_____________________________________________________________________________________________________________

Organizers

Michael Anderson (Stony Brook)

Sergiu Klainerman (Princeton)

 Philippe G. LeFloch (Paris)

Jared Speck (Cambridge, USA)


Location: Simons Center for Geometry and Physics

Date: One-month concentration period in January 2015

Workshop from January 20 to 23, 2015


Einstein’s field equation of general relativity is one of the most important geometric partial differential equations. Over the past decade, the mathematical research on Einstein equation has made spectacular progress on many fronts, including the Cauchy problem, cosmic censorship, and asymptotic behavior. These developments have brought into focus the deep connections between the Einstein equation and other important geometric partial differential equations, including the wave map equation, Yang-Mills equation, Yamabe equation, as well as Hamilton’s Ricci flow. The field is of growing interest for mathematicians and of intense current activity, as is illustrated by major recent breakthroughs concerning the uniqueness and stability of black hole models, the formation of trapped surfaces, and the bounded L2 curvature problem. The themes of mathematical interest that will be particularly developed in the present Program include the formation of trapped surfaces and the nonlinear interaction of gravitational waves. The new results are based on a vast extension of the earlier technique by Christodoulou and Klainerman establishing the nonlinear stability of the Minkowski space. This Program will be an excellent place in order to present the recent breakthrough on the bounded L2 curvature problem for the Einstein equation, which currently provides the lower regularity theory for the initial value problem, as well as the recently developed theory of weakly regular Einstein spacetimes with distributional curvature.

Long-term participants

Michael Anderson (Stony Brook)

Piotr Chrusciel (Vienna)

Mihalis Dafermos (Princeton)

Cécile Huneau (Paris)

Alexandru D. Ionescu (Princeton)

James Isenberg (Eugene)

Sergiu Klainerman (Princeton)

Philippe G. LeFloch (Paris)

Jared Speck (Cambridge, USA)

Jinhua Wang (Hangzhou)

Mu-Tao Wang (New York)

Qian Wang (Oxford)

Willie Wong (Lausanne)

Speakers during the Workshop

  • Tuesday January 20
    • Sung-Ji Oh (Berkeley) Linear instability of the Cauchy horizon in subextremal Reissner-Nordström spacetime under scalar perturbations
    • Volker Schlue (Toronto) Stationarity of time-periodic vacuum spacetimes
    • Alexandru D. Ionescu (Princeton) The Euler–Maxwell system for electrons: global solutions in 2D
    • Joachim Krieger (Lausanne) Concentration-compactness for the critical Maxwell-Klein-Gordon equation
  • Wednesday January 21
    • Xianliang An (Piscataway) Two results on formation of trapped surfaces
    • Tahvildar-Zadeh (Piscataway) The Dirac electron and the Kerr-Newman spacetime
    • Mihalis Dafermos (Princeton)
    • Jim Isenberg (Eugene) Asymptotically hyperbolic shear-free solutions of the Einstein constraint equations
  • Thursday January 22
    • Cécile Huneau (Paris) Stability in exponential time of Minkowski
    • Jacques Smulevici (Orsay) Vector field methods for transport equations with applications to the Vlasov-Poisson system
    • Mu-Tao Wang (New York) Quasi-local angular momentum and the limit at infinity
    • Spyros Alexakis (Toronto) The Penrose inequality for perturbations of the Schwarzschild exterior
  • Friday January 23
    • Mihai Tohaneanu (Statesboro) Pointwise decay for the Maxwell system on black holes
    • Qian Wang (Oxford)
    • Peter Blue (Edinburgh) Revisiting decay of fields outside a Schwarzschild black hole
    • Philippe G. LeFloch (Paris) Weak solutions to the Einstein equations in spherical or T2 symmetry

Attendees List

_____________________________________________________________________________________________________________

Organizers

 Philippe G. LeFloch (Paris)

Jérémie Szeftel (Paris)

Ghani Zeghib (Lyon)

ANR Project

“Mathematical General Relativity. Analysis and geometry of spacetimes with low regularity”

Wednesday June 17, 2015

Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris

Lecture room 15/25-326

 

11h Emmanuel Hebey (Cergy-Pontoise) Systèmes de Kirchhoff critiques stationnaires sur des variétés compactes

14h  Lydia Bieri (Ann Arbor) Gravitational radiation and two types of memory

Abstract.  We are believed to live on the verge of detection of gravitational waves, which are predicted by General Relativity. In order to understand gravitational radiation, we have to investigate analytic and geometric properties of corresponding solutions to the Einstein equations. Gravitational waves leave a footprint in the spacetime regions they pass, changing the manifold – and therefore displacing test masses – permanently. This is known as the memory effect. It has been believed that for the Einstein equations, being nonlinear, there exists one such effect with a small `linear’ and a large `nonlinear’ part. In this talk, I present some of my joint work with D. Garfinkle showing that these are in fact two different effects.

