Research

Current Research

I work in the analysis & PDEs group at UNC.

I study the oscillatory behavior of high-frequency (large eigenvalue) eigenfunctions, by restricting them to submanifolds and studying their growth there. To do so, I study their Fourier coefficients or generalized Fourier coefficients, which involve testing the sequence of restricted eigenfunctions against an arbitrary collection of functions living on the submanifold. I am particularly interested in understanding how the concentration of high-frequency eigenfunctions is affected by the geometry of the manifold. Using tools from semiclassical analysis, I have obtained an explicit bound on the generalized Fourier coefficients, which is "sensitive" to the geometry of the manifold.

I am supported by the NSF Graduate Research Fellowship Program.

Papers and Preprints:

Improved estimate on Fourier coefficients of restricted eigenfunctions via geodesic beams. Preprint available upon request (2023)
On the growth of generalized Fourier coefficients of restricted eigenfunctions. Comm. in PDEs, 48(2):252-285 (2023).
Spectrum of Kohn Laplacian on the Rossi sphere, with T. Abbas, A. Ramasami, and Y. Zeytuncu. Involve, a Journal of Mathematics (2018).
Thermoception of polygons, with E. Dryden and J. Langford. Preprint available upon request (2018).
Trends in Sustainable Transportation and Recreational Infrastructure: Transportation and Land Use Metrics for Public Health Comparisons, with M. Oswald-Beiler & G. Miller. Journal of Urban Planning and Development (2018).

Past research projects:

Heat content of polygons with Emily Dryden and Jeff Langford.

We considered a question similar to Kac's "Can you hear the shape of a drum?" Does the heat content (or thermal energy) determine the shape of a metal plate? Inspired by the result that one can "thermocept" the shape of a triangle [Meyerson & McDonald 2017] we studied quadrilaterals. We showed that you can "thermocept" parallelograms and acute trapezoids. This work was done for my undergraduate honors thesis at Bucknell in 2018.

Drums that sound the same [Gordon & Webb 1992]

Spectrum of the Kohn-Laplacian on the 3-Sphere with Tawfik Abbas, Allison Ramasami and Yunus Zeytuncu.

We studied the Kohn-Laplacian (a perturbed complex Laplacian) on the 3-sphere. In particular, we showed that the essential spectrum of the Kohn-Laplacian contains zero, which gives another proof of the global non-embeddability of the Rossi example. This work was done at the REU site in mathematical analysis and applications at UM-Dearborn in 2017.

The first few terms in the sequence of eigenvalues (depending on the size of the perturbation |t|) tending towards 0

Trends in sustainable transportational and recreational infrastructure with Michelle Beiler and Greg Miller.

We developed a process for analyzing sustainable transportation/recreation projects(such as trails, parks, and pools) and their impacts on public health using project and network metrics. We applied our methodology to a pilot study in four Pennsylvania counties. This project was conducted as part of the BGRI (Bucknell Geiginger research initiative) during 2016-2017.