# Institut de Recherche Mathématique Avancée, UMR 7501

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 IRMA, UMR 7501 7 rue René-Descartes 67084 Strasbourg Cedex Tél. 33 (0)3 68 85 01 29 Fax. 33 (0)3 68 85 03 28

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## 91e rencontre entre mathématiciens et physiciens théoriciens : Systèmes dynamiques et physique statistique

IRMA, 30 mai - 1er juin 2013

La 91ème rencontre entre mathématiciens et physiciens théoriciens aura pour thème : Systèmes dynamiques et physique statistique.

The 91th Encounter between Mathematicians and Theoretical Physicists will take place at Institut de Recherche Mathématique Avancée (University of Strasbourg and CNRS) on May 30 - June 1st, 2013. The theme will be : "Dynamical systems and statistical physics".

Organizers : Charles Boubel and Athanase Papadopoulos

The invited speakers include :
Jean-René Chazottes (Ecole Polytechnique)
Shrikrishna G. Dani (Bombay)
Bertrand Eynard (CEA)
Giovanni Gallavotti (Rome)
François Ledrappier (Paris)
Rémi Monasson (ENS Paris)
Samuel Petite (Amiens)
Hans Rugh (Orsay)
Klaus Schmidt (Vienna)
Maher Younan (Genève)

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.

Charles Boubel : charles.boubel@unistra.fr

## 30 mai 2013

### Hans-Henrik Rugh - Université Paris 11 - Orsay

Geometry and thermodynamics in the micro-canonical ensemble

The talk should be accessible to all people with a rudimentary knowledge of differential geometry.

Classical equilibrium statistical mechanics deals with the average behavior of a, typically large, Hamiltonian dynamical system. Differential calculus provides a mathematical tool to model this behavior in the so-called micro-canonical ensemble, and may be used to construct thermodynamic quantities like temperature and pressure. We show how such quantities may be observed numerically within the dynamical system itself under the ergodic hypothesis and without resorting to an axiomatic theory of thermodynamics. We also discuss more recent developments on non-equilibrium statistical mechanics where results at present are sparse and incomplete.

### Giovanni Gallavotti - Universita' di Roma La Sapienza

Perturbation theory for a "simple" non equilibrium state

Perturbative construction of the nonequilibrium steady state of a pendulum subject to torque, friction and under a stochastic forcing.

Coffea break

### François Ledrappier - University of Notre Dame

One application of dynamics and thermodynamical formalism to geometry

We consider the universal cover of a closed manifold with negative curvature. We are interested in the behaviour of the heat kernel as the time goes to infinity. In this talk, we show some estimates provided by the dynamics of the geodesic flow. This is part of a joint work in progress with Seonhee Lim.

### Klaus Schmidt - Universität Wien

Algebraic Z^d-actions and their homoclinic points

Algebraic $Z^d$-actions are actions of $Z^d$ by automorphisms of compact abelian groups. If $\alpha : n \to \alpha ^n$ is such a $Z^d$-action on a compact abelian group $X$ then a point $x\in X$ is homoclinic if $\lim_{\|n\|\to\infty }\alpha ^n(x) = 0$. This talk will be devoted to the question of existence of nonzero homoclinic points and to the dynamical consequences of existence or nonexistence of such points.

Coffea break

### Rémi Monasson - École Polytechnique and École Normale Supérieure, Paris

Interferences and transitions between spatial charts in a neuronal model of hippocampus

A great part of this talk is designed for a broad audience.

(joint work with S. Rosay, LPT-ENS, Paris)
Understanding the mechanisms by which space gets represented in the cortex and the hippocampus is a fundamental problem in neuroscience. The experimental discovery of so-called "place cells" and "grid-cells", encoding specific positions in space, provide essential elements in this context. How a spatial chart ie a relation between different points in space, may be built and memorised? In this talk, we will present a model involving binary neurons (that ay be either active or silent), making possible to store several spatial charts. This model may be solved with the help of the techniques of Statistical Physics of Disordered Systems. We will discuss the different possible phases of the system and the essential features of its dynamics (activated diffusion in one chart and transitions between charts).

### 17h00

Boat trip around the old city of Strasbourg. This trip is offered to the participants by the mayor of Strasbourg.

### 19h00

Dinner offered to the participants at the restaurant "Petit bois vert". The address is : Quai de la Bruche (Petite France region, in the historical center of Strasbourg).

## 31 mai 2013

### Jean-René Chazottes - École Polytechnique, Palaiseau

Concentration inequalities for dynamical systems

A great part of this talk is designed for a broad audience.

