I am a fourth year graduate student in the department of mathematics at Michigan State University. My research integrates analytic and geometric techniques to study partial differential equations. In particular my research investigates global/finite-in-time existence and stability results to geometric equations that model wave-like phenomena (i.e., relativistic membranes). My advisor is Willie Wong.
I am also working on a project with Jun Kitagawa on regularity theory of generated Jacobian equations.
These are notes that I wrote to prepare for my comprehensive exam. They cover topics that range from the positive energy theorem, global existence for quasilinear wave equations, generalized solutions to the Monge-Ampère, and GWP to the defocussing nonlinear Schrödinger equation.