Speaker:  Marie Meyer, U Kentucky

Title: Polytopes Associated to Graphs

Abstract:

There are many advantageous ways to associate a polytope to a graph. In this talk, we will discuss a couple of such constructions while highlighting some notable results. First we will look at edge polytopes introduced by Ohsugi and Hibi as well as applications through constructive examples from a paper by Lason and Michalek. Then we will look at Laplacian simplices associated to graphs and digraphs in the time remaining.

Speaker:  Joseph Cummings, U Kentucky

Title:  Cohen-Macaulay Stanley-Reisner Rings

Abstract:

It turns out that in order to solve many purely combinatorial problems relating to simplicial complexes, one needs to study algebraic properties of its Stanley-Reisner ring. For example, we will show the Cohen-Macaulay condition gives us sharp bounds on the complex’s $h$-vector. We will also discuss Reisner’s criterion which gives an equivalent combinatorial criterion for Cohen-Macaulyness, and time permitting, Stanley’s upper bound theorem for simplicial spheres.

Speaker:  Andy Wilson, U Penn

Title:  The combinatorics of symmetric quotient rings

Abstract: 

The coinvariant ring of the symmetric group is the quotient of the polynomial ring by the ideal generated by all symmetric polynomials without a constant term. Many properties of this ring are closely connected to the combinatorics of the symmetric group. What if, instead, we mod out by an ideal generated by some other set of polynomials? If the ideal is symmetric, can we use combinatorics to understand the properties of the resulting quotient ring? A variety of authors (Rhoades, Haglund, Shimozono, Huang, Scrimshaw, the speaker, and others) have discovered many well-behaved quotient rings this way. Furthermore, they have shown that the rings are connected to classical combinatorial objects like ordered set partitions and words. We will provide an overview of the work in this area and pose a conjecture that, if proven, would unify much of the existing work on this problem

Speaker:  Megan Bernstein, Georgia Tech

Title:  Progress in showing cutoff for random walks on the symmetric group

Abstract: 

Cutoff is a remarkable property of many Markov chains in which they rapidly transition from an unmixed to a mixed distribution. Most random walks on the symmetric group, also known as card shuffles, are believed to mix with cutoff, but we are far from being able to proof this. We will survey existing cutoff results and techniques for random walks on the symmetric group, and present three recent results: cutoff for a biased transposition walk, cutoff for the random-to-random card shuffle (answering a 2001 conjecture of Diaconis), and pre-cutoff for the involution walk, generated by permutations with a binomially distributed number of two-cycles. The results use either probabilistic techniques such as strong stationary times or diagonalization through algebraic combinatorics and representation theory of the symmetric group. 

Includes joint work with Nayantara Bhatnagar, Evita Nestoridi, and Igor Pak.

For further information, see Discrete CATS Seminar

Speaker:  McCabe Olsen, University of Kentucky

Title:  Level algebras and lecture hall polytopes

Abstract:

Given a family of lattice polytopes, a common question in Ehrhart theory is classifying the which polytopes in the family are Gorenstein. A less common question is classifying which polytopes in the family admit level semigroup algebras, a generalization of the Gorenstein property. In this talk, we consider these questions for lecture hall polytopes. We provide a characterization of the Gorenstein property for a large subfamily of lecture hall polytopes. Additionally, we also provide a complete characterization for the level property.

This is joint work with Florian Kohl

For further information, see Discrete CATS Seminar

Speaker:  Zhexiu Tu, Centre College

Title:  Topological Representations of Matroids and the cd-index

Abstract:

There are several different topological representations of non-orientable matroids. In this talk, inspired by Swartz's work, I will show an explicit fully partitioned homotopy sphere d-arrangement S that is a CW-complex whose intersection lattice is the geometric lattice of the corresponding matroid for matroids of rank < 5. Moreover S has a d-sphere in it that is a regular CW-complex. We will also look at enumerative properties, including how the flag f-vector formula of Billera, Ehrenborg and Readdy for oriented matroids applies to arbitrary matroids.

For further information, see Discrete CATS Seminar

Speaker:  Rafael González D'león, Universidad Sergio Arboleda and York University.

Title:  The Whitney dual of a graded poset

Abstract:

Two posets are Whitney duals to each other if the (absolute value of their) Whitney numbers of the first and second kind are switched between the two posets.   We introduce new types of edge and chain-edge labelings of a graded poset which we call Whitney labelings. We prove that every graded poset with a Whitney labeling has a Whitney dual and we show how to explicitly construct a Whitney dual using a technique that involves quotient posets. As an application of our main theorem, we show that geometric lattices, the lattice of noncrossing partitions, the poset of weighted partitions studied by González D'León-Wachs and the R*S-labelable posets studied by Simion-Stanley all have Whitney duals. We also show that a graded poset P with a Whitney labeling admits a local action of the 0-Hecke algebra on the set of maximal chains of P. The characteristic of the associated representation is Ehrenborg's flag quasisymmetric function of P. This is joint work with Josh Hallam (Wake Forest Universtity).

For further information, see Discrete CATS Seminar

Speaker:  Radmila Sazdanovic, NC State University

Title:  Chromatic homology theories

Abstract:

This talk is an entree to categorification through knot theory and graph theory. The focal point is the chromatic polynomial and is categorifications: chromatic graph homology over algebra defined by L. Helme-Guizon and Y. Rong, and the homology of a graph configuration space introduced by M. Eastwood, S. Huggett. Time permitting, we will discuss relations between these homology theories in the form of spectral sequences, as well as a new invariant of simplicial complexes inspired by the Eastwood and Huggett approach.

For further information, see Discrete CATS Seminar

Brian Davis, University of Kentucky.

Regular triangulations and Gröbner bases

In this talk we will give a friendly introduction to regular triangulations, which is a tool for breaking down an integer polytope into simpler pieces: high dimensional triangles!  In the second part of the talk we will present a context in which triangulations make a normally difficult computation much easier.   The main theorem, presented without proof, is really charming!  We assume no knowledge of commutative algebra beyond the prelim sequence.

For more details, see Discrete CATS website