Research

Einstein-Vlasov Equation as a formal PDE

Since I am now relatively comfortable with expressing PDEs as submanifolds of the jet bundle, upon Markus' recommendation I have started to look at the Einstein-Vlasov Equation. I did not realize that the E-V equation is not a purely differential equation, but is in fact an integro-differential equation. Trying to express this equation formally has provided an interesting problem, and I will definitely try to address this in my thesis.

Proving in detail involutivity of the Einstein Field Equation

In my reading, I have read several proofs of involutivity of EFE. However, in each of these proofs, I found key details having been either justified away or skipped over entirely. As a result, I spent a majority of this semester writing my own, very detailed, proof. In doing so, I have used Mathematica and Maple for large matrix computations, and have been familiarizing myself with some of Maple's differential geometry capabilities.

Involutivity from an algebraic perspective

In order to study the relationship between involutivity and formal integrability, I needed to understand more of the comodule structure of the Spencer/Koszul complexes. I started with Chapter 6 of Seiler's book on involution as a jumping off point.

Involutivity of Field Equations

For my Ph.D dissertation, I have settled on trying to clarify the relationship between involutivity and formal integrability of PDEs. In particular, I would like to study the Spencer cohomology of particular field equations in physics (such as Einstein, Yang-Mills, Klein-Gordon). I gave a talk in the Rocky Mountain Math Physics Seminar detailing how to write Einstein's Equation as a formal PDE.

Jet Theory

Over the summer of 2023, I began taking notes on a variety of texts on formal integrability of PDEs and jet bundles, with the aim of settling on a dissertation topic.

Infinite Dimensional Manifolds

For the Spring 2023 semester, the Rocky Mountain Math-Physics Seminar did a series of lectures on infinite dimensional manifolds. For this lecture series, I gave a talk on jet bundles (February) and profinite manifolds (April).

General Relativity

For my Ph.D. comprehensive exam (Fall 2022), I studied a variety of topics (see my syllabus), with the aim of understanding the geometry of general relativity. I presented my studies in the Rocky Mountain Math-Physics Seminar following the exam.

Spectra of Periodic Schrödinger Operators

For my M.S. at Virginia Tech, I studied the spectra of periodic Schrödinger operators on the octagonal lattice. I proved several results concerning gaps in the spectra, as well as proved the existence of a singularity. My thesis is available here, and my defense presentation is available here.

Math-Physics Education

I worked as an undergraduate researcher with Dr. Megan Wawro and Dr. Kaitlyn Serbin studying the language used by undergraduate physics students when talking about basis and change of basis in their quantum mechanics classes. Our findings were published in Physical Review Physics Education Research in June 2021, available here.