Du 16 au 18 juin 2011
IRMA
La 87ème rencontre entre physiciens théoriciens et mathématiciens aura pour thème la Géométrie de Lorentz en mathématiques et en physique. Elle est dédiée à Norbert A'Campo, pour son 70e anniversaire.
The 87th Encounter between Mathematicians and Theoretical Physicists will take place at Institut de Recherche Mathématique Avancée (University of Strasbourg and CNRS) on June 16-18, 2011. The theme will be : "Lorentz geometry in Mathematics and in Physics". The Encounter is dedicated to Norbert A'Campo for his 70th birthday.
Organizers: Charles Boubel and Athanase Papadopoulos.
List of registered participants is available.
The invited speakers include:
- Thierry Barbot (Avignon)
- Mauro Carfora (Pavia)
- Vladimir Chernov (Dartmouth College)
- Mihalis Dafermos (Cambridge)
- Charles Frances (Orsay)
- Jacques Franchi (Strasbourg)
- Armenak Gasparyan (Yaroslavl State University)
- Kirill Krasnov (Nottingham)
- Catherine Meusburger (Hamburg)
- Karim Noui (Tours)
- Vladimir Matveev (Jena)
- Jean-Marc Schlenker (Toulouse)
- Abdelghani Zeghib (ENS Lyon)
All talks are in English. Some of the talks will be survey talks intended for a general audience.
Graduate students and young mathematicians are welcome.
Registration is required (and free of charge), at the following link.
Hotel booking can be asked for through the registration link.
For questions please contact the organizers:
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Jeudi 16 juin 2011
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09:00 - 10:00
Mauro Carfora, Università di Pavia
Glimpses of Lorentz geometry and physics
[This talk is designed to be widely comprehensible by a broad audience.] Instead of a truly general review, I will make a selection and try to pinpoint also less obvious aspects. A more precise abstract will be disclosed later. -
10:00 - 11:00
Thierry Barbot, Université d'Avignon
Spacetimes in expansion
[This talk is designed to be widely comprehensible by a broad audience.] Let (M,g) be a spatially compact globally hyperbolic spacetime. By definition, there is a time function TM -> ]0+\infty[ whose levels are compact. Here we allow T to be weakly regular, for example only locally Lipschitz. For every t the level T^{-1}(t) is naturally a metric space (S_t,g_t). A natural challenge is to understand how these metrics degenerate for t -> 0, which can be interpreted as identifying the "Big-Bang geometry". An appealing situation is the case where the spacetime is in expansioni.ewhen the levels S_t are convex (the fundamental form is non-negative). I will review what is known on this problemwith emphasis in the case where (M,g) has constant sectional curvature. In this case one obtains in the (2+1)-dimensional case real trees as limit metric spaces. What happens in higher dimension is an open question under progress. -
11:00 - 11:30
Coffee break -
11:30 - 12:30
Vladimir Chernov, Dartmouth College
Low conjecture, linking and causality in global hyperbolic Lorentz spacetimes
Low conjecture and the Legendrian Lowc onjecture formulated by Natario and Tod say that for many spacetimes X, two events x,y in X are causally related if and only if the link ofspheres S_xS_y whose points are light rays passing through x and y is non-trivial in the contact manifold N of all light rays in X.We prove the Low and the Legendrian Low conjectures and show thatsimilar statements are in fact true in almost all $4$-dimensionalglobally hyperbolic spacetimes. The conjectures follow from the existence of the natural partial order on the space of Legendrian spheres in the contact manifold N of light rays in X. All the known examples of globally hyperbolic spacetimes where the Legendrian link S_xS_y does not determine causal relation between x and y areclosely related to the so called refocusing spacetimes.
If time permits we discuss which of the smooth 4-manifolds admit a globally hyperbolic Lorentz metric generalizing the results of Newman and Clarke.
This talk is based on joint work with Stefan Nemirovski.
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14:00 - 15:00
Catherine Meusburger, Universität Hamburg
Causality in 3d spacetimes with particles
We construct stationary flat three-dimensional Lorentzian manifolds with singularities that correspond to (2+1)-dimensional spacetimes containing massive particles with spin. We classify these spacetimes and give a systematic investigation of their causality properties. In particular we investigate the effect of the singularities on light signals exchanged between different observers. We derive an explicit condition that excludes causality violating signals.This is joint work with Thierry Barbot.
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15:00 - 15:30
Coffee break -
15:30 - 16:30
Karim Noui, Université de tours
Geometry and topology in three dimensions
The abstract will be announced later. -
17:00
Boat trip around the old city of Strasbourg (the boat trip is offered to the participants by the Mayor of Strasbourg) -
19:00
Dinner offered to the participants at the restaurant "petit Bois Vert" The address is : Quai de la Bruche (Petite France region in the historica center of Strasbourg) -
Vendredi 17 juin 2011
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09:00 - 10:00
Charles Frances, Université Paris 11 - Orsay
Rigidité des bords conformes en géométrie pseudo-Riemannienne
Le but de l'exposé est de montrer comment une construction classique de bord abstrait pour les géométries de Cartan peut-être utilisée pour prouver des résultats de rigidité des bords conformes d'espaces pseudo-Riemanniens. Nous donnerons également des applications de cette construction aux variétés conformément maximales. -
10:00 - 11:00
Abdelghani Zeghib, École Normale Supérieure de Lyon
The Minkowski problem in Minkowski space
[This talk is designed to be widely comprehensible by a broad audience.] In the classical Minkowski problem, one starts with a convex surface Sand associates to it its Gaussian curvature function k_SOne then composes with the inverse of the Gauss map and gets a function K_S on the 2-sphere. The Minkowski problem consists in characterizing the functions on the spherewhich have the form K_S for some convex surface S ?
