Du 2 au 4 juin 2016
IRMA
La 97ème rencontre entre mathématiciens et physiciens théoriciens aura pour thème : Autour de Poincaré
The 97th Encounter between Mathematicians and Theoretical Physicists will take place at Institut de Recherche Mathématique Avancée (University of Strasbourg and CNRS) on June 2-4, 2016. The theme will be: Around Poincaré
Organizers : Lizhen Ji (Ann Arbor), Athanase Papadopoulos (Strasbourg) and Sumio Yamada (Tokyo)
The invited speakers include :
- Norbert A'Campo (Bâle)
- Marie-Claude Arnaud (Avignon)
- Costas Bachas (Paris)
- Pierre Cartier (IHES)
- Marc Chaperon (Paris)
- Christos Charmousis (Orsay)
- Alain Chenciner (Paris)
- Chong-Qing Cheng (Nanjin)
- Bassam Fayad (Paris)
- Hidekazu Ito (Kanazawa)
- Yuri Manin (Bonn et Chicago)
- Ken'ichi Ohshika (Tokyo)
- Valentin Poenaru (Orsay)
- Alexey Sossinsky (Moscou)
- Muhammed Uludag (Istanbul)
- Jean-Christophe Yoccoz (Orsay)
Venue: Salle de conférences, IRMA building
The talks will be in English. Some of them will be survey talks intended for a general audience.
Graduate students and young mathematicians are welcome. Registration is free of charge but the potential participants are asked to contact one of the organisers:
- Lizhen Ji : lji@umich.edu
- Athanase Papadopoulos : athanase.papadopoulos@math.unistra.fr
- Sumio Yamada : yamada@math.gakushuin.ac.jp
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Jeudi 2 juin 2016
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10:00
Norbert A'campo, Basel
The Poincaré metric by a method of continuity
Abstract.--- Using pant decompositions instead of fundamental domains, and the Maskit inequality, the Poincaré metric on conformal surfaces will be constructed. -
11:00
Coffee Break
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11:30
Marc Chaperon, Paris
Avatars of of Poincaré's last theorem
Abstract.--- We shall talk about Poincaré's last theorem (proved by Birkhoff), of the remarkable Arnold conjectures which it inspired, of their proof by Conley, Zehnder, the speaker and Gromov, of their topolgical version in dimension 2 and of symplectic topology to which which they gave rise. -
14:00
Pierre Cartier, Bures-sur-Yvette
On the history of the Poincaré-Birkhoff-Witt theorem
Abstract.--- This theorem is basic in the theory of Lie algebras and Lie groups . Poincaré gave a proof using analysis . I shall describe the historical variations , the emphasis being sometimes on analytical methods alternating with a more algebraic version -
15:00
Coffee Break
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15:30
Costas Bachas, Paris
Non-local operators in quantum field theory
Abstract.--- Non-local operators, corresponding to extended external probes of a system, play an ever-increasing role in the study of quantum field theories. I will review several examples and some recent results, both in condensed-matter systems and in gauge field theories, and conclude with challenges for the future. -
16:30
Hidekazu Ito, Kanazawa
Poincaré center problem and normal form theory of vector fields
Abstract.--- Starting with the so-called Poincaré center problem, we discuss normal form theory for an analytic vector field near an equilibrium point of elliptic type. We focus on the relationship between the existence of a convergent normalization and integrability of vector fields. The normal form is expected to be solved explicitly. We survey several works on such relationship and present some new results on existence of a convergent normalization for commutative or non-commutative integrable systems near resonant equilibrium points. -
18:00
Mairie, Place Broglie, Reception At The City Hall By The Mayor Of Strasbourg
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Vendredi 3 juin 2016
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09:00
Alain Chenciner, Paris
A note by Poincaré
On November 30th 1896, Poincaré published in the “Comptes rendus de l’Académie des Sciences” a note entitled “On the periodic solutions and the least action principle”. He proposed to find periodic solutions of the planar Three-Body Problem by minimizing the Lagrangian action among loops in the configuration space which satisfy given constraints (the constraints amounted to fixing their homology class). For the Newtonian potential, proportional to the inverse of the distance, the “collision problem” prevented him from realizing his program ; hence he replaced it by a “strong force potential” proportional to the inverse of the squared distance. In the lecture, I shall explain the nature of the difficulties met by Poincaré and how, one century later, these difficulties have been partially resolved for the Newtonian potential, leading to the discovery of interesting new families of periodic solutions of the planar or spatial n-body problem. Finally, the surprizing relation of some of these solutions with the simplest n-body relative equilibria will be discussed. References: [1] H. Poincaré Sur les solutions p?eriodiques et le principe de moindre action, C.R.A.S. 1896, t. 123, 915-918 ; in Œuvres, tome VII. [2] Chenciner A note by Poincaré, RCD, V. 10, n02 (2005). [3] C. Marchal How the method of minimization of action avoids singularities, Celestial Mechanics, V. 83. 325353 (2002). [4] A. Chenciner Action minimizing solutions of the Newtonian n-body problem : from homology to symmetry, ICM Beijing (2002). [5] A. Chenciner & J. Féjoz Unchained polygons and the N-body problem,RCD Vol. 14, N01, 64-115 (2009). -
10:00
Coffee Break
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10:30
Muhammed Uludag, Istanbul
The outer automorphism of PGL(2,Z) and the induced 'modular' involution of the real line.
