Du 26 au 28 septembre 2013
IRMA
La 92e rencontre entre mathématiciens et physiciens théoriciens aura pour thème : Entropie en mathématiques et en physique.
The 92nd Encounter between Mathematicians and Theoretical Physicists will take place at Institut de Recherche Mathématique Avancée (University of Strasbourg and CNRS) on September 26-28, 2013. The theme will be : "Entropy in Mathematics and in Physics".
Organizers : Nicolas Juillet and Athanase Papadopoulos
The speakers are :
- Gérard Besson (Grenoble)
- Freddy Bouchet (Lyon)
- Mauro Carfora (Pavie)
- Djalil Chafaï (Paris)
- Stéphane De Gérando (Paris)
- Rémi Peyre (Nancy)
- Anders Karlsson (Genève)
- Christian Maes (Leuven)
- Gabriel Rivière (Lille)
- Nessim Sibony (Orsay)
- Karl-Theodor Sturm (Bonn)
- Constantin Vernicos (Montpellier)
The talks will be 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 this link. Hotel booking can be asked for through the registration link.
For practical and other questions please contact the organizers :
- Nicolas Juillet : nicolas.juillet@math.unistra.fr
- Athanase Papadopoulos : athanase.papadopoulos@math.unistra.fr
The following programm is still subjet to modification:
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Jeudi 26 septembre 2013
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09:30
Nessim Sibony, Orsay
Entropy in holomorphic dynamics.
This talk is designed for a general audience. The entropy of a dynamical system is a number measuring how much the system is chaotic. More precisely it measures the growth of the number of orbits one can distinguish before time n, at every scale. It is an important invariant at the center of the theory of dynamical systems.One of the goals is to construct an invariant measure on the most chaotic part, to study its geometry,and the rates of expansion-contraction (Lyapounov-exponents). The abstract theory is well estabished, Oseledec, Pesin...To decide wether it applies in concrete examples is another story, most of the time out of reach.
I will first recall Bowen's definition of entropy. Then I will discuss the notion in the context of holomorphic or meromorphic maps between compact Kähler manifolds.In particular it's relation with dynamical degrees, which describe the average expansion. This is strongly related to works by Gromov and Yomdin. Complex dynamics is a remarkable case where entropies can be computed explicitly in many situations.Measures of maximal entropy can be constructed, although the systems are not uniformly hyperbolic in general.
I will also develop a notion of entropy for laminations by hyperbolic Riemann surfaces. In that case, we use the hyperbolic time. The finiteness of entropy is related to the transverse Hölder continuity of the Poincaré metric. The theory can be extended to cover the case of singular foliations on complex surfaces. The lecture is based on Joint works with T.C Dinh and V.A. Nguyen. -
10:30
Coffee break -
11:00
Rémi Peyre, Nancy
Free energy functional in an optimal-transportation setting
We consider a toy model for gravitational spontaneous symmetry breaking: in this model, the distribution of matter evolves according to a first-order equation, with a short-range attractive force which tends to make matter collapse, versus a diffusion term which tends to homogeneize density. It can be shown that this evolution is tantamount to the gradient flow of the free energy (hence involving entropy) in some infinite-dimensional Riemannian manifold defined in terms of “optimal transportation”—all of that shall be explained in detail.
Hence our goal is to understand the behaviour of the free energy functional, in particular near the critical point (that is, when diffusion is just sufficient to prevent spontaneous symmetry breaking). We will show that the critical exponant of the activation energy is $(3-n/2)_+$, $n$ being the dimension of the space. This involves finding sharp estimates, both below and above, on the entropy of the system.
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12:00
Lunch -
15:30
Anders Karlsson, Genève
A heat kernel approach to tree entropy
The number of spanning trees of graphs is an invariant of interest in electrical networks, statistical physics, theoretical chemistry and mathematics. I will discuss the asymptotics of the number of spanning trees in families of graphs, especially discrete tori. The constant in the lead term is called the tree entropy. This number as well as other constants in the asymptotics have interest in number theory: they are expressed in terms of certain Mahler measures, zeta functions, modular forms, and their special values. When there is a limiting continuous torus, its determinant of Laplacian appears. Based on joint work with G. Chinta and J. Jorgenson. I will also mention recent works by F. Friedli and J. Louis. -
16:30
Coffe break -
17:00
Gérard Besson, Grenoble
On open 3-manifolds
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18:30
Dinner - restaurant "Le petit bois vert" (petite France quarter). All participans are invited. We start from the math institute at 18:00. -
Vendredi 27 septembre 2013
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09:30
Karl-Theodor Sturm, Bonn
Ricci Curvature and Entropy
We give a brief survey on Ricci curvature bounds and heat flow on metric measure spaces. Generalized bounds for the lower Ricci curvature on metric measure spaces (X; d;m) have been introduced by Lott & Villani and the author in terms of convexity properties of the relative entropy regarded as function on the Wasserstein space on the given space X. Besides Riemannian manifolds, the class of examples include Alexandrov spaces, Finsler spaces, path spaces as well as limits and constructions of such spaces (Euclidean cones, warped products). It turned out that - in great generality - the heat flow on metric measure spaces (X; d;m) can be defi ned equivalently as gradient flow on L^2(X) for the energy or as gradient flow on the L^2-Wasserstein space P^2(X) for the relative entropy. We will present recent results on the equivalence of the "entropic curvature-dimension condition" and Bochner's inequality. This also includes sharp gradient estimates and new contraction properties of the heat flows in L^2-Wasserstein metric. Among the applications are Li-Yau estimates on metric measure spaces satisfying a curvature-dimension condition. -
10:30
Coffee break -
11:00
Freddy Bouchet, Lyon
Large deviations and averaging for stochastic partial differential equations describing atmosphere dynamics.