We are going here to consider this problem in a Lorentzian settingby replacing the 3-Euclidean space by the 3-Minkowski spacerequiring the surface S to be spacelike and replacing the sphere by the hyperbolic plane.
This is a joint work with T. Barbot and F. Béguin.
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10:00 - 10:30
Coffe break -
11:30 - 12:30
Jacques Franchi, Université de Strasbourg
Diffusions relativistes covariantes
Dudley a montré en 1965 qu'il n'existe dans le cadre de l'espace de Minkowski qu'un seul processus de diffusion (c'est-à-dire markovien et continu) invariant par le groupe de Lorentz, et que sa vitesse est un mouvement brownien hyperbolique. Dans le cas d'une variété lorentzienne générique M, un processus de diffusion relativiste (Z_s) (nécessairement à valeurs dans le fibré pseudo-unitaire tangent T^1M, de loi notée P_x s'il est issu de x \in T^1M) est covariant lorsqu'il possède l'invariance relativiste suivante : pour toute isométrie f de M, le processus f(Z_s) a pour loi P_{f(x)}. On peut définir sur T^1M une famille de diffusions covariantes dont la variation quadratique est localement déterminée par la courbure de la variété, ce qui autorise l'interprétation de l'effet diffusif par les interactions d'une particule avec l'espace-temps ambiant. Un cas important dans lequel une analyse du comportement asymptotique de telles diffusions est raisonnablement envisageable est celui des pseudo-métriques de type produit semi-direct, dont en particulier les espaces de Robertson-Walker. Une telle étude peut être assez facilement faite dans l'exemple des variétés de type Einstein-de Sitter, au moins pour quelques fonctions simples de la courbure comme la courbure scalaire ou l'énergie. Il est également possible d'établir quelques critères généraux de non-explosion de telles diffusions, inspirés par la b-complétude de Schmidt, le test de Khasminsky, ou bien le cas plus simple et abondamment étudié des variétés riemanniennes, où la croissance du volume des boules est la notion clé. Références :
- DudleyLorentz-invariant Markov processes in relativistic phase spaceArkiv för Matematik 6no 14241-2681965.
- Franchi-Le JanCurvature Diffusions in General RelativityArXiv http://arxiv.org/abs/1003.38492010.
- Bailleul-FranchiNon-explosion criteria for relativistic diffusionsArXiv http://arxiv.org/abs/1007.1893v1 [math.PR]2010. -
14:30 - 15:30
Kirill Krasnov, University of Nottingham
Gravity as a diffeomorphism invariant Gauge theory
This talk will explain how Einstein's General Relativity can be reformulated as a particular diffeomorphism invariant SU(2) gauge theory. -
15:30 - 16:00
Coffee Break -
16:00 - 17:00
Mihalis Dafermos, University of Cambridge
The black hole stability problem
I will review recent progress on the black hole stability problem in general relativity. -
17:30
Reception at the City Hall - Address : Place Broglie -- We shall all go there after the last talK -
Samedi 18 juin 2011
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09:00 - 10:00
Vladimir Matveev, Universität Jena
How to reconstruct a metric by its unparameterized geodesics
We discuss whether it is possible to reconstruct a metric by its unparameterized geodesics and how to do it effectively. We explain why this problem is interesting for general relativity. We show how to understand whether all curves from a sufficiently big family are unparameterized geodesics of a certain affine connectionand how to reconstruct algorithmically a generic 4-dimensional metric by its unparameterized geodesics. The algorithm works most effectively if the metric is Ricci-flat. We also prove that almost every metric does not allow nontrivial geodesic equivalence and construct all pairs of 4-dimensional geodesically equivalent metrics of Lorenz signature.If time allows I will also explain how this theory helped tosolve two problems explicitly formulated by Sophus Lie in 1882 and the semi-Riemannian two-dimensional version of the projective Lichnerowicz-Obata conjecture.
The new results of the talk are based on the papers
arXiv:1010.4699arXiv:1002.3934 arXiv:0806.3169arXiv:0802.2344arXiv:0705.3592This is a joint work with Bryant, Bolsinov, Kiosak, Manno, Pucacco.
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10:00 - 10:30
Coffe break -
10:30 - 11:30
Jean-Marc Schlenker, Université de Toulouse
A cyclic extension of the earthquake flow
[This talk is designed to be widely comprehensible by a broad audience.] First we will introduce earthquakes. The earthquake flow is a flow on the product of Teichmüller space by the space of measured laminations. Among its key properties are Thurston's Earthquake Theorem (given two hyperbolic metrics, there is a unique left earthquake sending one to the other) and McMullen's complex earthquakes which extend earthquake lines as holomorphic disks. Then we will describe an extension of the earthquake flowwhich is a flow on the product of Teichmüller space by itself. When one of the factors tends projectively to a measured lamination, we recover the earthquake flow. Our extension shares many properties of the earthquake flow, in particular the two described above. The heuristics and some of the proofs use 3-dimensional AdS geometry. Joint work with Francesco Bonsante and Gabriele Mondello.