- PDE Seminar, Center for Mathematical Analysis, Geometry, and Dynamical Systems, Instituto Superior Técnico, Lisbon, Portugal, June 15, 2016.
- Analysis and PDE seminar, Imperial College, London, May 13, 2016.
- Oberseminar Geometrische Analysis, Universität Tübingen, February 12, 2016.
- Conference in Mathematical General Relativity, Tsinghua Sanya International Mathematics Forum in Sanya, Hainan, China, January 5-9, 2016. [Poster]
- Geometric Analysis and Partial Differential Equations Seminar, University of Cambridge, November 2, 2015.
- Dynamics of Self-Gravitating Matter, Workshop, Institute Henri Poincaré, Paris, October 26-29, 2015. [Poster]
- Vienna Relativity Seminar, University of Vienna, (and additional talk in the Geometric Analysis and Physics joint seminar of the University of Vienna and the University of Technology), Vienna, October 22, 2015.
- Mathematical Aspects of General Relativity, Workshop, Mathematisches Forschungsinstitut Oberwolfach, July 12-18, 2015. [Extended abstract, Preliminary report]
- Session A1: Mathematical General Relativity, GRG conference: A Centennial Perspective, Penn State, June 8, 2015.
- International Conference on Black Holes, Fields Institute, Toronto, June 2, 2015. [Video]
- Colloquium, University of Miami, February 2, 2015. [Slides]
- Mathematical Problems in General Relativity, Simons Center for Geometry and Physics, January 21, 2015. [Video]
- more ...
- Decay of linear waves on higher dimensional Schwarzschild black holes,
Analysis & PDE 6-3 (2013), 515--600.
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).
- Global results for linear waves on expanding Kerr and Schwarzschild de Sitter cosmologies,
Communications in Mathematical Physics: Volume 334, Issue 2 (2015), 977--1023.
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).
- (with Spyros Alexakis and Arick Shao) Unique continuation from infinity for linear waves, Advances in Mathematics 286 (2016) 481-544.
Published version: Elsevier, arXiv: 1312.1989
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.
- (with Spyros Alexakis) Non-existence of time-periodic vacuum space-times, Journal of Differential Geometry, to appear, 63 pages.
Video: Simons Center for Geometry and Physics (Volker Schlue)
Cf. 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.
- Decay of the Weyl curvature in expanding black hole cosmologies, 124 pages.
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.
- Linear waves on higher dimensional Schwarzschild black holes and Schwarzschild de Sitter spacetimes, Ph.D. Thesis, University of Cambridge (2012), available on DSpace, 223 pages.
- The asymptotics of the gravitational field and the memory effect, Diploma Thesis, ETH Zurich (2008), [expository, after D. Christodoulou, Nonlinear Nature of Gravitation and Gravitational-Wave Experiments, Phys. Rev. Letters (67), no 12, pp. 1486--1489, (1991)], [PDF], 48 pages.
- General Relativity, 153 pages.
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.
- Differential Geometry, based on lectures by Demetrios Christodoulou. [incomplete, last update 01/08: Chapter 2]
- General Relativity (APM426, MAT1700, Spring 2015, and 2013) at U of T.
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.
- Multivariable Calculus (MAT235, Fall 2012 & Spring 2013) at U of T.
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.]
- PDE Tutorial (Fall 2012) at CCA.
Additional reading: The lecture notes on Sobolev spaces by Willie Wong from the previous year may be useful for this course.
- My favorite conference in mathematical general relativity this year was held in tropical climate at the new Tsinghua Sanya International Mathematics Forum in Sanya, Hainan, China, form January 5-9, 2016.
It was organized by Lydia Bieri, Jim Isenberg, and Shing-Tung Yau, and attended by many people: [Group photo]. Notably, Demetri Christodoulou gave an inspiring talk about the formulation of the two-body problem in general relativity, similar to his lecture at the Institut Henri Poincaré in Paris. Moreover Abhay Ashtekar gave a fascinating talk on status of the theory of gravitational radiation with positive cosmological constant.
