Organizers

Lars Andersson (Potsdam)

Sergiu Klainerman (Princeton) 

Philippe G. LeFloch (Paris) 

September 14, 2015 to December 18, 2015

 

Main Themes of the Program

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 (Cauchy problem, cosmic censorship, asymptotic behavior). These developments have brought into focus the deep connections between the Einstein equation and other important geometric PDE’s, including the wave map equation, Yang-Mills equation, Yamabe problem, 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 breakthrough, concerning the uniqueness and stability of the Kerr black hole model, the formation of trapped surfaces, and the bounded L2 curvature problem. Specifically, the themes of mathematical interest that will be developed in the present Program and are currently most active include:

  •  The initial value problem for Einstein equation and the causal geometry of spacetimes with low regularity, formation of trapped surfaces
  • Techniques of Lorentzian geometry: injectivity radius estimates, geometry of null cones; construction of parametrices
  •  Geometry of black hole spacetimes: uniqueness theorems, censorship principles
  • Coupling of Einstein equation for self-gravitating matter models, weakly regular spacetimes with symmetry

Main Events

  • Sept. 14 to 18, 2015: Summer School INTRODUCTION TO MATHEMATICAL GENERAL RELATIVITY 

SPEAKERS (partial list):

Greg Galloway (Miami)

Gerhard Huisken (Tuebingen)

Alexandru Ionescu (Princeton)

Hans Ringstrom (Stockholm)

  • Sept. 23 to 25, 2015: Workshop MATHEMATICAL PROPERTIES OF SPACETIMES AND BLACK HOLES

SPEAKERS (partial list):

Greg Galloway  (Miami) *

Alexandru Ionescu (Princeton)

Hans Ringstrom (Stockholm)

  • Oct.: Workshop STATE OF THE ART ON BLACK HOLES

(This week hold in Montpelier and organized by Marc Herzlich and Erwann Delay)

SPEAKERS (partial list):

Piotr Chrusciel (Vienna)

Michael Eichmair (Zürich)

Mu-Tao Wang (New York)

  • Oct. 26 to 29, 2015: Workshop SELF-GRAVITATING MATTER MODELS
  • Nov. 16 to  20, 2015: International Conference MATHEMATICAL GENERAL RELATIVITY – A Celebration of the 100th Anniversary of General Relativity  

SPEAKERS (partial list):

Jean-Pierre Bourguignon (IHES)

Pieter Blue (Edinburgh)

Matthew Choptuik (Vancouver)

Demetrios Christodoulou (Zürich and Athens)

Mihalis Dafermos (Princeton)

Thibault Damour (Bures-sur-Yvette)

Richard Hamilton (New York) *

Jonathan Luk (Cambridge, US)

Juan Maldacena (Princeton) *

Roger Penrose (Oxford)

Igor Rodnianski (Cambridge) *

Richard Schoen (Stanford)

Jacques Smulevici (Orsay)

John Stachel (Boston)

Jérémie Szeftel (Paris)

Robert Wald (Chicago)

Qian Wang (Oxford)

* (to be confirmed)

  • Dec. 14 to 16, 2015: International Conference to CELEBRATE THE 100TH ANNIVERSARY OF THE BIRTH OF ANDRÉ LICHNEROWICZ 

(This week organized by Richard Kerner and Yvette Kosmann-Schwarzbach)


Program coordinated by the Centre Emile Borel at IHP. Financial support provided by the Institut Henri Poincaré and the ANR Project “Mathematical General Relativity. Analysis and geometry of spacetimes with low regularity”.
 
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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 Nov. 26, 2014

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

Lecture room 15-25–104

 


14h Qian Wang (Oxford)  A geometric approach to the sharp local well-posedness theory for quasilinear wave equations

Abstract. The commuting vector fields approach, devised for Strichartz estimates by Klainerman, was employed for proving the local well-posedness in the Sobolev spaces H^s with s>2+(2-\sqrt 3)/2 for general quasi-linear wave equation in (1+3) spacetime by him and Rodnianski. Via this approach they obtained the local well-posedness with s>2 for (1+3) vacuum Einstein equations. A proof of the sharp H^2+ local well-posedness result for general quasilinear wave equation was provided by Smith and Tataru by constructing a parametrix using wave packets. The difficulty of the problem is that one has to face the major hurdle caused by the Ricci tensor of the metric for the quasi-linear wave equations. I will present my recent work, which proves the sharp local well-posedness of general quasilinear wave equation in (1+3) spacetime by a vector field approach, based on geometric normalization and new observations on the mass aspect functions.

15h30 Jonathan Luk (Cambridge, UK) Stability of the Kerr Cauchy horizon

Abstract. The celebrated strong cosmic censorship conjecture in general relativity in particular suggests that the Cauchy horizon in the interior of the Kerr black hole is unstable and small perturbations would give rise to singularities. We present a recent result proving that the Cauchy horizon is stable in the sense that spacetime arising from data close to that of Kerr has a continuous metric up to the Cauchy horizon. We discuss its implications on the nature of the potential singularity in the interior of the black hole. This is joint work with Mihalis Dafermos.

