## 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++.

Wave Breaking Solutions to the Whitham Equation - UNC, Chapel Hill

Whitham’s equation is a partial differential equation used to model shallow water waves. Due to a term that is defined purely in Fourier space, I had to implement the Fast Fourier transform and a 4th order Runge-Kutta solver to obtain a solution. I then used dalising to clean up the solution and a shock capturing scheme to study its wave-breaking behavior.

Flow Enhanced Diffusion in a Circular Pipe - UNC, Chapel Hill

Investigated the diffusion in an infinitely long pipe with a circular cross-section and parabolic flow at large time scales using multiscale and homogenization on the advection equation.

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.