I am a Ph.D. student in Computational and Applied Mathematics at University of Chicago working on PDE’s and mathematical physics under the supervision of Guillaume Bal. My current projects are in the area of continuous topological insulators with applications in photonics and multi-layer graphene.
Cold Plasma Model
The cold plasma model is used to describe electromagnetic wave propagation in an electron cloud and has applications in both low-energy plasma physics and photonics. In our most recent work we studied its topological phases and constructed Bulk Difference Invariants (BDI’s) [Bal 2022], topological invariants which robustly characterize asymmetric transport along boundaries between phases, for the system. The 9×9 Hamiltonian describing EM waves in cold plasma presents a number of mathematical difficulties when compared to the 2d Dirac equation which is ubiquitous in the study of continuous topological insulators. Ongoing work includes the effects of the regularity of boundaries between topological phases on edge states.
Spectrum of transverse magnetic EM wave propagation in cold plasma containing a jump in the magnetic field.
Numerical Spectral Calculations
In order to validate the theory of topological insulators, we often wish to compute the spectrum of a differential operator. For models of realistic physical systems this can quickly become an intractable problem to solve analytically, so we turn to numerical spectral calculations. Finite difference discretiztions present a number of difficulties which require heuristic elimination of portions of the calculated spectrum and so we have developed a scanning method based on solving an ODE at each point in the energy/wavenumber plane which provides a much more accurate spectrum for continuous systems. The above spectra are outputs of the new method. See [Frazier and Bal 2025] for details.