/fɞlkʰœ ʃluɘ/
Laboratoire Jacques-Louis LionsI am currently a postdoc in the Laboratoire Jacques-Louis Lions at Université Pierre et Marie Curie (Paris 6), where I work with Jérémie Szeftel. 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 the University of Toronto, and at the Mathematical Sciences Research Institute in Berkeley. In Toronto, I worked with Spyros Alexakis, while at MSRI, I was mentored by Hans Lindblad. In 2014 I was also a Visiting Postdoctoral Research Associate at Princeton, and last year a postdoctoral laureate of the Fondation Sciences Mathématiques de Paris.
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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).
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).
Elsevier 1312.1989
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.
1504.04592
Simons Center for Geometry and Physics (Volker Schlue)
Oberwolfach Report: Extended abstract, 3 pages.
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.
This paper is the first in a series on the non-linear stability problem of the expanding region of Schwarzschild de Sitter cosmologies. Recently Hintz and Vasy (2016) proved the non-linear stability of Kerr de Sitter black hole exteriors on a domain bounded by the cosmological horizon. This paper approaches the characteristic initial value problem beyond the cosmological horizon. From my paper [2.] - which developed the relevant linear theory - we cannot expect the solutions to converge to the Kerr de Sitter geometry at future null infinity, but merely to a nearby geometry which is however a priori unknown. Here we show - under assumptions which are consistent with this expectation - that nonetheless the Weyl curvature of spacetime decays uniformly towards the future boundary at infinity. One expects this to be an essential part of a global existence result for the Einstein equations with positive cosmological constant in this region.
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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