Research
The study of quantum graphs is motivated by modeling physical phenomena and complex quantum systems with graphs. My research specifically pertains to the application of modeling waves propagating through thin branching structures. This can be accomplished by solving Schrodinger type wave equations on metric graphs. Given any graph G and spatial operator L, my goal is to numerically define L on G with spectral accuracy. The next step is to implement highly accurate time evolution solvers which are complicated by boundary conditions. Using highly accurate spatial and time dependent solvers, we will then apply the Adjoint Continuation Method to search for time periodic orbits on a variety of Quantum Graphs. All of these components will be adapted into a software package jointly with Roy Goodman.
Publications
Limiting Eigenfunctions of Sturm-Liouville operators Subject to a Spectral Flow by Tom Beck, Isabel Bors, Grace Conte, Graham Cox and Jeremy Marzuola, Annales mathématiques du Québec (2020).
Talks and Presentations
Searching for Time Periodic Solutions to the Nonlinear Schrödinger Equation [Slides]
Triangle Area Graduate Mathematics Conference at UNC, Chapel Hill, 8 Dec. 2020
Introduction to Bifurcation Theory and its Applications to the NLS Equation [Slides]
Mathematics Graduate Student Association Seminar at UNC, 29 Sept. 2020
Discretizing Schrödinger Type Operators on Quantum Graphs with Spectral Accuracy [Slides]
Mathematics Graduate Student Association Seminar at NCSU, 13 Mar. 2020
Graduate Mathematics Association Seminar at UNC, Chapel Hill, 2 Mar. 2020
Triangle Area Graduate Mathematics Conference at UNC, Chapel Hill, 9 Nov. 2019
Graduate Mathematics Association Seminar at UNC, Chapel Hill, 25 Mar. 2019
Introduction to Quantum Mechanics [Slides]
Girls Talk Math at UNC, Chapel Hill, 12 July 2019
Girls Talk Math at UNC, Chapel Hill, 10 July 2018
Girls Talk Math at UNC, Chapel Hill, 14 July 2017
Past Projects
Displacement Analysis of Neo-Hookean Elastic Materials - UNC, Chapel Hill
Derived and tested the behaviour of finite element formulations for incompressible plastic materials at finite strains using Cooks Membrane and the Elasto-Plastic Strip using FreeFem++.
Calculating the Stark Effect Energy Shift for the Hydrogen Atom - UNC, Chapel Hill
Found that the numerical eigenvalue methods produced very close estimates to the first-order perturbation theory corrections. However, the second-order estimates were slightly lower than the perturbation theory results. This is likely due to the fact that we truncated the Hamiltonian matrix and so we lost the effect of the higher level states on the eigenvalues.
Heterojunction-Assisted Impact Ionization - University of Oregon REU
Designed and performed a series of experiments to determine if it is possible to improve the efficiency of solar cells on a quantum level. Presented the theoretical background and experimental results to colleagues and undergraduate students.