My work is focused on the analysis of geometric hyperbolic partial
differential equations. More specifically, I am interested in the
Cauchy problem for the Einstein equations, in relation with the
following topics
Vacuum Einstein equations, Einstein-Vlasov, Einstein-Klein-Gordon system,
Structure and formations of singularities in symmetric solutions, loss
of global hyperbolicity,
Expanding solutions with weak regularity and their asymptotics,
Asymptotically Anti-de-Sitter solutions, linear wave equations on
asymptotically Anti-de-Sitter solutions, stability of asymptotically Anti-de-Sitter solutions.
Originally motivated by the study of the Einstein-Vlasov system, I have recently started working more on transport equations, including the study of linear kinetic transport operators and the Vlasov-Poisson or the Vlasov-Norström systems.
In collaboration with David Fajman and Jérémie Joudioux, The stability of the Minkowski space for the Einstein-Vlasov system 139 pages, July 2017, preprint available at arXiv:1707.06141.
In collaboration with David Fajman and Jérémie Joudioux, Sharp asymptotics for small data solutions of the Vlasov-Nordström system in three dimensions 74 pages, April 2017, preprint available at arXiv:1704.05353.
In collaboration with Gustav Holzegel, Jonathan Luk and Claude Warnick, Asymptotic properties of linear field equations in anti-de Sitter space, 56 pages, February 2015, preprint available at arXiv:1502.04965.
Publications
In collaboration with David Fajman and Jérémie Joudioux, A vector field method for relativistic transport equations with applications, accepted for publications in APDE, preprint available at arXiv:1510.04939.
In collaboration with Phillipe G. LeFloch, Future asymptotics and geodesic completeness of polarized T2-symmetric spacetimes., Anal. PDE 9 (2016), no. 2, 363-395.
In collaboration with Gustav Holzegel, Quasimodes and a Lower Bound on the Uniform Energy Decay Rate for Kerr-AdS Spacetimes, Anal. PDE 7 (2014), no. 5, 1057–1090.