Summer 2026: I successfully defended my dissertation! When all paperwork is finalized, I will attach a link here.
Spring 2026: The end is near. I have finalized the details of expressing the Einstein--Vlasov constraint equations, and similar PIDEs of the same class, formally. My focus has now shifted to writing my thesis, which in its current form is a collection of LaTex documents scattered across my desktop.
".... How do you write like you're running out of time? Write day and night like you're running out of time?
How do you write every second you're alive? Every second you're alive? Every second you're alive?!?"
Fall 2025: I have now established a way to express the Einstein--Vlasov constraint equation formally, using fiber integration. This process leads to a weaker form of formal integrability for PIDEs, which I am calling transversal formal integrability. This will be the main result of my thesis!
Summer 2025: I presented a general intro to formal theory of PDEs, as well as my current ideas for expressing Einstein--Vlasov formally, at the GR-Amaldi Conference in Glasgow.
Spring 2025: I have been thinking about how to express the Einstein--Vlasov equation formally. As this is a partial integro-differential equation, not a PDE, it is not immediately clear how to proceed using the existing theory.
Summer 2024: I want to understand the relationship between Spencer cohomology and formal integrability. I have been going through Chapters 6-7 of Seiler's textbook Involution in detail, taking copious notes and drawing pictures to help with this.
Fall 2023: I have starting reading through and trying to understand a few different proofs that Einstein's equations are formally integrable. Every proof that I have found is very technical, so I have been doing a lot of bouncing between the proofs themselves and background material. One of my goals for myself is to write up my own proof, with as many details as possible, and include it in my final thesis.
Spring 2023: As my interests lie in differential geometry and math physics, Markus recommended that I start looking into formal theory of PDEs. The Rocky Mountain Math-Physics Seminar is doing a semester on infinte dimensional manifolds, and so I am going to give one talk on jet bundles and one talk on profinite manifolds, which are relevant infinite dimensional manifolds in the formal theory of PDEs.
Fall 2022: I have spent the summer and fall studying for my comprehensive exam, which covers the topics of topology, differential geometry, and the mathematics of general relativity. See my syllabus. Following my comp exam, I gave an introductory talk on the math of general relativity in the Rocky Mountain Math-Physics Seminar.
Fall 2019-Spring 2020: For my master's degree at Virginia Tech, advised by Mark Embree and Jake Fillman, I studied the spectrum of the discrete Laplacian, with the following results:
Proof that the spectrum of the discrete Laplacian on the octagonal lattice is the interval [−3, 3].
Proof that perturbation of the Laplacian by a sufficiently small periodic potential Q (i.e., the Schrödinger operator) on the smallest possible fundamental domain can open at most two gaps in the resulting spectrum, and that these gaps may only open at eigenvalues ±1.
Existence of periodic potentials on larger fundamental domains that allow a third gap in the spectrum to open at 0.
Proof that, on the smallest fundamental domain, the dispersion relation of the Laplacian has singularities.
My thesis is available here.
Summer 2019-Fall 2019: Megan Wawro, Kaitlyn Steves-Serbin, and I investigated how undergraduate physics students understand the linear algebra concept of change of basis in the context of their introductory quantum mechanics course. Our findings are published here.