Kevin is a PhD student currently studying generalizations of two-player security games on graphs and matroids. He has also studied a hypergraph version of the chip-firing game (proposed by S. Backman).

**Ken Duna (Ph.D., 2019)**

Thesis title: *Matroid Independence Polytopes and Their Ehrhart Theory*

Ken studied matroid independence polytopes, which are trickier objects than their better known relations, matroid basis complexes. His dissertation contains a complete characterization of their 2-skeletons; very explicit equations for the special case of independence polytopes of shifted matroids; and Ehrhart polynomials for the very very special case of uniform matroids. Among other things, the complex zeros of these polynomials exhibit extremely beautiful geometry.

**Bennet Goeckner (Ph.D., 2018)**

Thesis title: *Decompositions of simplicial complexes*

Bennet, Art Duval, Caroline Klivans and I constructed a nonpartitionable Cohen-Macaulay simplicial complex, disproving a long-standing conjecture of Richard Stanley. Bennet's thesis studied other decomposition problems, including Stanley's conjecture that a $k$-acyclic complex decomposes into boolean intervals of rank $k$ and my own (unpublished) conjecture that the Duval-Zhang decomposition of a CM complex into boolean trees should admit a balanced version. Bennet is now (2018-2021) a postdoc at the University of Washington.

**Joseph Cummings (B.S. with honors, 2016)**

Joseph's honors project was on the Athanasiadis-Linusson bijection between parking functions and Shi arrangement regions.

**Robert Winslow (B.S. with honors, 2016)**

Robert's honors project was about matroids and combinatorial rigidity theory.

**Alex Lazar (M.A., 2014)**

Thesis title: *Tropical simplicial complexes and the tropical Picard group*

Alex studied *tropical simplicial complexes*, which
were introduced by Dustin Cartwright in this paper. Alex proved a
conjecture of Cartwright concerning tropical Picard groups (which
somewhat resemble critical groups of cell complexes). Here is a Sage worksheet Alex developed in the
course of his research.

**Keeler Russell (Undergraduate Honors Research Project, 2012-2013)**

Keeler studied a difficult problem proposed by Stanley: do there exist two nonisomorphic trees with the same chromatic symmetric function? Li-Yang Tan had previously ruled out a counterexample on \(n\leq 23\) vertices, using a brute-force search. Keeler developed parallelized C++ code to perform another brute-force search that ruled out a counterexample for \(n\leq 25\), thus reproducing and extending Tan's results. On the KU Mathematics Department's high-performance computing system, the \(n=25\) case (about 100 million trees) took about 90 minutes using 30 cores in parallel. Keeler's fully documented code (in C++) is freely available from GitHub or from my website.

**Brandon Humpert (Ph.D., 2011)**

Dissertation title: *Polynomials associated with graph coloring and orientations*

Brandon first invented a neat quasisymmetric analogue of Stanley's chromatic symmetric function. This project morphed into a study of the incidence Hopf algebra of graphs; Schmitt had given a general formula for the antipode on an incidence Hopf algebra, but Brandon came up with a much more efficient (i.e., cancellation-free) formula for this particular Hopf algebra, which became the core result of this joint paper.

**Tom Enkosky (Ph.D., 2011)**

Dissertation title: *Enumerative and algebraic aspects of slope varieties*

Tom tackled the problem of extending my theory of graph varieties to higher dimemsion. Briefly, fix a graph \(G=(V,E)\) and consider the variety \(X^d(G)\) of all "embeddings" of \(G\) in \(\mathbb{C}\mathbb{P}^d\) - i.e., arrangements of points and lines that correspond to the vertices and edges of \(G\) and satisfy containment conditions corresponding to incidence in \(G\) - how does the combinatorial structure of \(G\) control the geometry of this variety? In a joint paper, Tom and I figured out some answers to the question, including the component structure of \(X^d(G)\). Separately, Tom proved a striking enumerative result about the numner of pictures of the complete graph over the finite field of order 2.

**Jonathan Hemphill (M.A., 2011)**

Thesis title: *Algorithms for determining single-source, single-destination optimal paths on directed weighted graphs*

**Jenny Buontempo (M.A., 2008)**

Thesis title: *Matroid theory and the Tutte polynomial*

Last updated Fri 9/6/19