My work to this point has centered around applications of algebraic geometry to spectral theory. Listed below are descriptions of past and current projects. I also have a long-standing interest in applications of mathematics to biology and medicine.
Floquet isospectrality for discrete Schrödinger operators
Mentors: Wencai Liu, Rodrigo Matos, Matthew Faust
The specific goal of this project was to establish an isospectrality result for discrete Schrödinger operators with complex-valued, periodic potentials, over a sublattice of $\Z^d$ for some d. Under a parity assumption on one of the potential periods, we proved the existence of non-zero, complex-valued potentials for which the corresponding discrete Schrödinger operator is isospectral to the free (zero-potential) Schrödinger operator. This project was partially funded by the Texas A&M College of Arts & Sciences.
Results: Floquet Isospectrality of the Zero Potential for Discrete Periodic Schrödinger Operators, published in the The Journal of Mathematical Physics ; accompanying Macaulay2 code
A Bestiary of Bloch Varieties
Mentors: Frank Sottile, Matthew Faust
A discrete periodic operator is defined over an underlying $\Z^d$-periodic graph and serves as a discrete model of electron transport in a crystal. Its spectrum is encoded by an algebraic hypersurface called its Bloch variety. We utilize tools of computational and combinatorial algebraic geometry to study discrete invariants of Bloch varieties over families of periodic graphs, with the goal of understanding various geometric and asymptotic properties. With the help of Macaulay2 and the Texas A&M Mathematics Department’s computational cluster, we are conducting a large-scale investigation for millions of periodic graphs.
Results: Experimental paper and Macaulay2 package forthcoming
The Critical Point Degree of a Periodic Graph
Mentors: Frank Sottile, Matthew Faust
This project is an outgrowth of our larger experimental investigation, and is the focus of our current work. Certain questions about the spectrum of a discrete periodic operator may be addressed by understanding the critical points of the Bloch variety. Previous work of Faust and Sottile provides a bound on the number of critical points of the Bloch variety, modulo periodicity. This is in terms of the volume of the Newton polytope of the variety’s defining polynomial. Through our experimental work, we developed an finer, structural understanding of the critical points. This has led to an improved bound involving combinatorial features of the underlying graph as well as an analysis of the Bloch variety’s asymptotic behavior.
Results: Paper currently in preparation
Conference Presentations
- SIAM TX-LA 7-th Annual Meeting, Minisymposium on Periodic Operators, October 2024
- “The Critical Point Degree of a Periodic Graph”, slides
- TEXAS ALGEBRAIC GEOMETRY SYMPOSIUM, April 2024
- “Discrete Invariants of the Dispersion Relation”, poster
- TX-LA Undergraduate Mathematics Conference, March 2024
- “Floquet Isospectrality of the Zero Potential for Discrete Periodic Schrödinger Operators”, slides (joint)
- SIAM TX-LA 6-th Annual Meeting, Minisymposium on Applications of Combinatorial and Computational Algebraic Geometry, October 2023
- “Invariants of the Dispersion Relation for Discrete Periodic Operators”, slides
- Joint Mathematics Meetings, Pi Mu Epsilon Student Poster Session, January 2023
- “Discrete Invariants of the Dispersion Relation”, poster