Scholarship

Most of my technical mathematical work involves problems in geometric analysis and mathematical general relativity. More specifically, I have worked on theory for solving partial differential equations related to the Einstein equations. I have also supervised a large number of undergraduate projects.

Projects with Undergraduate Students

I have supervised projects for undergraduate students on these topics:

• Reaction-diffusion equations. I have supervised several projects related to numerical simulations of reaction-diffusion equations. Alison and Jenny completed a fun project on the Belousov-Zhabotinsky reaction. Sarah and Nuzhat both completed projects related to the FRAP imaging method.

• Discrete analogs of differential equations. I have supervised two different projects related to discretizations of differential equations. Karlie and Sara worked on a project that established solutions to boundary value problems via discrete approximations. Eli and Mack studied flows of polygons that are semi-discrete analogs of the curve shortening flow.

• Parabolic geometric flows. I have supervised two different projects related to parabolic geometric flows. Katie and Adam worked on the curve shortening flow subject to Dirichlet boundary conditions. Colin worked on a variant of the curve shortening flow arising from rotationally-symmetric mean curvature flow.

Papers

• Paul T. Allen, John M. Lee, and David Maxwell. Sobolev-class asymptotically hyperbolic manifolds and the Yamabe problem. Under review. https://arxiv.org/abs/2206.12854

• Paul T. Allen, James Isenberg, John M. Lee and Iva Stavrov Allen. Asymptotic Gluing of Shear-Free Hyperboloidal Initial Data Sets. Annales Henri Poincaré (2021). https://arxiv.org/abs/1912.02839

• 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 Poincaré (2017) 18: 2789. arXiv:1506.05842

• Paul T. Allen, James Isenberg, John M. Lee, Iva Stavrov Allen. Weakly asymptotically hyperbolic manifolds. Communications in Analysis and Geometry, Volume 26 (2018), Number 1. arXiv:1506.03399

• Paul T. Allen, Adam Layne, Katharine Tsukahara. The Dirichlet problem for curve shortening flow. (unpublished) arXiv:1208.3510

• Paul T. Allen, Lars Andersson, Alvaro Restuccia. Local well-posedness for membranes in the light cone gauge. Comm. Math. Phys. 301 (2011), no. 2, 383–410. http://arxiv.org/abs/0910.1488

• Paul T. Allen, Alan D. Rendall. Asymptotics of linearized cosmological perturbations. J. Hyperbolic Differ. Equ. 7 (2010), no. 2, 255–277. http://arxiv.org/abs/0906.2517

• Paul T. Allen, Adam Clausen, James Isenberg. 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. http://arxiv.org/abs/0710.0725

• Paul T. Allen, Lars Andersson, James Isenberg. Timelike minimal submanifolds of general co-dimension in Minkowski space time. J. Hyperbolic Differ. Equ. 3 (2006), no. 4, 691–700. http://arxiv.org/abs/math/0512036

Select Presentations

• April 2026: Talk at Rocky Mountain Section Meeting of the MAA
• September 2023: Talk at Simons Center workshop “Mass, the Einstein Constraint Equations, and the Penrose Inequality Conjecture”
• April 2023: Talk at the American Physical Society April Meeting session “Geometric Analysis and General Relativity”
• March 2020: Local organizer for the Pacific Northwest Geometry Seminar
• October 2019: Geometric Analysis Seminar at University of Oregon
• February 2019: Oregon Liberal Arts Mathematics Colloquium at Linfield College
• January 2019: Colloquium and Topology-Geometry Seminar at Oregon State University
• January 2018: Talk in AMS Special Session on Mathematical Relativity and Geometric Analysis at JMM
• October 2017: Talk at Pacific University
• March 2017: Talk at the Charleston AMS meeting
• February 2017: Short talk to math students at Lewis & Clark
• July 2016: Talk at BIRS workshop “Geometric Analysis and General Relativity”
• March 2016: Colloquium at University of Connecticut
• January 2016: Short talk at the Joint Math Meetings
• Summer 2015: Mathematisches Forschungsinstitut Oberwolfach
• Spring 2015: Lewis & Clark Physics Colloquium
• Fall 2014: Seattle University Mathematics Colloquium
• Summer 2014: University of Science & Technology of China, Geometric Analysis and Relativity
• Spring 2014: Willamette University Physics Colloquium