Committee on Computational and Applied Mathematics
University of Chicago
mjfrazer (at) uchicago (dot) edu
Aspiring physicist/applied mathematician passionate about applying advanced methods in partial differential equations to solve problems in mathematical physics and materials science. Current research interests involve topological insulators and applications of topological physics to cold plasma. Also amateur cellist, avid outdoorsman, and former Naval officer.
Current Projects:
Topological Properties of Cold Magnetized Plasma
Applying methods from topological insulators- novel electronic materials with edge states protected from scattering by their topological properties- to cold magnetized plasma. At least one protected edge state has already been demonstrated (Topological Langmuir Cyclotron Wave-TLCW)[Qin & Fu, Science Advances 2023] , and my goal is to fully characterize the topological properties of the system through Chern numbers and Bulk Difference Invariants (BDI) and establish conditions when the Bulk-Edge Correspondence (BEC) exists in cold plasma. Analytic calculations of Chern numbers for the entire system confirm past numerical results and show that the difference in Chern numbers for the TLCW corresponds to the number of edge states. Link to current results:
Numerical calculations of the spectra of two edge separate edge states in cold magnetized plasma. Right shows the TLCW.
Scanning Methods for Eigenvalue Calculations in Topological Physics
Current numerical calculations of spectra for systems in topological physics rely on applying a finite-difference scheme to the associated ODE formed by taking the Fourier transform in time and all but one spatial variable and applying a standard eigenvalue solver. Although used with some success in multiple systems, this method often involves somewhat arbitrary boundary conditions and requires heuristic elimination of eigenvalues which spuriously arise as a result. I have been experimenting with methods that instead can calculate eigenvalues and their associated mode (edge or bulk) by scanning values of energy and parallel wave number with exact boundary conditions and no need for elimination of eigenvalues. Recent result using atmospheric waves shown below:
Past Projects:
Crystal Orientation Mapping in Carbonate Biominerals
Developed a method for mapping crystal orientation of Carbonate (CaCO3) grains using optical microscopy by extending the use of the Liquid-Crystal (LC)-Polscope to high retardance samples. Experimental setup requires taking data with LC-PolScope at multiple wavelengths and applying post-processing methods I developed using MATLAB, hence it is called Multi-wavelength PolScope (MPS) technique. Code available on request. Part of my work within the Gilbert Group at UW-Madison studying coral skeletons and other marine biominerals.
