Students of Jeremy L. Martin

Current Students

Emma Colaric

Emma is a MA student studying scheduling problems and Ehrhart theory.

Mark Denker

Mark is a PhD student interested in algebraic combinatorics in general.

Kevin 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.

Alumni/ae

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including Stanley's conjecture that a $k$-acyclic complex decomposes into boolean intervals of rank $k$ and my own (unpublished) conjecture that the Duval-Zhang decomposition of a CM complex into boolean trees should admit a balanced version.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 Mon 2/24/19