— Research —
A short tour of what I am working on, what I have worked on, and the questions I keep returning to.
My principal interest is the regularity theory of nonlinear elliptic equations — the study of when, and to what degree, solutions to variational problems can be expected to be smooth. The motivating object is the minimizer of an energy functional, and the questions are old ones: does a minimizer exist; if so, in what space; and how much can we say about its behaviour beyond merely belonging to that space?
These questions were given their most influential modern formulation by Hilbert, in the nineteenth of the famous twenty-three problems he posed in Paris in 1900. Hilbert asked whether all solutions of regular variational problems are necessarily analytic. The answer, settled affirmatively in the scalar case by De Giorgi (and independently by Nash) in the 1950s, opened a tradition of work that continues today.
For my undergraduate honors thesis, supervised by Eduardo Teixeira at UCF, I developed a self-contained exposition of De Giorgi's resolution of Hilbert's 19th problem. The thesis treats:
The aim was to write a document a graduate student could read end-to-end without external references — at least in the linear setting — and to use that exposition as a launching pad for the questions I plan to pursue at Purdue: fully-nonlinear equations, free-boundary problems, and the geometry of obstacle-type singularities.
Since 2021 I have been part of Kerri Donaldson-Hanna's planetary science group at UCF. The Moon's primordial crust is dominated by anorthosite, a feldspar-rich rock whose exposure on crater walls and rims offers a glimpse of the lunar interior preserved through billions of years of impact bombardment.
My contributions have been principally observational and computational. I built a photometric-analysis pipeline for LRO Narrow Angle Camera observations, generated digital terrain models for twenty craters of interest, assembled Moon Mineralogy Mapper mosaics for twenty-five of them, and helped produce Christiansen-feature maps from Diviner data. A manuscript reporting the high-resolution photometric properties of pure-anorthosite exposures is in preparation.
In parallel, with Adrienne Dove's group, I have contributed to the design and execution of Strata-2P experiments — including schematic design, electronics troubleshooting, and analysis of thermal-conductivity measurements of lunar regolith simulant.
In the summer of 2023 I joined the Summer@ICERM program at Brown, working in the group of L. A. Velazquez, A. Harsy, C. Johnson, and J. Sorrells on the flexible-tile model of DNA self-assembly. The model translates a problem in nanotechnology — assembling DNA molecules into prescribed nanostructures — into a question of graph theory: given a target graph G, what is the minimum number of tile types, bond-edge types, and overall tiles needed to realize G under various permissiveness constraints?
Our group catalogued the optimal constructions for graphs of order at most six in scenarios one, two, and three, and extended the analysis to several infinite families: stacked-prism graphs, hexagonal lattices, and web graphs. Two manuscripts are in progress, on counter-examples in scenario two and on relationships between the invariants $T_i(G)$ and $B_i(G)$ for $i=2,3$ on low-order graphs.
A non-exhaustive list of texts I am working through this year:
A list of presentations, publications, and awards is on the publications page.