- A Celebration of the 100th Anniversary was held in Paris as part of the semester program on mathematical general relativity.
- A large conference on General Relativity & Gravitation: A Centennial Perspective was held at Penn State in June 2015. It brought together a broad spectrum of both physicists and mathematicians; most of the plenary sessions we recorded.
- The Fields Institute in Toronto hosted a Focus Program for the Centenary of Einstein's Equations of General Relativity in May & June, 2015. The six weeks (!) long program offered lectures and discussions in essentially all areas of mathematical general relativity, beginning with a workshop on constraint equations (with very instructive survey talks by Mu-Tao Wang and Sergio Dain), followed by two weeks on perturbation methods (with interesting talks on the post-Newtonian approximation by Luc Blanchet). The program of the conference on black holes held inspiring talks by Shing-Tung Yau on the history of mathematical relativity, Tsvi Piran on observational evidence for black holes, and Mihalis Dafermos on the stability of black holes; among many others, see in particular the lectures of Andras Vasy, and Rick Schoen. The remaining three weeks were very actively devoted to the black hole stability problem (also with both positive and negative cosmological constant, as discussed e.g. by Semyon Dyatlov and Claude Warnick), the nature of singularities in general relativity (with interesting talks by Jan Sbierski, Greg Fournodavlos, Jonathan Luk, and Jim Isenberg), and finally non-linear wave equations more generally.
- In January I participated in the Mathematical Problems in General Relativity workshop at the Simons Center for Geometry and Physics in Stony Brook. Many young people attended, among them Sung-Jin Oh and Cecile Huneau who gave great talks.
- In September 2014 I traveled to the Heidelberg Laureate Forum and had a chance to see Michael Atiyah speak about beauty. I thought the lectures by Yoccoz, and and Werner were inspiring.
- In August 2014 I visited Seoul, South Korea, where the International Congress of Mathematicians 2014 was held. It was a conference of enormously broad scope; I thought the lectures by Etienne Ghys, John Milnor, and Manjul Bhargava were fantastic, but really I enjoyed most of the lectures. (Photo with Jonathan Luk and Hans Ringstrom.)
- The 17th Riviere-Fabes Symposium which was held at the University of Minnesota in April 2014 hosted lectures on analysis & PDEs by Alice Chang, Alex Ionescu, Frank Merle, and Maciej Zworski. (Group photo.)
- In January 2014 I attended a conference in honor of Yvonne Choquet Bruhat's 90th birthday. She set the stage for the evolution problem in general relativity in 1952, and remains active more than 60 years later. (Photo of Mme Choquet at the conference.)
- I was very happy to be at MSRI in the fall semester of 2013 for the program on mathematical general relativity. The lectures given in the Initial Data and Evolution Problems workshop were recorded, and provide a great cross-section of current research in general relativity.
- The Oberwolfach Report 37/2012 is a great overview of current research activity in general relativity. It is a collection of extended abstracts of the talks given at the workshop "Mathematical Aspects of General Relativity" which took place at MFO from July 29 -- August 4. There are also some notes taken by Willie Wong.
- The lecture notes of the course by Mihalis Dafermos and Igor Rodnianski at the Clay Mathematics Summer School 2008 are a seminal introduction to the wave equation on black hole spacetimes.
Other Interests and Writings
- I maintain an interest in photography. In recent years I mainly used a Mamiya RB67 for medium format film photography; see my collection of photos taken on trips to Italy, Israel and Korea. Currently, I am using a Fuji X100; follow my photostream on:
- I contributed to the development of the 8pen, a writing system for small devices which received some attention in the media. Here is a short note which describes some of the conceptual ideas behind the 8pen; see also this video, and these blog posts.
- An essay on entropy that I wrote as a student at King's for the John Rose prize, a yearly essay competition for the explanation of a scientific principle.
- In October 2012 I participated in the UNICEF Prof-in-a-box campaign, and my class raised funds for UNICEF Canada. (Photo)
- In July 2012 I graduated from Cambridge University. (Photo with Stefanos Aretakis)