I will start with a broad introduction intended for a general audience. Then I will specialize to dynamical systems and present some applications of concentration inequalities. Finally, I will sketch how these inequalities can be proven.

### Viviane Baladi - Københavns Universitet

On the Whitney-Holder regularity of the SRB measure in the quadratic family

A great part of this talk is designed for a broad audience.

This is a joint work with M. Benedicks and D. Schnellmann.
For a smooth one-parameter family of smooth hyperbolic discrete-time dynamics (i.e., Anosov systems, which are structurally stable), the "physical" (SRB) measure depends differentiably on the parameter, say t, and the derivative is given by an explicit "linear response" formula (Ruelle, 1997). When structural stability does not hold, the linear response may break down. This was first observed for piecewise expanding interval maps, where linear response holds for tangential families, but where a modulus of continuity t log(t) is possible for transversal families (Baladi-Smania, 2008). The case of smooth unimodal maps is much more delicate. Ruelle (Misiurewicz case) and Baladi-Smania (slow recurrence case) recently obtained linear response for fully tangential families (confined within a topological class). In this talk we focus on the transversal case (e.g. the quadratic family), where we obtain Holder upper and lower bounds (in the sense of Whitney, along suitable classes of parameters).

Coffea break

### Maher Younan - Université de Genève

Topological Glasses

I will review the work I did during my graduate studies on the subject of topological glasses, starting with the 2-dimensional model, then moving to the generalisation to 3 dimensions. In both cases, the phase space is the set of triangulations of the sphere ($S^2$ or $S^3$), the dynamics are given by the Pachner moves that conserve the number of nodes (T1 in 2d, 2-3 and 3-2 in 3d) and the energy is local and a function of the neighbourhood of a node. I will point out some interesting properties of the phase space and I will show that these models exhibit glassy behaviour.

### Bertrand Eynard - IPHT, CEA Saclay

From random matrices to geometry : the topological recursion in mathematical physics

This talk is intended for non-specialists, and is suitable for phd students.

Computing the large N expansion of the eigenvalue statistics of a random matrix, has been an important question for many years, with applications ranging from quantum gravity to electronics or even finance. Recently was found a recursion relation which allows to compute recursively all terms in the asymptotic expansion of any correlation function. The initial data for the recursion is a Riemann surface embedded in CxC, called spectral curve. Beyond random matrices, this recursion can be applied to any embedded Riemann surface, and generates a sequence of differential forms, called "invariants" of the surface. Those invariants posses fascinating mathematical properties (quasi-modular forms, satisfy Hirota equations, special geometry relations,...), and have many applications in physics and mathematics. In particular, many known invariants of enumerative geometry, can be recovered as specializations, for instance Gromov-Witten invariants of Calabi-Yau manifolds, or knot polynomials (Jones polynomial). The talk will introduce this recursion, and present some examples of applications.

Coffea break

### Shrikrishna G. Dani - Indian Institute of Technology, Bombay

Dynamics of the geodesic flow on the modular surface and applications in number theory

We discuss various dynamical properties of the geodesic flow associated with the modular surface ,and what they signify in number theory, especially with regard to values of binary quadratic forms.

### 17h30

Reception offered to the participants by the Mayor of Strasbourg. The reception takes place at the City Hall. Address : Place Broglie - We shall all go there a while after the last talk.

## 1er juin 2013

### Samuel Petite - Université de Picardie - Amiens

The Frenkel-Kontorova model for almost-periodic environments

A great part of this talk is designed for a broad audience.

The Frenkel-Kontorova model describes how a chain of atoms minimizes its total energy on interaction with a substrat. We consider in a joint work with E. Garibaldi and P. Thieullen the case where the environment is almost periodic. A minimizing configuration in the Aubry sense may not detect the configurations at the lowest energy. We introduce a stronger notion of calibrated configurations and prove in the case of a one dimensional quasi-periodic environment,  (e.g. the Fibonacci case), their existence. The main tool we use is the Mather set, the set of minimizing measures, and the space of tilings.

Coffea break

### Vladimir Fock - Université de Strasbourg

Dimer models and integrable systems

A. B. Goncharov and R. Kenyon defined a class of integrable systems on cluster varieties enumerated by convex polygons on a plane with integral vertices. The generating function for the commuting Hamiltonians is given by a partition function of a dimer model on a bipartite graph. Every such system has a family of commuting continuous flows enumerated by integral points inside the  polygons and discrete flows enumerated by integral points on the boundary. We will present our study of their construction and show that their integrable systems coincide with well known ones on affine Poisson-Lie groups; however we will try to show that construction of Goncharov and Kenyon gives a rather new point of view on integrable systems and admit several generalizations.

Dernière mise à jour le 13-05-2013