Institut de Recherche Mathématique Avancée, UMR 7501

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30 mai 2013

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09h00

Hans-Henrik Rugh - Université Paris 11 - Orsay

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.

10h00

Giovanni Gallavotti - Universita' di Roma La Sapienza

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

11h00

Coffea break

11h30

François Ledrappier - University of Notre Dame

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.

14h00

Klaus Schmidt - Universität Wien

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.

15h00

Coffea break

15h30

Rémi Monasson - École Polytechnique and École Normale Supérieure, Paris

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).

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31 mai 2013

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09h00

Jean-René Chazottes - École Polytechnique, Palaiseau

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.

10h00

Viviane Baladi - Københavns Universitet

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).

11h00

Coffea break

11h30

Maher Younan - Université de Genève

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.

14h30

Bertrand Eynard - IPHT, CEA Saclay

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.

15h30

Coffea break

16h00

Shrikrishna G. Dani - Indian Institute of Technology, Bombay

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.

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1er juin 2013

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09h00

Samuel Petite - Université de Picardie - Amiens

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.

10h00

Coffea break

10h30

Vladimir Fock - Université de Strasbourg

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.