## Students of Jeremy L. Martin

### Current Students

• Mark Denker is a PhD student interested in algebraic combinatorics in general.
• Kevin Marshall is a PhD student currently studying Hopf monoids of set families. He has previously studied generalizations of two-player security games on graphs and matroids.

### Former Students

• Kevin Marshall (PhD, 2022) studied Hopf monoids. Kevin's dissertation Hopf Monoids of Set Families constructed a commutative Hopf structure on a very large species: set families $$\mathcal{F}\subseteq 2^I$$, subject only to the requirement $\0\in\mathcal{F}$. The Hopf structure is inspired by operations on antimatroids and admits a nice antipode formula when $\mathcal{F}$ is the family of order ideals of a poset on $I$. Kevin is working as a software engineer at Epic Systems in Madison, WI.
• Emma Colaric (MA, 2020) studied a paper of Felix Breuer and Caroline Klivans on the connections between scheduling problems and Ehrhart theory. Emma is a data scientist at SelectQuote Insurance in Kansas City, MO.
• Ken Duna (PhD, 2019) studied matroid independence polytopes, which are trickier objects than their better known relations, matroid basis complexes. Ken's dissertation Matroid Independence Polytopes and Their Ehrhart Theory 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. Ken is teaching high school mathematics in Cloverdale, CA.
• Bennet Goeckner (Ph.D., 2018) studied the structure 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 dissertation Decompositions of Simplicial Complexes studied related 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including Stanley's conjecture that a $$k$$-acyclic complex decomposes into boolean intervals of rank $$k$$ and my own (unpublished) conjecture that the Duval-Zhang decomposition of a CM complex into boolean trees should admit a balanced version.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 was a postdoc at the University of Washington and is now an assistant professor at the University of San Diego.
• Joseph Cummings (BS with honors, 2016) studied the Athanasiadis-Linusson bijection between parking functions and Shi arrangement regions. Joe received his PhD from Kentucky in 2022 and is now a postdoc at Notre Dame.
• Robert Winslow (BS with honors, 2016) studied matroids and combinatorial rigidity theory.
• Alex Lazar (MA, 2014) studied tropical simplicial complexes, which were introduced by Dustin Cartwright in this paper. In his thesis, Tropical simplicial complexes and the tropical Picard group, Alex proved a conjecture of Cartwright on tropical Picard groups (which somewhat resemble critical groups of cell complexes) and eventually published his work as an article in the Electronic Journal of Combinatorics. Alex went on to earn a PhD at the University of Miami.
• Keeler Russell (Undergraduate Honors Research Project, 2012-2013) 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 (PhD, 2011) started by inventing 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 and his dissertation Polynomials associated with graph coloring and orientations.
• Tom Enkosky (PhD, 2011) 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. Tom's dissertation was entitled Enumerative and algebraic aspects of slope varieties.
• Jonathan Hemphill (MA, 2011) studied the $$A^*$$ algorithm, which is a heuristic version of the Dijkstra's classical algorithm for finding shortest paths in graphs.
• Jenny Buontempo (MA, 2008) studied matroid theory and the Tutte polynomial. Jenny went on to earn a PhD in STEM Education at the University of Texas.

Last updated 8/17/22