Xinliang (Adam) An

Department of Mathematics
University of Toronto
Bahen Centre
40 St.George Street
Toronto, Ontario M5S 2E4

Email: xinliang.an@utoronto.ca

I am a Postdoctoral Fellow in Mathematics at University of Toronto. I got my PhD from Princeton University in 2014. My advisor is Professor Sergiu Klainerman.

Research Interests:

I am interested in geometric PDEs. My main research direction is singularity formation in general relativity. With tools from hyperbolic PDE, quasilinear elliptic equations, geometric analysis, dynamical system and numerical approach, I am working to answer questions arising from physics. Together with my co-authors, we have worked on and provided answers for the following four physical questions:
  • Can “black holes” form dynamically in vacuum? (link)
  • To form a “black hole”, what is the least size of initial data? (link)
  • Can we find the boundary of a “black hole” region? Can we show that a “black hole region” is emerging from a point? (link)
  • For Einstein vacuum equations, could singularities other than black hole type form in gravitational collapse? (link)
Besides general relativity, I am also exploring metastable states and instability mechanism in quantum mechanics. (link)

Papers and Preprints:

  • Polynomial blow-up upper bounds for Einstein-scalar field system, in preparation
  • (with Willie Wong ) Warped product space-times, 31 pages
  • Emergence of apparent horizon in gravitational collapse, 51 pages
  • An iteration scheme in gravitational collapse II, 120 pages, preprint
  • An iteration scheme in gravitational collapse I, 60 pages, preprint
  • (with Avy Soffer ) Fermi’s golden rule and H^1 scattering for nonlinear Klein-Gordon equations with metastable states, 36 pages
  • (with Xuefeng Zhang and Hong Lu) Examples of naked singularity formation in higher-dimensional Einstein-vacuum spacetimes, 16 pages
  • (with Xuefeng Zhang) Naked singularity and black hole formation in self-similar Einstein-scalar fields with exponential potentials, 24 pages
  • (with Jonathan Luk ) Trapped surfaces in vacuum arising dynamically from mild incoming radiation, 93 pages
  • Formation of trapped surfaces in General Relativity, PhD thesis, 309 pages
  • Formation of trapped surfaces from past null infinity, 125 pages

Selected Conference Talks:

  • International Conference on Nonlinear Waves and General Relativity, The Chinese University of Hong Kong, December 11-17, 2017
  • 27th Midwest Relativity Meeting, University of Michigan, October 12-14, 2017
  • The IV Applied Mathematics, Modeling and Computational Science (AMMCS) International Conference, University of Waterloo, August 20-25, 2017
  • International Conference on Evolution Equations (in conjunction with the 31st annual Shanks Lecture), Vanderbilt University, May 16-20, 2016
  • Tsinghua Sanya International Mathematics Forum, Mathematical Sciences Center, Tsinghua University, January 5-9, 2016
  • Nonlinear Wave equations and their numerical study, Fields Institute, June 22-26, 2015
  • Focus Program on 100 Years of General Relativity (Black Hole Stability), Fields Institute, June 8-12, 2015
  • General Relativity and Gravitation: A Centennial Perspective, Pennsylvania State University, June 7-12, 2015
  • Rutgers-CUNY Symposium on Geometric Analysis, CUNY-Rutgers, March 27-29, 2015
  • Mathematical Problems in General Relativity, Simons Center for Geometry and Physics, January 21, 2015

Selected Seminar Talks:

  • Black Hole Initiative Colloquium, Harvard University, November 14, 2017
  • Mathematical Physics Seminar, University of Vienna, April 26, 2017
  • Montreal Analysis Seminar, McGill University, April 7, 2017
  • PDE Seminar, Peking University, May 30, 2016
  • Calderon-Zygmund Analysis Seminar, University of Chicago, Feb 29, 2016
  • New York Area Mathematical General Relativity Seminar, Stony Brook University, December 12, 2014
  • Analysis/PDE Seminar, Johns Hopkins University, January 27, 2014
  • General Relativity Seminar, Columbia University, November 30, 2012

Past Teaching:

  • Fall 2014: Partial Differential Equations
  • Spring 2015: Advanced Calculus I and Honors Calculus III
  • Fall 2015: Advanced Calculus I and Honors Calculus III
  • Spring 2016: Differential Equations for Engineering and Physics
  • Fall 2016: Calculus I
  • Spring 2017: Calculus II