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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 Sept. 17, 2014

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

Lecture room 15-25-321

 


14h Arick Shao (Imperial College) Unique continuation from infinity for linear waves

Abstract. We prove various unique continuation results from infinity for linear waves on asymptotically flat space-times. Assuming vanishing of the solution to infinite order on suitable parts of future and past null infinities, we derive that the solution must vanish in an open set in the interior. The parts of infinity where we must impose a vanishing condition depend strongly on the background geometry; in particular, for backgrounds with positive mass (such as Schwarzschild or Kerr), the required assumptions are much weaker than in Minkowski spacetime. These results rely on a new family of geometrically robust Carleman estimates near null cones and on an adaptation of the standard conformal inversion of Minkowski spacetime. Also, the results are nearly optimal in many respects. This is joint work with Spyros Alexakis and Volker Schlue.

15h30 Claude Warnick (Warwick)  Dynamics in anti-de Sitter spacetimes

Abstract. When solving Einstein’s equations with negative cosmological constant, the natural setting is that of an initial-boundary value problem. Data is specified on the timelike conformal boundary as well as on some initial spacelike hypersurface. Questions of local well-posedness and global stability are sensitive to the choices of boundary conditions. I will present recent work exploring the effects of non-trivial boundary data for the asymptotically AdS initial-boundary value problem, including a recent result in collaboration with Holzegel. I will also outline some interesting open problems in the area.

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Seminar on

Mathematical General Relativity

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 25, 2014

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

Lecture room 1525-321

 


14h Jan Sbierski (Cambridge, UK) A Zorn-free proof of the existence of a maximal Cauchy development for the Einstein equations

Abstract. In 1969, Choquet-Bruhat and Geroch showed that there exists a unique maximal Cauchy development of given initial data for the Einstein equations. Their proof, however, has the unsatisfactory feature that it relies crucially on the axiom of choice in the form of Zorn’s lemma. In particular, their proof ensures the existence of the maximal development without actually constructing it. In this talk, we present a proof of the existence of a maximal Cauchy development that avoids the use of Zorn’s lemma and, moreover, provides an explicit construction of the maximal development.

15h15 Sergiu Klainerman (Princeton)  Remarks on the stability of the Kerr solution in axial symmetry

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Seminar on

Mathematical General Relativity

Organizers:

 Philippe G. LeFloch (Paris)

Jérémie Szeftel (Paris) 

Ghani Zeghib (Lyon)

ANR Project

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


May 28, 2014

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

Lecture room  1525-321

 


14h Erwann Delay (Avignon) A study of some curvature operators near the Euclidian metric

Abstract. We will show that some curvature operators of Ricci (or Einstein) type are locally invertible, in some weighted Sobolev spaces on Rn, near the euclidian metric. In the smooth case, we then deduce that the image of some Riemann-Christoffel type operators are smooth submanifolds in the neighborhood of the Euclidian metric.

15h30 Mahir Hadzic (London) Stability problem in the dust-Einstein system with a positive cosmological constant

Abstract. The dust-Einstein system models the evolution of a spacetime containing a pressureless fluid, i.e. dust. We will show nonlinear stability of the well-known Friedman-Lemaitre-Robertson-Walker (FLRW) family of solutions to the dust-Einstein system with positive cosmological constant. FLRW solutions represent initially a quiet fluid evolving in a spacetime undergoing accelerated expansion. We work in a harmonic-type coordinate system, inspired by prior works of Rodnianski and Speck on Euler-Einstein system, and Ringstrom’s work on the Einstein-scalar-field system. The main new mathematical difficulty is the additional loss of one degree of differentiability of the dust matter. To deal with this degeneracy, we commute the equations with a well-chosen differential operator and derive a family of elliptic estimates to complement the high-order energy estimates. This is joint work with Jared Speck.

 
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Seminar on

Mathematical General Relativity

Organizers:

 Philippe G. LeFloch (Paris)

Jérémie Szeftel (Paris) 

Ghani Zeghib (Lyon)

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ANR Project

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


February 12, 2014

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

Lecture room  1525-103


14h  Florian Beyer  (Dunedin)  Graceful exit from inflation for minimally coupled Bianchi A scalar field models

Abstract. We consider the dynamics of Bianchi A scalar field models which undergo inflation. The main question is under which conditions does inflation come to an end and is succeeded by a decelerated epoch. This so-called ‘graceful exit’ from inflation is an important ingredient in the standard model of cosmology, but is, at this stage, only understood for restricted classes of solutions. We present new results obtained by a combination of analytical and numerical techniques.

