Scholarship

Most of my current mathematical work involves problems in geometric analysis that involve addressing geometric questions by means of studying an associated partial differential equation. The things I spend most of my time thinking about are related to mathematical relativity. I am also interested in applications of (partial) differential equations in a variety of other disciplines, including physics, life-sciences, and economics/finance.

Summer projects with undergraduates

  • Summer 2017: I worked with students Karlie Schwartzwald and Sara Stout on discrete approximations of boundary value problems.
  • Summer 2016: I worked with students Eli Barnes and Mack Beveridge on geometric flows for polygons.
  • Summer 2014: Two students did independent research projects under my supervisition: Colin Gavin worked on mean curvature flow; Sam Stewart worked on numerical models of blowup phenomena for nonlinear waves.
  • Summer 2011: I worked with Alison Fankhauser and Jenny Louthan creating numerical models for reaction-diffusion equations arising in chemistry. Inspired by conversations during this project, I wrote a short introduction to the Fredholm alternative, based on analyzing finite-difference approximations. These notes, in turn, lead to a paper in the College Mathematics Journal.
  • Summer 2010: I worked with Katie Tsukahara and Adam Layne studying the Dirichlet problem for curve shortening flow on spheres.

Papers

Some presentations

A very incomplete list of a few (somewhat recent) talks….

Jim-PNGS