Research

My research interests lie chiefly in algebraic combinatorics, especially the study of polytopes and in particular flow polytopes. I am also interested in combinatorial quantum field theory, combinatorial algebraic geometry, and tropical geometry. Below you can find a brief description of each of my publications and preprints. Click on the image to be taken to the arXiv version of the paper. Links to published versions are included after each description where applicable.

Generalized Pitman-Stanley polytope: vertices and faces (with Maura Hegarty, Alejandro H. Morales, and Annie Raymond)

In 1999, Pitman and Stanley introduced the polytope bearing their name along with a study of its faces, lattice points, and volume. The Pitman-Stanley polytope is well-studied due to its connections to probability, parking functions, the generalized permutahedra, and flow polytopes. Its lattice points correspond to plane partitions of skew shape with entries 0 and 1. Pitman and Stanley remarked that their polytope can be generalized so that lattice points correspond to plane partitions of skew shape with entries 0,1,…,m. Since then, this generalization has been untouched. We study this generalization and show that it can also be realized as a flow polytope of a grid graph. We give multiple characterizations of its vertices in terms of plane partitions of skew shape and integer flows. For a fixed skew shape, we show that the number of vertices of this polytope is a polynomial in m whose leading term, in certain cases, counts standard Young tableaux of a skew shifted shape. Moreover, we give formulas for the number of faces, as well as generating functions for the number of vertices.

Sequences of Trees and Higher-Order Renormalization Group Equations (with Loïc Foissy and Karen Yeats)

We define a notion of higher order renormalization group equation and investigate when a sequence of trees satisfies such an equation. In the strongest sense, the sequence of trees satisfies a kth order renormalization group equation when applying any choice of Feynman rules results in a Green function satisfying a kth order renormalization group equation, and we characterize all such sequences of trees. We also make some comments on sequences of trees which require special choices of Feynman rules in order to satisfy a higher order renormalization group equation.

Published in the Journal of Mathematical Physics.

Tiered trees, weights, and q-Eulerian numbers (with Sam Glennon, Paul E. Gunnells, and Einar Steingrimsson)

Maxmin trees are labeled trees with the property that each vertex is either a local maximum or a local minimum. Such trees were originally introduced by Postnikov, who gave a formula to count them and different combinatorial interpretations for their number. In this paper we generalize this construction and define tiered trees by allowing more than two classes of vertices. Tiered trees arise naturally when counting the absolutely indecomposable representations of certain quivers, and also when one enumerates torus orbits on certain homogeneous varieties. We define a notion of weight for tiered trees and prove bijections between various weight 0 tiered trees and other combinatorial objects; in particular order n weight 0 maxmin trees are naturally in bijection with permutations on n-1 letters. We conclude by using our weight function to define a new q-analogue of the Eulerian numbers.

Published in Journal of Combinatorial Theory, Series A.