Research and Slides

Research Interests

I am interested in discrete geometry, especially polytopes and their relationships to other areas of mathematics. This has most recently brought me to applied algebraic geometry, in the context of studying algebraic formulations of the Kuramoto equations, which describe networks of interconnected oscillators, and power-flow equations, which describe networks of electrical generators and consumption sites. I frequently use techniques from toric algebra and combinatorics to decompose polytopes in ways that encode desirable properties of the underlying network.

I also have strong interest in Ehrhart theory, which studies the distribution of lattice points inside of dilations of polytopes. Many times, the Ehrhart series of a polytope can reveal information about the polytope that may have been previously opaque, such as the normalized/relative/discrete volumes, existence of triangulations with desirable properties, face numbers, and so forth.

Slides of Selected Talks

1. Identifying the State Polytopes of Certain Combinatorial Neural Codes, UC San Diego Combinatorics Seminar, San Diego, CA, October 2018

2.  The Pinnacle Set of a Permutation, 5th Lake Michigan Workshop on Combinatorics and Graph Theory at University of Notre Dame, South Bend, IN, April 2018

3.  A Polyhedral Proof of a Wreath Product Identity, Joint Mathematics Meetings in San Diego, CA, January 2018

4.  Detecting the Integer Decomposition Property in Reflexive Simplices, CanaDAM at Ryerson University, Toronto, Canada, June 2017

5.  Pattern-Avoiding Polytopes, AMS Special Session at the University of St. Thomas, Minneapolis, MN, October 2016

Publications and Preprints (*beside a name denotes undergraduate student coauthor)

1. "Weighted Ehrhart Theory: Extending Stanley's nonnegativity theorem", with Esme Bajo, Jesús A. De Loera, Alexey Garber, Sofía Garzón Mora, Katharina Jochemko, and Josephine Yu. Submitted for publication. Preprint available at arXiv:2303.09614.

2. "Perfectly Matchable Set Polynomials and h*-polynomials for Stable Set Polytopes of Complements of Graphs", with Florian Kohl. Submitted for publication. Preprint available at arXiv:2207.14759.

3. "Normalized Volumes of Type-PQ Adjacency Polytopes for Certain Classes of Graphs", with Joakim Jakovleski* and Qizhe Pan*. Accepted for publication in Involve. Preprint available at arXiv:2202.09853.

4. "Facets and facet subgraphs of adjacency polytopes," with Tianran Chen and Evgeniia Korchevskaia. Discrete Applied Mathematics, 328:139-153, 2023. 

5. "Projective Normality and Ehrhart Unimodality for Weighted Projective Space Simplices," with Benjamin Braun, Derek Hanely, Morgan Lane, and Liam Solus. Submitted for publication. Preprint available at arXiv:2103.17156.

6. "State Polytopes Related to Two Classes of Combinatorial Neural Codes." Advances in Applied Mathematics, 142:102418, 2023

7. "A toric deformation method for solving Kuramoto equations," with Tianran Chen. Nonlinear Dynamics, 109, 08 2022.

8. "Computing Volumes of Adjacency Polytopes via Draconian Sequences," with Tianran Chen. Electronic Journal of Combinatorics, 29(1):Paper 1.61, 2022.

9. "Further Combinatorics and Applications of Two-Toned Tilings," with Greg Simay. Fibonacci Quarterly, 58(4):300-320, 2020.

10. "Universal Gröbner Bases of Toric Ideals of Combinatorial Neural Codes," with Melissa Beer*, Thomas Elgin*, Matthew Hertel*, Kira Laws*, Rajinder Mavi, Paula Mercurio, and Alexandra Newlon*. Involve, Vol. 14 (2021), No. 5, 723-742.

11. "Descents and des-Wilf Equivalence of Permutations Avoiding Certain Non-Classical Patterns," with Caden Bielawa*, Daniel Greeson*, and Qinhan Zhou*. Involve, Vol. 12 (2019), No. 4, 549–563.

12. "A Polyhedral Proof of a Wreath Product Identity," with Bruce Sagan. Journal of Combinatorics, 10(4), 711-723, 2019 (Special issue in memory of Jeff Remmel).

13. "A Product Formula for the Normalized Volume of Free Sums of Lattice Polytopes," with Tianran Chen. In Advances in Algebra: Proceedings of the Southern Regional Algebra Conference 2017, pages 111–119, 2019. 

14.  "Counting Equilibria of the Kuramoto Model using Birationally Invariant Intersection Index," with Tianran Chen and Dhagash Mehta. SIAM Journal on Applied Algebra and Geometry (SIAGA), 2(4), 489–507, 2018.

15. "The pinnacle set of a permutation," with Sarah A. Nelson, T. Kyle Petersen, and Bridget E. Tenner. Discrete Mathematics, 341(11):3249-3270, 2018.

16. "Pattern-Avoiding Polytopes," with Bruce Sagan. European Journal of Combinatorics, 74:48-84, 2018. Extended abstract for FPSAC 2017 available here.

17. "Detecting the Integer Decomposition Property and Ehrhart Unimodality in Reflexive Simplices," with Benjamin Braun and Liam Solus. Advances in Applied Mathematics, 100:122-142, 2018.

18. "Width-k Generalizations of Classical Permutation Statistics." J. Integer Sequences, 20(6):Article 17.6.3, 14, 2017.

19. "Ehrhart series, Unimodality, and Integrally Closed Reflexive Polytopes," with Benjamin Braun. Annals of Combinatorics, 20(4):705–717, 2016.

20. "Ehrhart Series of Polytopes Related to Symmetric Doubly-Stochastic Matrices," The Electronic Journal of Combinatorics, 22(2):Paper 2.17, 2015.

21. "Compositions Constrained by Graph Laplacian Minors," with Benjamin Braun, Ashley Harrison, Jessica McKim*, Jenna Noll*, and Clifford Taylor. INTEGERS, 13 (2013), A41.