My mathematical research interests are in algebraic, geometric, and topological combinatorics, including:
Integer points in rational polytopes and cones
Quasi-polynomials and generating functions
Integer partitions and compositions, integer sequences
Simplicial and cell complexes arising in combinatorics
Graphs and associated matrices
Graded algebras and associated resolutions
Groebner bases and triangulations
Applications of combinatorics
I am also interested in aspects of experimental mathematics, for which I use the following software:
SageMath, a free open source system based on Python
LattE, software for counting lattice points in lattice polytopes
Macaulay2, software for computational commutative algebra
Normaliz, software for polyhedral computations
Online Encyclopedia of Integer Sequences entries resulting from my work:
A Brief Survey on Lattice Zonotopes, (joint with Andrés R. Vindas-Meléndez), in Algebraic and Geometric Combinatorics on Lattice Polytopes, Proceedings of the Summer Workshop on Lattice Polytopes, T. Hibi and A. Tsuchiya (eds), World Scientific, New Jersey, 2019, pp 101-116.
Unimodality Problems in Ehrhart Theory, in Recent Trends in Combinatorics, Beveridge, A., et al. (eds), Springer, 2016, pp 687-711, doi 10.1007/978-3-319-24298-9_27
Equatorial Flow Triangulations of Gorenstein Flow Polytopes (joint with Alvaro Cornejo), submitted.
Volume Inequalities for Flow Polytopes of Full Directed Acyclic Graphs (joint with James Ford McElroy), Combinatorial Theory, 5(2) 2025, #14.
Minimal Free Resolutions of Numerical Semigroup Algebras via Apery Specialization (joint with Tara Gomes, Ezra Miller, Christopher O'Neill, and Aleksandra Sobieska), Pacific Journal of Math, Vol. 334 (2025), No. 2, 211–231, https://doi.org/10.2140/pjm.2025.334.211
Local h*-Polynomials for One-row Hermite Normal Form Simplices (joint with Esme Bajo, Giulia Codenotti, Johannes Hofscheier, and Andrés R. Vindas-Meléndez), Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, 2025, https://doi.org/10.1007/s13366-025-00784-z
Ehrhart Limits (joint with McCabe Olsen), Ars Math. Contemp. 25 (2025), #1.02, https://doi.org/10.26493/1855-3974.3092.4af
Facets of Random Symmetric Edge Polytopes, Degree Sequences, and Clustering (joint with Kaitlin Bruegge and Matthew Kahle), Discrete Mathematics and Theoretical Computer Science, Vol. 25:2 (2023)
Triangulations of Flow Polytopes, Ample Framings, and Gentle Algebras (joint with Matias von Bell, Kaitlin Bruegge, Derek Hanely, Zachery Peterson, Khrystyna Serhiyenko, and Martha Yip), Selecta Mathematica, 30(55), 2024.
Facets of Symmetric Edge Polytopes for Graphs with Few Edges (joint with Kaitlin Bruegge), Journal of Integer Sequences, Vol. 26 (2023), Article 23.7.2
The Integer Decomposition Property and Weighted Projective Space Simplices (joint with Robert Davis, Derek Hanely, Morgan Lane, and Liam Solus), INTEGERS, 24 (2024), Article A60.
Triangulations, Order Polytopes, and Generalized Snake Posets (joint with Matias von Bell, Derek Hanely, Khrystyna Serhiyenko, Julianne Vega, Andrés R. Vindas-Meléndez, and Martha Yip), Combinatorial Theory, 2(3), 2022.
A Regular Unimodular Triangulation of Reflexive 2-Supported Weighted Projective Space Simplices (joint with Derek Hanely), Annals of Combinatorics, 25, 935–960 (2021).
Decompositions of Ehrhart h*-polynomials for Rational Polytopes (joint with Matthias Beck and Andrés R. Vindas-Meléndez), Discrete and Computational Geometry, (68), 50-71, (2022).
Extended abstract version for FPSAC published in Sém. Lothar. Combin. 85B (2021), Art. 38, 13 pp.
Base-b Representations (joint with Santiago de León, Kyle Franz, and Sami Sultan), Amer. Math. Monthly 126 (2019), no. 9, 859.
Antichain Simplices (joint with Brian Davis), Journal of Integer Sequences, Vol. 23 (2020), Article 20.1.1.
Hajós-Type Constructions and Neighborhood Complexes (joint with Julianne Vega), SIAM J. Discrete Math. 34 (2020), no. 4, 2424–2447
Homomorphism Complexes and Maximal Chains in Graded Posets (joint with Wesley K. Hough), European Journal of Combinatorics, Volume 81, October 2019, Pages 178-194.
h*-Polynomials With Roots on the Unit Circle, (joint with Fu Liu), Experimental Mathematics, 30:3 (2021), 332-348, DOI: 10.1080/10586458.2018.1538912
Laplacian Simplices, (joint with Marie Meyer), Adv. in Appl. Math. 114 (2020), 101976
Counting Arithmetical Structures on Paths and Cycles, (joint with Hugo Corrales, Scott Corry, Luis David Garcia Puente, Darren Glass, Nathan Kaplan, Jeremy L. Martin, Gregg Musiker, and Carlos E. Valencia), Discrete Mathematics, Volume 341, Issue 10, October 2018, Pages 2949-2963.