15h30  Cécile Huneau (ENS, Paris)  Vacuum constraint equations for asymptotically flat space-times with a translational Killing field

Abstract. In the presence of a space-like translational Killing field, vacuum Einstein equations in 3+1 dimensions reduces to 2+1 Einstein equations with a scalar field. Minkowski space-time is a trivial solution of vacuum Einstein equation with a translational Killing field. A natural question is therefore the nonlinear stability of Minkowski solution in this setting. A first step in addressing this problem is the study of the constraint equations. The main difficulty in that case is due to the delicate inversion of the Laplacian. In particular, we have to work in the non constant mean curvature setting, which enforces us to consider the intricate coupling of the Einstein constraint equations.

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9th DFG–CNRS WORKSHOP

 Micro-Macro Modeling and Simulation

of Liquid-Vapor Flows

organized with financial support from DFG and CNRS

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Tuesday February 25, 2014 at 1:30pm

to

Thursday February 27, 2014 at 1:00pm

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Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 4, Place Jussieu, Paris.

Subway station: Jussieu

Lecture room 15-16 — 309

Schedule and abstracts here !

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INVITED SPEAKERS

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Main organizer

Philippe G. LeFloch (Paris)

Co-organizers

Benjamin Boutin (Rennes)

Frédéric Coquel (Palaiseau)

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PRACTICAL INFORMATIONS 

How to come to the Laboratoire Jacques-Louis Lions

Hotels near the University Pierre et Marie Curie

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EARLIER WORKSHOPS “Micro-Macro Modeling and Simulation of Liquid-Vapor Flows”

Eight Workshop, Berlin, February 2013

Seventh Workshop, Paris, February 2012

Sixth Workshop, Stuttgart, January 2011

Fifth Workshop, Strasbourg, April 2010

Fourth Workshop, Aachen, February 2009

Third Workshop, Strasbourg, January 2008

Second Workshop, Bordeaux, November 2007

Opening Workshop, Kirchzarten, November 2005

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Conference on

Nonlinear Wave Equations at IHP

Organizers: 

 Sergiu Klainerman (Princeton)

Philippe G. LeFloch (Paris)

Jérémie Szeftel (Paris)

Fondations des Sciences Mathématiques de Paris

ANR Project

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

May 21 to May 24, 2013

Institut Henri Poincaré, Paris

Schedule available here

Further informations available here

Poster of the conference  here

INVITED SPEAKERS

Lars Andersson (Potsdam)

Stefanos Aretakis (Princeton)

Nicolas Burq (Paris-Sud)

Pieter Blue (Edinburgh)

Mihalis Dafermos (Princeton)

Jean Marc Delort (Paris-Nord)

Gustav Holzegel (London)

Alexandru Ionescu (Princeton)

Joachim Krieger (EPFL)

Jonathan Luk (UPenn)

Franck Merle (Cergy & IHES)

Sung-Jin Oh (Princeton)

Fabrice Planchon (Nice)

Pierre Raphael (Nice)

Igor Rodnianski (MIT)

Chung-Tse Arick Shao (Toronto)

Jacques Smulevici (Paris-Sud)

Jacob Sterbenz (San Diego)

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Seminar on

Mathematical General Relativity

Organizers:

 Philippe G. LeFloch (Paris)

Ghani Zeghib (Lyon)

ANR Project

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


February 20, 2013

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

Lecture room  1525-103

 


14h  Florian Beyer  (Dunedin)  Asymptotics and conformal structures of solutions to Einstein’s field equations

Abstract. Roger Penrose’s idea that the essential information about the asymptotics of solutions of the Einstein’s field equations is contained in the conformal structure and the associated conformal boundary has led to astonishing successes. In his original work, he provided several examples which made the importance of his idea evident. However, the question whether general solutions of Einstein’s field equations are compatible with this proposal remained unanswered. Motived by this, Helmut Friedrich has initiated a research programme to tackle this problem based on his so-called conformal field equations. In this talk I report on the status of this work and some of Friedrich’s results, but also on joint work with  collaborators at the University of Otago.

15h30  Julien Cortier (IHES, Bures-sur-Yvette)  On the mass of asymptotically hyperbolic manifolds

Abstract. By analogy with the ADM mass of asymptotically Euclidean manifolds, a set of global charges can be defined for asymptotically hyperbolic manifolds. We will review their various definitions and , in particular, focus on the notion of “mass aspect” tensor, which gives rise to the  energy-momentum vector and arises  in the hyperbolic formulation of the positive mass theorem. We will compute these quantities for examples such that the Schwarzschild-anti de Sitter metrics, and we will present a family of counter-examples with “non-positive” mass when completeness is not assumed.

 
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Seminar on

Mathematical General Relativity

Organizers: 

 S. Klainerman (Princeton)

P.G. LeFloch (Paris)

A. Zeghib (Lyon)

Fondations des Sciences Mathématiques de Paris

ANR Project

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

Thursday January 17, 2013

Laboratoire J-L Lions

Université Pierre et Marie Curie, Paris

Lecture room (see below)

.

 11h (Room  15-25- 104)    Sergiu Klainerman (Princeton)   On  the formation of trapped surfaces

Abstract. I will talk about a new result obtained in collaboration with J. Luk and I. Rodnianski in which we relax significantly Christodoulou’s main condition for the formation of trapped surfaces in vacuum.

