Research Interests

My research interests continue to evolve. A list of publications is available on my Publications page. Here is a brief summary of my research interests, both past and present:

Quantum computing, quantum algorithms, and quantum information.

The area of quantum computing has exploded in the last decade. In 2022 I was named a QuForce Innovation Fellow. I was fortunate to work with Elizabeth Campolongo (Ohio State University) and Hardik Routray (Rutgers University) under the guidance of Alex Kahn on a project having to do with the BB84 quantum key distribution protocol. Among the highlights of the project was the opportunity to work on a quantum computer with eleven qubits at IonQ. This project is ongoing and I am on the lookout for other problems in this area and in quantum computing more generally.

Low regularity well-posedness for nonlinear dispersive partial differential equations.

Though I was primarily interested in the problems of proving that the family of generalized Korteweg-de Vries (KdV) equations are well-posed (locally and globally), I also spent time thinking about nonlinear Schrödinger equations and their similar well-posedness issues. This remains an area of interest for me.

Soliton stability.

Ultimately my PhD thesis was centered on the problem of showing that soliton solutions for generalized KdV equations are stable in the low regularity setting. What I actually showed was that any instability of solitons for subcritical generalized KdV equations is bounded above by a polynomial. There’s been a good deal of progress on these problems in the last four years; the stability problems for the KdV equation and the modified KdV equation appear to have been completely solved using the inverse scattering transform. As far as I know, the problem for the quartic generalized KdV equation remains open below the energy norm.

Periodic solutions to nonlinear wave equations on tori.

Since coming to Wofford College in 2014 I’ve had the pleasure of working with Nem Kosovalic (formerly of the University of South Alabama and currently a data scientist at Shopify) on problems having to do with the existence of periodic-in-time solutions to nonlinear wave equations on tori. This work is ongoing.