We consider the formation of large scale structures (zonal jets and vortices), in planetary atmosphere turbulence, within the barotropic quasi-geostrophic equations (the Ising model of geophysical turbulence). This model includes as a special case the 2D stochastic Navier-Stokes equations. We develop a theory based on tools of statistical physics (large deviations, averaging, kinetic theory) and field theory (instantons). We study the limit of a time scale separation between inertial dynamics on one hand, and the effect of forces and dissipation on the other hand. Using a kinetic theory approach, we prove that stochastic averaging can be performed explicitly in this problem, which is unusual in turbulent systems. It is then possible to integrate out all fast turbulent degrees of freedom, and to get explicitly an equation that describes the slow evolution of zonal jets. For some range of parameters, the dynamics has several attractors, with extremely rare and abrupt transition from one attractor to another. Using large deviation results, derived either from path integrals or from generalization of Freidlin--Wentzell theory, we compute the transition rates and transition trajectories between two attractors. -
12:00
Lunch -
14:00
Stéphane De Gérando, Paris
Introduction au Labyrinthe du temps.
30 minutes talk. Le labyrinthe du temps est à la fois une œuvre dynamique (``work in continuous progress") poly-artistique et technologique (image, son…) et un projet de recherche en cours de développement. Stéphane de Gérando présentera l'oeuvre ainsi que des pistes pour des collaborations possibles avec des mathématiciens, sur le thème de cette pensée labyrinthique, fragmentation de la mémoire jusqu’au seuil entropique d’une œuvre chaotique ou « décomposée ». Stéphane de Gérando, auteur du Labyrinthe du temps est compositeur. Il enseigne la composition et les nouvelles technologies au Conservatoire du XIXe arrondissement de Paris, docteur HDR, Premier Prix et 3eme Cycle de composition du CNSMDP, ancien directeur du CFMI de Strasbourg (département Arts de l'UdS) et du département musique du CEFEDEM d'Aquitaine. -
14:30
Constantin Vernicos, Montpellier
Volume entropy of Hilbert geometries and approximability of convex bodies
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15:30
Coffee break -
16:00
Gabriel Rivière, Lille
Eigenmodes of the wave equation, chaotic dynamical systems and entropy.
This talk is designed for a general audience.
We will discuss some properties of stationary solutions of the wave equation on a bounded euclidean domain (or more generally on a compact manifold). In other words, we will be concerned with the properties of the eigenfunctions of the Laplacian.
Our main focus will be on domains for which the "free motion" of a particle (the geodesic flow) enjoys chaotic properties. We will explain how this chaotic feature influences the properties of the eigenfunctions in the high frequency limit. Finally, we will de! scribe some recent progresses have that been obtained on these questions thanks to the use of the so-called Kolmogorov-Sinai entropy. -
17:00
Djalil Chafaï, Paris
Some probabilistic aspects of Boltzmann entropy.
Starting from the motivation of Boltzmann, we will follow the entropy along the central limit theorem of classical and of free probability, and we will end up with random matrices and Coulomb type gases. -
18:30
Reception at the city hall of Strasbourg by the Mayor of Strasbourg. All participants are invited. We start at 18:00 from the place of the conference. -
Samedi 28 septembre 2013
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10:00
Mauro Carfora, Pavia
The Ubiquitous Entropy: from Geometrical Enumeration to Quantum Field Theory
This talk is designed for a general audience.
I will review the ubiquitous and somewhat unexpected role that entropy plays in a number of geometrical and physical situations. -
11:00
Coffee break -
11:30
Christian Maes, Leuven
Nonequilibrium entropies
In contrast to the quite unique entropy concept useful for systems in (local) thermodynamic equilibrium, there is a variety of quite distinct nonequilibrium entropies, reflecting different physical points. We disentangle these entropies as they relate to heat, fluctuations, response, time-asymmetry, variational principles, monotonicity, volume contraction or statistical forces. However, not all of those extensions yield state quantities as understood thermodynamically. At the end we sketch how aspects of dynamical activity can take over for obtaining an extended Clausius relation.