Research

I am primarily interested in problems about lattice points that arise when studying semigroups, discrete geometry, and discrete optimization. Lattice points in convex sets are fundamental objects at the intersection of discrete geometry, integer optimization, and number theory.  My approach combines tools from geometry, algebra, combinatorics, number theory, and optimization with computational experimentation using tools like SageMath and LP solvers.

Research:

• "Optimal Polygonal Chess," joint work with  Jesús A. De Loera and Christopher O'Neill. In Preparation. 

• "An Arithmetic Measure of Width for Convex Bodies," joint work with  Jesús A. De Loera and Christopher O'Neill. Submitted. You can view a preprint here. 

TL;DR: We  introduce a discrete alternative to lattice width that measures the number of distinct values a linear functional attains on the lattice points of a convex body, prove structural results that generalize the theorem for sets of length from semigroup theory, and establish Ehrhart-type asymptotic results for rational polytopes.  

• "Integer Points in Arbitrary Convex Cones: The Case of the PSD and SOC Cones,"  joint work with Jesús A. De Loera, Luze Xu, and Shixuan Zhang.   Mathematical Programming.  You can view it here. 

TL;DR: We introduce an analogue of the Hilbert basis for non-polyhedral convex cones, called (R,G)-finite generation, that uses the action of a finitely generated group G on a finite set of roots R to generate the lattice points of the cone, and show that both the positive semidefinite cone and the second-order cone are finitely generated in this way.

• "Optimizing Polytopal Norms with Respect to Numerical Semigroups." Masters Thesis.  You can view it here. 

TL;DR: I study the behavior of polytopal norms on the sets of factorizations of elements in numerical semigroups, and establish Ehrhart-type asymptotic results describing their optimal behavior as we increase the element in the semigroup. 

Talks Given:

• "Arithmetic Width"

  • at The Graduate Online Combinatorics Colloquium (GOCC) in Oct 2025

  • at The Algebra & Discrete Mathematics Seminar (ADM, UC Davis) in September 2025

  • at The Student Run Research Seminar   (SRRS, UC Davis) in March 2025

• "Generating Conical Semigroups: SOC and PSD" 

  • at The 25th Conference on Integer Programming and Combinatorial Optimization (IPCO) in July 2024

  • at The 2024 INFORMS Optimization Society (IOS) in March 2024

  • at the Student Run Research Seminar at U.C. Davis in December 2023

•  "Optimizing polytopal norms with respect to numerical semigroups" 

  • at The Graduate Student Combinatorics Conference (GSCC) in April 2021

  • at The Underrepresented Students in Topology and Algebra Research Symposium (USTARS) in April 2021