We study the involution of the real line induced by the outer automorphism of the extended modular group PGL(2,Z). This ‘modular’ involution is discontinuous at rationals but satisfies a surprising collection of functional equations. It preserves the set of real quadratic irrationalities mapping them in a highly non-obvious way to each other. It commutes with the Galois action on the set of real quadratic irrationals. More generally, it preserves set-wise the orbits of the modular group, thereby inducing an involution of the moduli space of real rank-two lattices. It induces a duality of Beatty partitions of the set of positive integers. It also induces a subtle symmetry of Lebesgue's measure. This involution conjugates (though not topologically) the Gauss’ continued fraction map to an intermittent dynamical system on the unit interval with an infinite invariant measure. The transfer operator (resp. the functional equation) naturally associated to this dynamical system is closely related to the Mayer transfer operator (resp. the Lewis’ functional equation). We give a description of this involution as the boundary action of a certain automorphism of the infinite trivalent tree. We prove that its derivative exists and vanishes almost everywhere. It is conjectured that algebraic numbers of degree at least three are mapped to transcendental numbers under this involution. In a paper of his on binary quadratic forms, Poincaré states that "it is not possible, for the indefinite quadratic forms to find invariants, in the sense that we gave to this word..." Our study can be understood as a study of what can be done by modifying the meaning of the word "invariant". -
11:30
Valentin Poenaru, Bures-sur-Yvette
Around the 4d smooth Schoenflies conjecture
Abstract.--- I will sketch my proof that all 4d smooth Schoenflies balls are geometrically simply connected, a piece of the Schoenflies Conjecture.I will also explain how these things fit in the context of the smooth 4d Poincaré Conjecture,the only still missing piece of the general PC. -
14:00
Christos Charmousis, Orsay
black holes in scalar tensor theories
Abstract.--- We will review modified gravity theories and in particular scalar tensor theories, where we have an additional scalar field coupling to the metric tensor. By means of a theorem given by Horndeski back in 1974 we will briefly discuss the most general of these theories with second order field equations. We will examine a particular sub-class of Horndeski theory which has interesting properties with respect to the cosmological constant problem. We will then find black hole solutions of this subclass which in some cases will be identical to GR solutions. The novel ingredient will be the presence of a time and space dependent scalar field. As we will see time dependence and higher order Galileon terms will bifurcate no hair theorems and provide scalar tensor black holes with a non trivial scalar field. -
15:00
Coffee Break
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15:30
Bassam Fayad, Paris
Double exponential stability of elliptic fixed points in Hamiltonian dynamics : prevalence and optimality of the phenomenon
Abstract.--- The stability (or instability) of an equilibrium in Hamiltonian dynamics can be studied from various points of view : the classical Lyapunov stability or topological stability, the stability from a probabilistic point of view addressed by KAM theory, or the (even more classical!) effective stability. By effective stability, we mean the time spent by solutions that start close to the equilibrium in a neighborhood of the equilibrium. Combining KAM theory, Nekhoroshev theory and estimates of Normal Birkhoff forms we will explain why typically (in a very general measure theoretic and topological sense) equilibria as well as invariant quasi-periodic tori of Hamiltonian systems are double exponentially stable, in the sense that nearby solutions remain close to the equilibrium or invariant torus for an interval of time which is doubly exponentially large with respect to the inverse of the initial distance to the point or torus. Using Herman synchronized diffusion, we show that the double-exponential stability is optimal, even for fixed points and tori with definite torsion. -
16:30
Chong-Qing Cheng, Nanjin
The mechanism of Arnold diffusion in a priori stable system