Detecting the Integer Decomposition Property and Ehrhart Unimodality in Reflexive Simplices, (joint with Robert Davis and Liam Solus), Advances in Applied Math, Volume 100, September 2018, 122-142.
Euler-Mahonian Statistics and Descent Bases for Semigroup Algebras, (joint with McCabe Olsen), European J. Combin. 69 (2018), 237-254.
Matching and Independence Complexes Related to Small Grids, (joint with Wesley K. Hough), Electron. J. Comb., 24(4) (2017), #P4.18
Generating Functions and Triangulations for Lecture Hall Cones, (joint with Matthias Beck, Matthias Koeppe, Carla Savage, and Zafeirakis Zafeirakopoulos), SIAM J. Discrete Math., 30(3), 2016, 1470-1479.
r-Stable Hypersimplices, (joint with Liam Solus), Journal of Combinatorial Theory, Series A , 2018, Vol.157, p.349-388
Ehrhart series, unimodality, and integrally closed reflexive polytopes, (joint with Robert Davis), Ann. Comb. 20 (2016), no. 4, 705-717.
Hyperoctahedral Eulerian Idempotents, Hodge Decompositions, and Signed Graph Coloring Complexes, (joint with Sarah Crown Rundell), Electron. J. Comb., 21(2) (2014), #P2.35
s-Lecture Hall Partitions, Self-Reciprocal Polynomials, and Gorenstein Cones, (joint with Matthias Beck, Matthias Koeppe, Carla Savage, and Zafeirakis Zafeirakopoulos), The Ramanujan Journal, February 2015, Volume 36, Issue 1-2, pp 123-147.
Lattice Point Generating Functions for Symmetric Cones, (joint with Matthias Beck, Thomas Bliem, and Carla Savage), Journal of Algebraic Combinatorics 38 (2013), 543-566.
Compositions constrained by graph Laplacian minors, (joint with Robert Davis, Jessica Doering, Ashley Harrison, Jenna Noll, and Clifford Taylor), INTEGERS 13 (2013), Article A41.
Euler-Mahonian Statistics via Polyhedral Geometry, (joint with Matthias Beck), Advances in Mathematics 244 (2013), 925-954.
Mahonian Partition Identities via Polyhedral Geometry, (joint with Matthias Beck and Nguyen Le), From Fourier Analysis and Number Theory to Radon Transforms and Geometry: In Memory of Leon Ehrenpreis, (H. Farkas, R. Gunning, M. Knopp, and B. A. Taylor, eds.), Developments in Mathematics 28 (2013), 41--54.
Cellular Resolutions of Ideals Defined by Simplicial Homomorphisms, (joint with Jonathan Browder and Steven Klee), Israel J. Math. 196 (2013), no. 1, 321-344. DOI: 10.1007/s11856-012-0149-2.
Deformation Retracts of Neighborhood Complexes of Stable Kneser Graphs, (joint with Matthew Zeckner), Proc. Amer. Math. Soc. 142 (2014), 413-427.
Independence Complexes of Stable Kneser Graphs, Electronic Journal of Combinatorics, 18, no. 1 (2011), P118.
Nowhere-Harmonic Colorings of Graphs, (joint with Matthias Beck), Proc. Amer. Math. Soc. 140 (2012), 47-63.
Symmetries of the Stable Kneser Graphs, Adv. in Appl. Math., 45 (2010), no. 1, 12 - 14.
The Complex of Non-Crossing Diagonals of a Polygon, (joint with Richard Ehrenborg), J. Combin. Theory Ser. A, 117 (2010), no. 6, 642 - 649.
Ehrhart Polynomial Roots and Stanley's Non-negativity Theorem, (joint with Mike Develin), Integer Points in Polyhedra--Geometry, Number Theory, Representation Theory, Algebra, Optimization, Statistics, AMS Contemporary Mathematics 2008, Volume: 452, pp 67-78.
Norm Bounds For Ehrhart Polynomial Roots, Discrete and Computational Geometry, 39 (2008), no. 1-3, 191-193.
An Ehrhart Series Formula For Reflexive Polytopes, Electronic Journal of Combinatorics, 13, no. 1 (2006), N 15.
Rationality of Poincare Series for a Family of Lattice Simplices, (joint with Brian Davis). This paper gives a combinatorial proof for the rationality of Poincare series for certain semigroup algebras associated with lattice polytopes.
Ehrhart Theory for Lattice Polytopes
My thesis is a combination of the papers "Norm Bounds...," "An Ehrhart Series Formula...," and "Ehrhart Polynomial Roots..." shown above.