 14h (Room 15-25-326) Chung-Tse Arick Shao (Toronto)   Null cones to infinity, curvature flux, and Bondi mass

Abstract. In general relativity, the Bondi mass in an asymptotically flat spacetime represents, roughly, the mass remaining in the system after some has radiated away. In this talk, we make sense of and control the Bondi mass for a single null cone in an Einstein-vacuum spacetime under minimal assumptions. In terms of regularity, we assume only small weighted curvature flux along the null cone and small data on an initial sphere of the cone. Furthermore, we make no global assumptions on the spacetime, as all our conditions deal only with the single null cone under consideration. This work is joint with S. Alexakis.

15h30  (Room 15-25-326)  Gustav Holzegel  (Princeton) Existence of dynamical vacuum black holes

Abstract. This is joint work with Mihalis Dafermos and Igor Rodnianski. We prove the existence of a large class of non-stationary vacuum black holes whose exterior geometry asymptotes in time to a fixed Schwarzschild or Kerr metric. The spacetimes are constructed by solving a backwards scattering problem for the vacuum Einstein equations with characteristic data prescribed on the horizon and at null infinity. The data admits the full functional degrees of freedom to specify data for the Einstein equations. An essential feature of the construction is that the solutions converge to stationarity exponentially fast with their decay rate intimately related to the surface gravity of the horizon and hence to the strength of the celebrated redshift effect which, in our backwards construction, is seen as a blueshift.

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Seminar on

Mathematical General Relativity

Organizers:

 Philippe G. LeFloch (Paris)

Ghani Zeghib (Lyon)

ANR Project

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


Friday December 21, 2012

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

Lecture room  1525-3-21


Speaker


11h15 –  José A. Font  (Valencia)  Simulations of neutron star mergers and black hole-torus systems

Abstract. Merging binary neutron stars are among the strongest sources of gravitational waves and have features compatible with the events producing short–hard gamma-ray bursts. Numerical relativity has reached a stage where a complete description of the inspiral, merger and post-merger phases of the late evolution of binary neutron star systems is possible. This talk presents an overview of numerical relativity simulations of binary neutron star mergers and the evolution of the resulting black hole–torus systems. Such numerical work is based upon a basic theoretical framework which comprises the Einstein’s equations for the gravitational field and the hydrodynamics equations for the evolution of the matter fields. The most well-established formulations for both systems of equations are briefly discussed, along with the numerical methods best suited for their numerical solution, specifically high-order finite-differencing for the case of the gravitational field equations and high-resolution shock-capturing schemes for the case of the relativistic Euler equations. A number of recent results are reviewed, namely the outcome of the merger depending on the initial total mass and equation of state of the binary, as well as the post-merger evolution phase once a black hole–torus system is produced. Such system has been shown to be subject to non-axisymmetric instabilities leading to the emission of large amplitude gravitational waves.

 

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Seminar on

Mathematical General Relativity

Organizers:

 Philippe G. LeFloch (Paris)

Ghani Zeghib (Lyon)

ANR Project

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


Wed. November 14, 2012

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

Lecture room  15–16 309 (third level) 


Speakers


10h – Miguel Sánchez Caja (Granada) Recent interrelated progress in Lorentzian, Finslerian and Riemannian geometry

Abstract.    Recently, a correspondence between the conformal structure of a class of Lorentzian manifolds (stationary spacetimes) and the geometry of a class of Finsler manifolds (Randers spaces) has been developed. This correspondence is useful in both directions. On one hand, it allows a sharp description of geometric elements on stationary spacetimes in terms of Finsler geometry. On the other hand, the geometry of spacetimes suggests, both, new geometric elements and new results, for any Finsler manifold, including the Riemannian case. Here, three levels of this correspondence will be explained: (1) Causal structure of spacetimes: properties of Finslerian distances: 0903.3501.  (2) Visibility and gravitational lensing: convexity of Finsler hypersurfaces: 1112.3892, 0911.0360. (3) Causal boundaries: Cauchy, Gromov, and Busemann boundaries in Riemannian and Finslerian settings: arXiv:1011.1154.

11h30 Vladimir Matveev (Jena) Geodesic degree of mobility of Lorentzian metrics

Abstract. The degree of mobility of a metric can be defined as the dimension of the space of solutions of a certain linear PDE system of finite type whose coefficients depend on the metric, and, for a given metric, there are standard algorithms to determine it. The standard algorithms strongly depend on the metric and in most cases it is possible to find the maximal and sub-maximal values of the degree of mobility, only. I will show that the degree of mobility of a manifold is closely related to the space of parallel symmetric tensor fields on the cone over the manifold. In the case the metric is Einstein, it is essentially the tractor cone. I will use it to describe all possible values of the degree of mobility (on a simply connected manifold) for Riemannian and Lorentzian metrics. I will also consider the case when the metric is Einstein and, as a by-product, solve the classical Weyl-Petrov-Ehlers conjecture, and also show applications. Most these results are based on joint projects with  A. Fedorova and S. Rosemann.