Abstract.--- In the study of Arnold diffusion in nearly integrable systems with 3 degrees of freedom, after the problem for a priori unstable system was solved about ten years ago, it has been widely known a difficult problem how to cross double resonance. In this talk, I shall sketch the proof how to solve this problem and its application for the problem of multiple resonance which appears in the systems with arbitrarily many degrees of freedom. -
19:00
Dinner At The Restaurant Le Petit Bois Vert (petite France)
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Samedi 4 juin 2016
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09:00
Marie-Claude Arnaud, Avignon
A multi-dimensional Birkhoff theorem for non autonomous Hamiltonian flows
Abstract.--- A lot of problems coming from the physics are symplectic, as the N-body problem and other classical mechanical systems. Close to the complectely elliptic periodic orbits of symplectic dynamics, using some change of coordinates called normal form, we are often lead to study the time 1 map of a so-called Tonelli Hamiltonian vector field. This kind of diffeomorphisms was introduced for example by Poincaré in the study of the circular restricted 3-body problem. For Hamiltonian and symplectic Dynamics, the invariant Lagrangian submanifolds are a central object of study ( Poincaré knew that when he tried to find such invariant submanifolds by using Linstedt series in his famous book « les méthodes nouvelles de la mécanique céleste » ) . Here we will focus on the so-called time-dependent Tonelli Hamiltonian flows that are defined on the cotangent bundle of a manifold, and explain that under some topological appropriate assumptions, an invariant Lagrangian submanifold has to be a graph. This is a joint work with Andrea Venturelli. -
10:00
Coffee Break
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10:30
Alexey Sossinsky, Moscou
One of Poincaré's last ideas about the philosophy of science and tolerance space theory
Abstract.--- In his "Last Essays'', Poincaré stresses the fact that in the "physical continuum", (physical measurement), as opposed to the "mathematical continuum" (the real numbers), there is no notion of exact equality, only the relation of approximate equality, which is reflexive, symmetric, but not transitive. Poincaré clearly pointed out the importance of such a relation, but never got around to working out the mathematical structures that it leads to. This was first done in the 1960ies by Christopher Zeeman, who rediscovered Poincaré's idea, and called (rather unfortunately) the main notion -- a set of points with a reflexive symmetric relation on it -- a tolerance space. It is amazing that such a simple object as a set with reflexive and symmetric relation on it has such varied and rich mathematical structures attached to it, and these structures turn out to be simpler and more efficient for the applications than the corresponding structures in the categories of topological, simplicial, and cell spaces, or on smooth manifolds. These structures, in particular, the homology and homotopy of tolerance spaces, were developed by the speaker in the 1970ies and applied, in particular, to prove existence theorems for numerical solutions of PDE via what I call "almost fixed point" theorems. . The applications to numerical analysis were not noticed by the community of applied mathematicians, and after the speaker's 1985 paper, nothing essential was published. Tolerance space theory was forgotten -- undeservedly, in my opinion -- for almost half a century. Recently, there has been some renewed interest in the topic, and one can hope that this will result in demonstrating that Poincaré was right in stressing the importance of the idea both from the point of view of theory and practice. -
11:30
Ken'ichi Ohshika, Osaka
3-manifolds fibring over the circle: From Poincaré’s Analysis Situs to today
Abstract.--- In a short paper entitled “Sur l’analysis situs”, Poincré constructed torus bundles over the circle, which he used to show that there are different 3-manifolds with the same Betti numbers. This should be regarded as the beginning of the 3-manifold theory. From this time on, surface bundles over the circle have been playing important roles in the 3-manifold theory. I will survey a history of the 3-manifold theory focusing on surface bundles, and touch upon my recent work on asymptotic behaviour of hyperbolic 3-manifolds fibring over the circle.