Ken is a PhD student currently studying generalizations of the chip-firing model and connections to toric varieties.

Bennet is a PhD student studying structural problems on simplicial complexes. He, Art Duval, Caroline Klivans and I constructed a nonpartitionable Cohen-Macaulay simplicial complex, disproving a long-standing conjecture of Richard Stanley. Bennet is currently working on the balanced case of the partitionability conjecture, which remains open.

**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 Tue 6/7/16