Academic Appointments


Administrative Appointments


  • Professor, Department of Mathematics, Stanford University (2008 - Present)
  • Associate Professor, Department of Mathematics, Stanford University (2005 - 2008)
  • Visiting Associate Professor, Department of Mathematics, Northwestern University (2004 - 2005)
  • Associate Professor, Department of Mathematics, Massachusetts Institute of Technology (2003 - 2006)
  • Enseignant invité, Département de Mathématiques, Université de Nantes (2002 - 2002)
  • Local organizer, Programme in Scattering Theory, Erwin Schrödinger Institute, Vienna, Austria (2001 - 2001)
  • Assistant Professor, Department of Mathematics, Massachusetts Institute of Technology (1999 - 2003)
  • Morrey Assistant Professor, Department of Mathematics, University of California, Berkeley (1997 - 2000)
  • Teaching Assistant, Department of Mathematics, Massachusetts Institute of Technology (1993 - 1996)
  • Alfred P. Sloan Research Fellowship, Alfred P. Sloan Foundation (2002 - 2004)
  • Alfred P. Sloan Doctoral Dissertation Fellowship, Alfred P. Sloan Foundation (1996 - 1997)
  • Clay Research Fellowship, Clay Mathematics Institute (2004 - 2006)
  • Chambers Fellowship, Stanford University (2008 - 2009)

Honors & Awards


  • Elected as a member, American Academy of Art and Sciences (2019)
  • Simons Fellowship, Simons Foundation (2017-2018)
  • Bocher Prize, American Mathematical Society (2017)
  • Invited speaker for the Partial Differential Equations section, International Congress of Mathematicians, Seoul (2014)
  • Clay Research Fellowship, Clay Mathematics Institute (2004-2006)
  • Alfred P. Sloan Research Fellowship, Alfred P. Sloan Foundation (2002-2004)

Boards, Advisory Committees, Professional Organizations


  • Director of Graduate Studies, Stanford University, Department of Mathematics (2007 - 2010)
  • Co-organizer, Programme on the ‘Modern Theory of Wave Equations’, Erwin Schrödinger Institute, Vienna, Austria
  • Co-organizer, ‘A conference on Inverse Problems in honor of Gunther Uhlmann’, University of California, Irvine (2012 - 2012)
  • Co-organizer, conference ‘Microlocal methods in spectral and scattering theory’, Northwestern University (2011 - 2011)
  • Co-organizer, conference ‘Microlocal methods in mathematical physics and global analysis’, Tübingen, Germany (2011 - 2011)
  • Co-organizer, conference ‘From wave propagation to K-theory’, Stanford University (2008 - 2008)
  • Co-organizer, semester-long Program on Analysis of Singular Spaces, Mathematical Sciences Research Institute, Berkeley, CA (2008 - 2008)
  • Co-organizer, conference on Spectral Theory and Global Analysis, University of Oldenburg, Germany (2006 - 2006)
  • Co-organizer, Scattering Theory and Singular Spaces, Northwestern University (2005 - 2005)
  • Co-organizer, conference on Scattering Theory and Singular Spaces, Northwestern University (2005 - 2005)
  • Co-organizer, Geometric Analysis: a conference in Honor of Richard Melrose, Massachusetts Insitute of Technology (2002 - 2002)
  • Local organizer, semester-long Programme in Scattering Theory, Erwin Schrödinger Institute, Vienna, Austria (2001 - 2001)
  • Member, Natural Sciences Curriculum Review Committee, Stanford University, School of Humanities and Sciences (2012 - 2013)
  • Editor, Analysis & PDE (journal) (2007 - Present)

Professional Education


  • Ph.D., Massachusetts Institute of Technology, Mathematics (1997)
  • M.S., Stanford University, Mathematics (1993)
  • B.S., Stanford University, Physics (1993)

Current Research and Scholarly Interests


My research concentrates on topics in two broad areas of applications of microlocal analysis in which, partly with collaborators, I introduced new ideas in recent years: non-elliptic linear and non-linear partial differential equations (PDE), typically concerning wave propagation or other related phenomena, and inverse problems for X-ray type transforms along geodesics and related problems for determining the metric tensor from boundary measurements.

Some of my work describes the long-time or far field behavior, including existence, of waves on curved space-times. Physically these arise in general relativity, including electromagnetic waves on a curved background. The microlocal approach to analysis on these spaces has made breakthroughs possible on linear and non-linear problems on asymptotically hyperbolic (AH) spaces as well as Kerr-de Sitter (KdS) space (rotating black holes in a cosmological spacetime), culminating in the proof of the stability of slowly rotating KdS spaces with Peter Hintz. More recently, with Dietrich Hafner and Peter Hintz, I extended some of these tools to the vanishing cosmological constant case (Minkowski, Kerr). Ongoing work aims to extend these tools to further spaces, such as perturbations of Kerr spacetimes, and also to study other equations on cosmological spacetimes. Other projects study basic objects in quantum field theory, in particular the Feynman propagator.

Another main area of my interest is inverse problems, where, together with Gunther Uhlmann, I introduced new tools for the spatially localized inversion of the geodesic X-ray transform, and with Plamen Stefanov and Gunther Uhlmann we extended this to the boundary rigidity problem, namely determining a Riemannian metric on a manifold with boundary from the length of geodesic segments connecting boundary points. Ongoing work involves extensions of this to anisotropic elasticity which plays an important role in the interior of the Earth. Another project with Yiran Wang studies the light ray transform with potential applications to imaging by the cosmic background radiation.

2021-22 Courses


Stanford Advisees


All Publications