14h30 Philippe LeFloch (Paris)  Injectivity radius and canonical foliations of Einstein spacetimes

Abstract. I will discuss recent results on the local geometry of spacetimes with low regularity, when no assumption on the derivatives of the curvature tensor is made, obtained in collaboration with Bing-Long. Chen. Specifically, I will establish that, under geometric bounds on the curvature and injectivity radius, only, there exist local foliations by CMC (constant mean curvature) hypersurfaces, as well as CMC–harmonic coordinates. Importantly, these coordinates are defined in geodesic balls whose radii depend on the assumed bounds, only, and the components of the Lorentzian metric have the best possible regularity.

14h45 Mehdi Belraouti (Avignon) Asymptotic behavior of level sets of a convex time function


15h00 Ghani Zeghib (Lyon) Actions on the circle and isometry groups of globally hyperbolic Lorentz surfaces (after D. Monclair)

Abstract. Let M be a globally hyperbolic spatially compact spacetime with dimension 1+1. A Cauchy surface in it is diffeomorphic to the circle and, more canonically, its family of lightlike geodesics is diffeomorphic to two copies of the circle and, under mild conditions, M embeds as an open set of the 2-torus.  The isometry group G of M acts naturally on these circles, so that G is a subgroup of Diff(S1). We will establish here that G tends to be included in PSL(2, R), the group of projective transformations of the circle S1, up to a global conjugacy by an element of the circle.

15h30 Eduardo Garcia-Rio (Santiago de Compostela) Quasi-Einstein and Ricci soliton Lorentzian metrics

Abstract. Quasi-Einstein metrics are natural generalizations of Einstein metrics and gradient Ricci solitons.  Moreover, they are closely related to the existence of warped product Einstein metrics. Such metrics are defined by an overdetermined equation involving the Ricci curvature and the Hessian of a potential function. I will present some results on the geometry of Lorentzian quasi-Einstein metrics by focusing primary on those which are locally conformally flat. In this setting the Ricci curvature determines the whole curvature tensor and thus the different possibilities depend on the geometry of the level sets of the potential function: warped product metrics and pp-waves appear in a natural way.

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Seminar on

Mathematical General Relativity

Organizers: 

 S. Klainerman (Princeton)

P.G. LeFloch (Paris)

A. Zeghib (Lyon)

Fondations des Sciences Mathématiques de Paris

ANR Project

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

Thursday June 14, 2012

Laboratoire J-L Lions

Université Pierre et Marie Curie, Paris

Lecture room 15-25 104

.

Jean-Philippe Nicolas (Brest) Perspectives in conformal scattering

Abstract. The origins of conformal scattering are to be found in a paper by Friedlander in 1980 “Radiation fields and hyperbolic scattering theory”, in which he realized that the Lax-Phillips theory was in fact providing an interpretation of scattering theory as the well-posedness of the characteristic Cauchy problem for the conformally rescaled wave-equation on null infinity. He clearly saw that the method provided an interesting geometrical short-cut to define a scattering operator with the advantage that all the analytical structure can be recovered a posteriori. The true power of the conformal approach to scattering theory lies in its complete indifference to time dependence. This talk will review the essential features of Lax-Phillips theory and its intimate link with conformal infinity via the Radon transform and the Whittaker formula, then describe the pinciples of conformal scattering with the crucial importance of the precise resolution of the Goursat problem; we will present some results (actual scattering constructions and studies of the Goursat problem) and explain the necessary steps of the extension of the method to black hole spacetimes, which is currently under development.

Jérémie Szeftel (Paris) The bounded L2 curvature conjecture in general relativity

Abstract.  In order to control locally a spacetime which satisfies the Einstein equations, what are the minimal assumptions one should make on its curvature tensor? The bounded L2 curvature conjecture roughly asserts that one should only need an L2 bound on the curvature tensor on a given space-like hypersurface. This conjecture has its roots in the remarkable developments of the last twenty years centered around the issue of optimal well-posedness for nonlinear wave equations. In this context, a corresponding conjecture for nonlinear wave equations cannot hold, unless the nonlinearity has a very special nonlinear structure. I will present the proof of this conjecture, which sheds light on the specific null structure of the Einstein equations. This is joint work with S. Klainerman and I. Rodnianski.

Thierry Barbot (Avignon) Particles in flat spacetimes in expansion
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Seminar on

Mathematical General Relativity

Organizers: 

 S. Klainerman (Princeton)

P.G. LeFloch (Paris)

A. Zeghib (Lyon)

Fondations des Sciences Mathématiques de Paris

ANR Project

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


Thursday May 31, 2012

Laboratoire J-L Lions

Université Pierre et Marie Curie, Paris

Lecture room 15-25 104


2 pm Pieter Blue   (Edinburg)

Decay for the Maxwell field outside a Kerr black hole

Abstract.  This talk will repeat some of the material from last year on the same topic (January 12, 2011) and present some new results. The goal of this talk is to prove uniform energy bounds and Morawetz (integrated decay) estimates.  In the exterior of a Kerr black hole, one of the components of the Maxwell system satisfies a wave equation with a complex potential. Trapping and the complex potential interact to provide surprisingly difficult challenges. Pseudodifferential techniques can treat a model problem with both features. However, because of the structure of the original Maxwell system, a new idea suggests classical derivatives alone should be sufficient.

