I am presently a postdoctoral fellow at the University of Toronto. My research interests are in problems related to mathematical general relativity.
I received my Ph.D. in 2012 as a student of Mihalis Dafermos in Cambridge.
Since then I have been a Postdoctoral Fellow at MSRI in Berkeley, where I was mentored by Hans Lindblad, and a Visiting Postdoctoral Research Associate at Princeton. Here in Toronto I work with Spyros Alexakis.
Next year I will be in Paris as a Postdoc de la Fondation Sciences Mathematiques de Paris.
Curriculum Vitae: CV.
Published version: [pdf] (screen) [pdf] (printing)
In this paper I prove several decay statements for solutions to the wave equation on Schwarzschild black hole spacetimes in all dimensions. While the quantitative decay rates had been established in the 3+1-dimensional case, c.f. Luk (2009), the main interest in this paper lies in its method of proof, which uses and extends the "new physical-space approach to decay" of Dafermos and Rodnianski (2009).
Published version: Springer, arXiv: 1207.6055v2
In this paper I develop the global study of linear waves on Kerr de Sitter spacetimes. I am particularly interested here in the so-called cosmological region that is bounded in the past by the cosmological horizons and to the future by a spacelike hypersurface at infinity. It is shown that the expansion of that region provides a stability mechanism that manifests itself in a global redshift effect; moreover this effect persists in a large class of expanding cosmologies near the Schwarzschild de Sitter geometry. Global boundedness and decay results are obtained when our estimates are combined with earlier work concerning the stationary region by Dafermos and Rodnianski (2007) and Dyatlov (2010).
Video: Banff, Geometry and Inverse problems workshop (Spyros Alexakis).
We explore in this paper the question if solutions to wave equations are completely determined from their radiation towards infinity. We are led to show that if the radiation fields of two solutions to a wave equation with suitably fast decaying coefficients conincide to all orders on suitable parts of null infinity, then these solutions indeed coincide in a neighborhood of infinity. The size of this neighborhood depends strongly on the geometry of the asymptotically flat background spacetime; in particular we find that a positive mass of the spacetime works strongly in our favor.
Video: Simons Center for Geometry and Physics (Volker Schlue)
In this paper we address an older question in general relativity: Is it possible for an isolated self-gravitating relativistic system to be in periodic motion? We prove that any asymptotically flat spacetime, which is assumed to be time-periodic and a solution to the Einstein vacuum equations far away from the sources, must be stationary, at least near infinity. Thus genuinely time-periodic solutions do not exist. This problem has been repeatedly studied, first by Papapetrou [Annalen der Physik, 1957], and most recently by Bicak, Scholz and Tod [arXiv:1003.3402], whose approach yields a symmetry "at infinity". Our proof relies crucially on the uniqueness results for linear waves obtained in our previous [3.] for the extension of this symmetry to the spacetime.
Contents: 1. The equivalence principle and its consequences, 2. Einstein's field equations in the presence of matter, 3. Spherical Symmetry, 4. Dynamical Formulation of General Relativity, Slow Motion Approximation, Gravitational Radiation.
Description: A course on general relativity theory for advanced undergraduates and beginning graduate students in mathematics and physics alike. Please refer to the Syllabus for details.
Lectures will be in the winter on Tue & Thu 1-3PM in BA 6183, Tutorials are Wed 4-5PM in BA 4010, and my office hours are Thu 4PM - 5PM in BA 6120.
Course materials: Lecture notes, Problem sets [1: Lorentz transformations] , [2: Receding observers], [3: Tidal forces], [4: Geodesic coordinates], [5: Electromagnetic field], [6: Perfect fluids], [7: Surface gravity], [8: Radiation field]; Projects.
A course on vector calculus with applications in the natural sciences. We follow the Syllabus, but please check black board for frequent announcements. I teach the Thursday 6PM - 9PM session in MP 103, and my office hours are Tuesdays 4PM - 6PM. [Some nice complementary notes at the level of the course on Kepler's Laws can be found on Gilbert Weinstein's website.]
Additional reading: The lecture notes on Sobolev spaces by Willie Wong from the previous year may be useful for this course.
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