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

  • Paul T. Allen. Boundary Value Problems and Finite Differences. College Math Journal 47 (2016), no. 1, 34 – 41.
  • Paul T. Allen, James Isenberg, John M. Lee, Iva Stavrov Allen. The shear-free condition and constant-mean-curvature hyperboloidal initial data. Classical and Quantum Gravity, Volume 33, Number 11.  arXiv:1506.06090.
  • Paul T. Allen and Iva Stavrov Allen. Smoothly compactifiable shear-free hyperboloidal data is dense in the physical topology. Annales Henri Poincare (2017) 18: 2789. https://doi.org/10.1007/s00023-017-0565-2 (arXiv:1506.05842).
  • Paul T. Allen, James Isenberg, John M. Lee, Iva Stavrov Allen. Weakly asymptotically hyperbolic manifolds. arXiv:1506.03399. Communications in Analysis and Geometry, Volume 26 (2018), Number 1.
  • Paul T. Allen, Adam Layne, Katharine Tsukahara. The Dirichlet problem for curve shortening flow. arXiv:1208.3510
  • Allen, Paul T.; Andersson, Lars; Restuccia, Alvaro. Local well-posedness for membranes in the light cone gauge. Comm. Math. Phys. 301 (2011), no. 2, 383–410.arXiv, MathSciNet
  • Allen, Paul T.; Rendall, Alan D. Asymptotics of linearized cosmological perturbations. J. Hyperbolic Differ. Equ. 7 (2010), no. 2, 255 – 277.arXiv, MathSciNet
  • Allen, Paul T.; Clausen, Adam; Isenberg, James Near-constant mean curvature solutions of the Einstein constraint equations with non-negative Yamabe metrics. Classical Quantum Gravity 25 (2008), no. 7, 075009, 15 pp.arXiv, MathSciNet
  • Allen, Paul; Andersson, Lars; Isenberg, James Timelike minimal submanifolds of general co-dimension in Minkowski space time. J. Hyperbolic Differ. Equ. 3 (2006), no. 4, 691 – 700.arXiv, MathSciNet

Some presentations

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

Jim-PNGS