3:30 pm Jonathan Kommemi (Cambridge) 

Global structure of spherically symmetric spacetimes

Abstract. At the heart of the (weak and strong) cosmic censorship conjectures is a statement regarding singularity formation in general relativity. Even in spherical symmetry, cosmic censorship seems, at the moment, mathematically intractable. To give a framework in which to address these very difficult problems, we will introduce a notion of spherically symmetric ‘strongly tame’ Einstein-matter models, an example of which is given by Einstein-Maxwell-Klein-Gordon (self-gravitating charged scalar fields). We will demonstrate that for any ‘strongly tame’ model there is an a priori characterization of the spacetime boundary. In particular, for any ‘strongly tame’ Einstein-matter model, a ‘first singularity’ must emanate from a spacetime boundary to which the area-radius r extends continuously to zero.

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Seminar on

Mathematical General Relativity

Organizers: 

 S. Klainerman (Princeton)

P.G. LeFloch (Paris)

A. Zeghib (Lyon)

Fondations des Sciences Mathématiques de Paris

ANR Project

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


Thursday May 24, 2012

Laboratoire J-L Lions

Université Pierre et Marie Curie, Paris

Lecture room 15-25 104


11:00 am A. Shadi Tahvildar Zadeh (Rutgers) Zero-gravity limit of Kerr-Newman spacetimes and their electromagnetic fields

Abstract.  We discuss the limit of vanishing G (Newton’s constant of universal gravitation) of the Kerr–Newman electrovacuum spacetimes. We investigate the topologically nontrivial spacetime emerging in this limit and show that it consists of two copies of flat Minkowski spacetime glued at a timelike cylinder. The electromagnetic fields of the Kerr–Newman spacetimes converge to nontrivial solutions of Maxwell’s equations on this background spacetime. We show how to obtain these fields by solving Maxwell’s equations with singular sources supported only on a circle in a spacelike slice of the manifold. These sources do not suffer from any of the pathologies that plague the alternate sources found in previous attempts to interpret the Kerr–Newman fields on the topologically simple Minkowski spacetime.

2:00 pm James Isenberg (Eugene)  AVTD behavior in smooth solutions of Einstein’s equations

Abstract.  One of the more useful approaches to studying the Strong Cosmic Censorship conjecture in a family of solutions of Einstein’s equations is to first verify that generic solutions in that family exhibit AVTD (asymptotically velocity term dominated) behavior near their singular regions. It has been proven (by Ringstrom) that AVTD behavior occurs in generic Gowdy spacetimes, and it has also been shown that it occurs in at least some vacuum spacetimes with T2 isometry, and in some with U(1) isometry. These T2 and U(1) results have been proven using Fuchsian techniques, and have the unfortunate feature that, like many Fuchsian-based results, they require that the spacetimes be analytic. In work done with Florian Beyer, Philippe LeFloch, and Ellery Ames, we show that the analyticity condition can be removed, at least for the T2 case. To prove this result, we have developed a variant of the Fuchsian technique which does not require analyticity. It is very likely that this variant can be applied to U(1) symmetric vacuum spacetimes as well as to those with T2 symmetry.

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Seminar on

Mathematical General Relativity

Organizers:

 Philippe G. LeFloch (Univ. Pierre et Marie Curie)

Ghani Zeghib (Ecole Normale Supérieure, Lyon)

ANR Project

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


Thursday April 5, 2012

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

Lecture room 15-25 101 (first level)


10:00 am Lars Andersson (Potsdam) Hidden symmetries and conserved charges

Abstract.  A proof of decay estimates for test fields with non-zero spin, e.g. Maxwell and linearized gravity, on the Kerr background is an important step towards understanding the black hole stability problem. Fields with non-zero spin on Kerr admit non-radiating modes which must be eliminated in order to prove decay. In this talk I will discuss the relation between conserved charges and hidden symmetries for linearized gravity on Minkowski space and vacuum spaces of Petrov type D, and outline the application of these ideas in proving estimates for the higher spin fields on the Kerr background.

11:30 am  François Béguin (Orsay) On the BKL conjecture for vacuum spatially homogeneous models

Abstract. About forty years ago, Belinski, Kalatnikov, and Lifschitz proposed a rather complete description (based on heuristic arguments) of the asymptotic behavior of spatially homogeneous spacetimes close to their initial singularity. In particular, a “generic” vacuum spatially homogeneous spacetime is expected to exhibit a “chaotic” oscillatory behavior closed to its initial singularity. It was only in 2010 that the first rigorous mathematical results going in this direction were established. I will review here what is known (from a rigorous mathematical viewpoint) on the behavior of vacuum spatially homogeneous spacetimes close to their initial singularity. I will also try to explain what one can (or cannot) hope to prove about this asymptotic behavior and will sketch the proofs.

