basis for the coinvariant algebra of S_n and an appropriate Hilbert series calculation. If time permits, we will briefly discuss some representation theoretic results of Adin, Brenti, and Roichman, as well as discuss the prospect of research in relating the ideas behind these proofs.

October 19

Brian Davis, University of Kentucky

An Introduction to Syzygies of Lattice Ideals
The goal of the talk is to develop a passing acquaintance with vocabulary like syzygy, minimal free resolution, toric ideal, and Hilbert series. We give a combinatorial method of computing the generating function of a semi-group, then we tell a parallel story in the more general world of commutative algebra. Although the connection won't be our focus, the generating functions we calculate are central objects in Ehrhart theory.

October 16 (note this is a FRIDAY). The talk is at 8:30am in POT 745.

Marie Meyer (Masters Exam), University of Kentucky

Reflexive Polytopes and the Reflexive Dimension
Reflexive polytopes were first introduced in the context of theoretical physics and have since played a role in Mirror Symmetry, construction of Calabi-Yau varieties and Gorenstein polytopes. Applications aside, reflexive polytopes are interesting combinatorial objects. In this talk we define what it means for a polytope, P, to be reflexive and characterize P according to its Ehrhart series. Then we look at the work done by Haase and Melnikov in defining the reflexive dimension of a polytope and producing lower and upper bounds.

October 12

Dustin Hedmark, University of Kentucky

Homology of Filters in the Partition Lattice
Starting with the computation of the Mobius function of the even partition lattice by Sylvester in 1976, there has been much interest in understanding the topology and representation theory of filters in the partition lattice. In this talk I will speak on current work with Dr. Ehrenborg where we are able to compute the homology groups, as well as the S_{n-1} action on these homology groups, for arbitrary filters in the partition lattice Pi_n using the Mayer Vietoris Sequence. We will spend most of our time looking at examples of computations of homologies in the partition lattice, notably a derivation of Wach's well known results on the d-divisible partition lattice.

October 05

No talk this week; TLC October 3-4.

September 28

Sarah Nelson, University of Kentucky

Convex polytopes, h-vectors, and Gale diagrams
For any convex d-polytope P, we may describe P as the convex hull of n points in R^d. Associated with P is its flag-f-vector, which enumerates the numbers of chains of faces of the various possible types. The toric h- and g-vectors are certain linear transformations of this vector. Algebraists and topologists care about these statistics, because they measure the dimension of the intersection cohomology of certain toric varieties related to polytopes. For a simplicial polytope P, Lee defined the winding number w_k in a Gale diagram corresponding to P. He showed that w_k in the Gale diagram equals g_k of the corresponding polytope. After discussing the simplicial case and its significance, we will extend these results by explaining how to determine g_k of the polytope in certain cases by only considering the corresponding Gale diagram. In particular, we determine g_k for any two-dimensional Gale diagrams.

September 21

Martha Yip, University of Kentucky

Generalized Kostka polynomials
Kostka numbers appear in several areas of mathematics, including combinatorics, and representation theory. In the first half of this talk, we will review the combinatorics of Kostka numbers, and introduce one- and two-parameter generalizations of these numbers. In the second half, we give an overview of the connection between Macdonald polynomials and the double affine Hecke algebra, and discuss a different two-parameter generalization of Kostka numbers.

September 14

Rafael Gonzalez d'Leon, University of Kentucky

A family of symmetric functions associated with Stirling permutations
We present exponential generating function analogues to two classical identities involving the ordinary generating function of the complete homogeneous symmetric function. After a suitable specialization the new identities reduce to identities involving the first and second order Eulerian polynomials. These results led us to consider a family of symmetric functions associated with the Stirling permutations introduced by Gessel and Stanley.