_____________________________________________________________________________________________________________

Organizers

 Philippe G. LeFloch (Paris)

Jérémie Szeftel (Paris)

Ghani Zeghib (Lyon)

ANR Project

“Mathematical General Relativity. Analysis and geometry of spacetimes with low regularity”

Wednesday May 27, 2015

Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris

Lecture room 15-25–326

 

14h Thierry Barbot (Avignon) Surfaces polygonales fuchsiennes et espace de Teichmüller décoré

Abstract. Dans l’article “Fuchsian polyhedra in Lorentzian space-forms, Mathematische Annalen 350, 2, pp. 417-453, 2011″, F. Fillastre a montré que toute métrique euclidienne avec singularités coniques d’angles > 2 pi sur une surface compacte se réalise de manière unique comme une surface de Cauchy polygonale dans un espace-temps globalement hyperbolique localement plat radial (i.e. dont le groupe d’holonomie fixe un point de l’espace de Minkowski). Dans cet exposé, j’évoquerai le travail de L. Brunswic dans son travail de thèse sous ma direction, qui vise à reprouver ce résultat et à l’étendre au cas des surfaces polygonales dans un espace-temps localement plat mais admettant des particules massives. Le but est de montrer qu’il y a encore existence et unicité une fois prescrit la masse des particules massives (le cas régulier montré par Fillastre correspondant au cas où l’angle singulier des particules massives est 2pi). Je montrerai aussi que la situation étudiée par R. Penner dans l’article “The Decorated Teichmϋller Space of Punctured Surfaces, Commun. Math. Phys. 113, 299-339 (1987)” est un cas limite de la situation étudiée par Brunswic, et correspond au cas où les particules sont d’angle conique nul. Je montrerai aussi comment répondre positivement à la question dans le cas où il n’y a qu’une singularité.

15h30 Andrea Seppi (Pavia) Convex surfaces in (2+1)-dimensional Minkowski space

Abstract.  It is known that the hyperbolic plane admits an isometric embedding into Minkowski space; in 1983 Hanu and Nomizu first observed the existence of non-equivalent isometric embeddings, thus showing a relevant difference with the Euclidean case. In this talk, I will introduce some natural properties of a convex surface in Minkowski space, concerning causality and asymptotic behavior. I will then explain some new results (jointly with Francesco Bonsante) on the classification of constant curvature surfaces with bounded principal curvatures and on the solvability of Minkowski problem in (2+1)-dimensional Minkowski space. If time permits, I will give the main ideas of the proof and especially the relation to some type of Monge-Ampere equations.

_____________________________________________________________________________________________________________

Organizers

 Philippe G. LeFloch (Paris)

Jérémie Szeftel (Paris)

Ghani Zeghib (Lyon)

ANR Project

“Mathematical General Relativity. Analysis and geometry of spacetimes with low regularity”

Wednesday April 15, 2015

Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris

Lecture room 15-25–326

 

14h Shiwu Yang (Cambridge) Decay properties of solutions of Maxwell Klein-Gordon equations

Abstract.  I will present some recent progress on the asymptotic behavior of global solutions to Maxwell-Klein-Gordon equations. I will show that the integrated local energy and the energy flux through the outgoing null hypersurfaces decays polynomially in the retarded time in Minkowski space with data merely bounded in some gauge invariant weighted Sobolev space. This in particular includes the case with large charge. One novelty of this work is that these decay estimates precisely capture the asymptotic properties for the non-linear fields with arbitrarily large data. If in addition that the initial data for the scalar field is sufficiently small, then we show the pointwise decay of the solutions. This result improves the previous result of Lindblad and Sterbenz in which smallness is required for both the scalar field and the Maxwell field.

15h30 Gustav Holzegel (London) Local and global dynamics in asymptotically anti de Sitter spacetimes

Abstract.  Asymptotically anti de Sitter (aAdS) spacetimes play a prominent role in theoretical physics and mathematics.  Due to the presence of a timelike hypersurface at infinity these spacetimes are not globally hyperbolic, a fact that leads to intricate initial boundary value problems when studying global dynamics of hyperbolic equations on these backgrounds. In this talk, I will present several local and global results for the massive wave equation on aAdS spacetimes (including black hole spacetimes) with emphasis on how different boundary conditions (Dirichlet, Neumann or dissipative) influence the global dynamics. In particular, I will outline a recent proof (obtained in collaboration with J. Luk, J. Smulevici and C. Warnick) of linear stability and decay for gravitational perturbations on anti de Sitter space under dissipative boundary conditions. The proof unravels an interesting trapping phenomenon near the conformal boundary which necessarily leads to a degeneration in the decay estimates. Time permitting some future applications will also be discussed.

_____________________________________________________________________________________________________________

Organizers

 Philippe G. LeFloch (Paris)

Jérémie Szeftel (Paris)

Ghani Zeghib (Lyon)

ANR Project

“Mathematical General Relativity. Analysis and geometry of spacetimes with low regularity”

Wednesday March 4, 2015

Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris

Lecture room 15-25–326

 

14h François Fillastre (Cergy-Pontoise) Minkowski problem in Minkowski space

Abstract. T. Barbot, F. Beguin and A. Zeghib solved a smooth Lorentzian version of the Minkowski problem in dimension (2+1). More precisely they proved that if M is a flat 3-dimensional maximal globally hyperbolic spatially compact spacetime, then there exists a unique strictly convex smooth space-like surface in M with a prescribed smooth positive Gauss curvature. We will look at this problem for any dimensions. The existence part is solved in a generalized way (a measure is prescribed rather than a function). Concerning the regularity of the solution, the 2+1 case is specific. The arguments are based on tools from the geometry of convex sets. Joint work with Francesco Bonsante (Pisa).

Philippe LeFloch, DIRECTOR OF RESEARCH AT CNRS Email address: pglefloch [at] gmail.com

IHP PROGRAM

Enter your email address to follow this blog and receive notifications of new posts by email.

Join 77 other followers

Archives