2:00 pm Simone Calogero (Granada) Dynamics of spatially homogeneous cosmological models

Abstract.  Understanding the asymptotic dynamics of the Universe is one of the main goals of theoretical cosmology. In the context of spatially homogeneous (SH) cosmological models, where the Einstein equations of general relativity reduce to a system of ordinary differential equations, our knowledge on this problem has increased substantially over the years, thanks to the application of methods from the theory of finite dimensional dynamical systems. After reviewing the results known for the dynamics of perfect fluid cosmological models, this talk will focus on the more complicated case of a space-time filled with anisotropic matter. In the latter case the qualitative global behavior of SH solutions depends strongly on the asymptotic values of the principal pressures in the limit toward the singularity. Such behavior has been completely characterized for locally rotationally symmetric (LRS) solutions in the class A of Bianchi models. It has been found that there exist matter models, compatible with the standard energy conditions, for which Bianchi type IX LRS solutions are singularity-free for an initial data set with positive measure, and others for which the approach toward the singularity of generic solutions is oscillatory. This talk is based on a series of works in collaboration with Mark Heinzle.

3:30 pm Ghani Zeghib (Lyon) Projective transformation groups

Abstract.  To pseudo-Riemannian (in particular Riemannian or Lorentzian) metric one naturally associates its Levi-Civita connection. It is also natural to ask whether, conversely, the connection determines the metric, that is, (essentially) whether two metrics having the same geodesics coincide?  More strongly, two metrics are said to be projectively equivalent if their geodesics coincides as geometric non-parametrized curves. One then asks when projectively equivalent metrics coincide? The answer to this classical problem is `no’, but only for very special metrics.  This confirms a real interest for this problem since it allows the emergence of special structures. We will deal here with a parallel (i.e. weaker) rigidity question: classify metrics admitting an essential projective transformation, that is, diffeomorphism preserving, non-parametrized geodesics without being an isometry?

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Seminar on

Mathematical General Relativity

Organizers:

 Philippe G. LeFloch (Univ. Pierre et Marie Curie)

Ghani Zeghib (Ecole Normale Supérieure, Lyon)

ANR Project

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


Thursday March 22, 2012

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

Lecture room 15-25 102 (first level)


14h   Mihalis Dafermos (Cambridge) Black holes without spacelike singularities

Abstract.   It is shown that for small, spherically symmetric perturbations of asymptotically flat two-ended Reissner-Nordstroom data for the Einstein-Maxwell-real scalar field system, the boundary of the dynamic spacetime which evolves is globally represented by a bifurcate null hypersurface across which the metric extends continuously. Under additional assumptions, it is shown that the Hawking mass blows up identically along this bifurcate null hypersurface, and thus the metric cannot be extended twice differentiably, in fact, cannot be extended in a weaker sense characterized at the level of the Christoffel symbols. The proof combines estimates obtained in previous work with an elementary Cauchy stability argument. There are no restrictions on the size of the support of the scalar field, and the result applies to both the future and past boundary of spacetime. In particular, it follows that for an open set in the moduli space of solutions around Reissner-Nordstrom, there is no spacelike component of either the future or the past singularity.

15h30  Rabah Souam (Paris)  Harmonic diffeomorphisms and maximal surfaces

Abstract.  We study the existence (or the non-existence) of harmonic diffeomorphisms between certain domains in the Euclidean  two-sphere. In particular, we construct harmonic diffeomorphisms from circular domains in the complex plane onto finitely punctured spheres, with at least two punctures. This result follows from a general existence theorem for maximal graphs with isolated singularities in the Lorentzian product M x R, where M is an arbitrary n-dimensional compact Riemannian manifold (with n larger than 1).  In contrast, we show that there is no harmonic diffeomorphism from the unit complex disc onto the (once) punctured sphere, and no harmonic diffeomeorphisms from finitely punctured spheres onto circular domains in the Euclidean two-sphere. This is a joint work with Antonio Alarcon.

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Seminar on

Mathematical General Relativity

Organizers:

 Philippe G. LeFloch (Univ. Pierre et Marie Curie)

Ghani Zeghib (Ecole Normale Supérieure, Lyon)

With the financial support of the ANR Project

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


Thursday February 9, 2012

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

Lecture room 15-25 101 (first level)


14h   Alan Rendall (AEI, Potsdam) Singularity formation in solutions of the Einstein-Vlasov system

Abstract.  Important questions in mathematical relativity are when singularities form in solutions of the Einstein equations coupled to matter and, in cases where they do form, what their qualitative nature is. A type of matter model which apparently rarely loses smoothness in the absence of black hole formation is collisionless matter modelled by the Vlasov equation. This contrasts with dust, a type of matter popular among relativists. In this talk I describe recent work with Juan Velazquez where we try to obtain new insights about the dynamics of the Einstein-Vlasov system by interpolating between smooth Vlasov and dust in a suitable way. We have shown that for certain mildly singular initial data a curvature singularity can form. It is constructed by means of a shooting argument for a system of ordinary differential equations. From the point of view of physics it would be desirable to improve this solution in various ways and I will report briefly on work in progress on doing this.

