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
rstable (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 facetnormals of the cut polytope of G.
Along the way, we will discover that realrooted polynomials are lurking in
the background of all of these problems.

November 16 Thomas Barron, University of Kentucky
nsemigroups and covering semigroups
A semigroup is a set with an associative binary operation. In this talk we
give an introduction to semigroups and nsemigroups (the higherarity analog
of semigroups), leading to the notion of covering semigroups, which are binary
semigroups containing the given nsemigroup 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
nbccomplex and its shellability, the BilleraJiaReiner quasisymmetric
function, and many others. We then discuss the hvectors of complexes that
satisfy our relaxed version of the exchange axiom, extend Stanley's pure
Osequence conjecture about the hvector 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 EulerMahonian Identity
Originally due to Carlitz in 1975, the EulerMahonian identity is a qanalogue
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
semigroup, 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 CalabiYau
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_{n1} 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 ddivisible partition lattice.

October 05 No talk this week; TLC October 34.

September 28 Sarah Nelson, University of Kentucky
Convex polytopes, hvectors, and Gale diagrams
For any convex dpolytope P, we may describe P as the convex hull of n points
in R^d.
Associated with P is its flagfvector, which enumerates the numbers of chains
of faces of the various possible types.
The toric h and gvectors 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 twodimensional 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 twoparameter 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 twoparameter 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.