15h30 François Filastre (Cergy-Pontoise) Brunn–Minkowski theory in Minkowski spacetime 

Abstract.  The Brunn–Minkowski theory deals with the relations between the addition and the volume of convex bodies of the Euclidean space. Convex bodies are described by function on the sphere. The main result of the theory is that the volume is log-concave. We establish an analog result for a class of convex sets in the Minkowski spacetime. The compactness is replaced by a global invariance property under the action of particular groups of linear isometries. In particular, these convex sets can be described by functions on compact hyperbolic manifolds and, in this case, the volume is convex.

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7th DFG–CNRS WORKSHOP

Two-Phase Fluid Flows. Modeling and Computational Methods 

Main organizer:    

 Philippe G. LeFloch (Univ. Pierre et Marie Curie, Paris)

Co-organizers:

Christophe Berthon (Nantes) and Philippe Helluy (Strasbourg)

With financial support from the DFG and the CNRS


Tuesday Feb. 14, 2012 at 2pm to Thursday Feb. 16 at noon

Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 4, Place Jussieu, Paris.

Subway station: Jussieu

Lecture room 15-16 — 309


SCHEDULE, list of participants, and abstracts


INVITED SPEAKERS


Gonca Aki
 
(Berlin) An incompressible diffuse flow with phase transition

Mathieu Bachmann (Aachen) Numerical simulation of shock wave-bubble interactions using laser-induced cavitation bubbles

Frank Boyer (Marseille)  Numerical methods for a three-component phase field model

Sergey L. Gavrilyuck (Marseille) Diffuse interface model for compressible fluid-compressible elastic-plastic solid interaction

Maren Hantke (Magdeburg) Exact solutions to the Riemann problem for compressible isothermal Euler equations for two phase flows, with and without phase transition

Jonathan Jung (Strasbourg) Computing bubble oscillations on GPU (graphics processing unit)

Mirco Kraenkel (Freiburg) Numerics for phase field models

Hélène Mathis (Nantes) Model adaptation for hyperbolic systems with relaxation 

Khaled Saleh (Paris) A splitting method for the isentropic Baer-Nunziato two-phase flow model 

Nicolas Seguin (Paris)  Model adaptation in hierarchies of hyperbolic systems

Gabriele Witterstein (Munich) Existence of transition profiles for compressible flows

Christophe Zeiler (Stuttgart) Curvature driven liquid-vapor flow of compressible fluids

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PRACTICAL INFORMATIONS 

LIST OF PARTICIPANTS

How to come to the Laboratoire Jacques-Louis Lions ?

Hotels near the University Pierre et Marie Curie ?

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EARLIER WORKSHOPS “Micro-Macro Modeling and Simulation of Liquid-Vapour Flows”

Sixth Workshop, Stuttgart, Jan. 2011

Fourth Workshop, Aachen, Feb. 2009

Second Workshop, Bordeaux

Opening Workshop, Kirchzarten, Nov. 2005

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Seminar on Compressible Fluids

Tuesday January 10, 2012

Laboratoire Jacques-Louis Lions

Université Pierre et Marie Curie

4 Place Jussieu, 75258  Paris

Building 15/16. Lecture room 309

With the support of LRC MANON

  • 14h00 : Philippe Helluy (Strasbourg) Résolution des équations de Maxwell-Vlasov sur GPU

Abstract.  Je présenterai un couplage d’une méthode Galerkin-Discontinu et d’une méthode PIC (Particle-In-Cell) pour la résolution des équations de Vlasov-Maxwell. Ces méthodes ont déjà été implémentées à de nombreuses reprises. La nouveauté consiste ici à le faire sur une carte graphique avec le langage OpenCL, ce qui conduit à des façons différentes d’organiser l’algorithme de couplage.

  • 15h30 : Christophe Berthon (Nantes)  Schémas hydrostatiques décentrés pour les équations shallow-water

Abstract.  We consider the numerical approximation of the shallow–water equations with non–flat topography. We introduce a new topographic discretization which makes all schemes to be well–balanced and robust. In contrast with the well–known hydrostatic reconstruction, the proposed numerical procedure does not involve any cut–off. Moreover, the proposed scheme is able to deal with dry areas. Several numerical benchmarks are presented to assert the interest of the method.

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Organizers: Frédéric Coquel, Edwige Godlewski, et Philippe LeFloch

Philippe LeFloch, DIRECTOR OF RESEARCH AT CNRS contact@philippelefloch.org philippelefloch.org

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