# Program

### Mini lectures:

• Serge Aubry:
Localized Dynamical Structures in infinite nonlinear networks and open problems.

I will first talk on energy localization as Discrete Breathers for spatially periodic systems whose linear spectrum is absolutely continuous. I will also talk on similar problems for nonlinear systems with discret linear spectrum (observed for instance in disordered systems with Anderson localization) bringing new problems related to KAM theory and Arnold diffusion in infinite dimension.

The main lecture in pdf

• Stéphane Gaubert:
Tropical methods for ergodic control and zero-sum games

In the tropical world, the maximum and the addition are thought of as additive and multiplicative laws. Then, the dynamic programming operators associated to deterministic optimal control problems become linear. This observation has led to the development by several schools of results analogous to linear functional analysis or convex analysis, which met recent works in tropical geometry. I will survey here the application of tropical methods to optimal control, and more generally, to zero-sum games. In the latter situation, I will pay a special attention to the finite state space case, which is better understood although still open from the computational complexity point of view. The topics covered will include: tropical convex sets and cones, their extreme points and rays; Poisson-Martin type representation of stationary solutions and abstract boundary'' of control problems; deformation of Perron-Frobenius operators, limit eigenvalues and eigenvectors and related inequalities; stationary solutions of stochastic one player problems; combinatorial aspects of tropical convexity; equivalence between tropical convexity and zero-sum games with mean payoff; algorithms.

The main lecture: part I
The main lecture: part II
The main lecture: part III

• Rafael de la Llave:
KAM and numerical methods for 1-D models. Numerical results at breakdown.

We present some KAM theorems based on the variational formulation, specially for 1-Dimensional dynamics. The methods apply to systems with long range interactions as well as to systems with quasi-periodic potential.

The methods lead to very fast and realiable algorithms which are suitable for implementations and can be used to study the breakdown of smooth circles. We will present implementations.

We present the standard picture based on renormalization group (developed in the 80's by many people) as well as some more recent developments suggested by the numerical computations.

The main lecture: part I
The main lecture: part II
The main lecture: part III

• Panagiotis Souganidis:
Recent advances in random homogenization of first and second order fully nonlinear pde.

In these lectures I will present an overview of the recent developments to the theory of random homogenization for Hamilton-Jacobi, "viscous" Hamilton-Jacobi and uniformly elliptic second order pde in stationary ergodic environments.

### Schedule of the talks:    (each talk lasts 50 mn + 10 mn of questions)

Monday 12 Tuesday 13 Wednesday 14 Thursday 15 Friday 16 Saturday 17
9:00 -- 10:00 Marie-Claude Arnaud Grigory Litvinov Claude Viterbo Allessio Figali Diogo Gomes Antonio Siconolfi
10:00 -- 11:00 Yaron Ostrover Marianne Akian Michael Bialy Nicola Gigli Régis Monneau Hiroyoshi Mitake
11:00 -- 11:30 break break break break break break
11:30 -- 12:30 Serge Aubry
Lecture I
Rafael de la Llave
Lecture I
Artur Lopes Rafael de la Llave
Lecture II
Serge Aubry
Lecture III
Panagiotis Souganidis
Lecture I
12:30 -- 13:00 lunch at IMB
12:30 -- 14:00
13:00 -- 14:30 lunch at IMB lunch at IMB lunch at IMB lunch at IMB Wine and Cheese
tasting lunch at
IMB
14:30 -- 15:00 Parallel sessions
 Anthony Quas Olivier Ley
Parallel sessions
 Xifeng Su Martino Bardi
Visit of a
Châteaux at
St Emilion
14:00 -- 19:00
Serge Aubry
Lecture II
Rafael de la Llave
Lecture III
15:00 -- 15:30 Free
afternoon
15:30 -- 16:00 Parallel sessions
 Julien Brémont Marco Rorro
Parallel sessions
 Blaz Mramor Andrea Davini
16:00 -- 16:30 break break
16:30 -- 17:00 break break Ugo Bessi
 Oliver Jenkinson Levon Nurbekian
17:00 -- 17:30 Bernard Bonnard Lilian Wong
17:30 -- 18:00 Andrea Venturelli Eduardo Garibaldi
18:00 -- 18:30

 Monday 19 Tuesday 20 Wednesday 21 9:00 -- 10:00 Hitoshi Ishii Toshio Mikami Renato Iturriaga 10:00 -- 11:00 William Mceneaney Rémi Abgrall Rafael Ruggiero 11:00 -- 11:30 break break break 11:30 -- 13:00 Stéphane Gaubert Lecture I Panagiotis Souganidis Lecture III Stéphane Gaubert Lecture III 13:00 -- 14:30 Lunch at IMB Lunch at IMB Lunch at IMB 14:30 -- 15:30 Panagiotis Souganidis Lecture II Stéphane Gaubert Lecture II Renaud Leplaideur 15:30 -- 16:00 Rodrigo Bissacot Ricardo Freire 16:00 -- 16:30 break break 16:30 -- 17:00 Enrico Valdinoci Bob Rink break 17:00 -- 17:30 17:30 -- 18:00 Naoyuki Ichihara Kohei Soga 18:00 -- 18:30

### Titles of the lectures:

1. Name: R. Abgrall
Institution: INRIA Bordeaux Sud Ouest and Universite de Bordeaux

Title: numerical approximation on firts order HJ equations on general meshes.

Abstract: In this talk I will provide a general technique to approximate first order HJ equations on general meshes. The emphasis will be given on high order approximation with the most possible compact stencil. Example on the Eiconal equations or related examples, that plays an important role in computational physics.

2. Name: Marianne Akian
Institution: Ecole Polytechnique, France

Title: Fixed points of discrete convex monotone dynamical systems: from ergodic to negative discount stochastic control problems.

Abstract: Convex, order preserving maps of $R^n$ coincide with the dynamic programming operators of stochastic control problems with $n$ states, discrete time, and possibly negative discount rate. Hence, fixed points of such maps are value functions of infinite horizon stochastic control problems, whereas nonlinear additive eigenvalues are value functions of ergodic stochastic control problems.

We shall present several results concerning the structure of the sets of fixed points of such maps, generalizing max-plus spectral theory. The undiscounted case inspired similar results about the set of stationary solutions of Hamilton-Jacobi-Bellman equations (which correspond to continuous time and state stochastic control problems). One may thus ask if the same can occur for the possibly negative discount case.

This talk covers joint work with Stéphane Gaubert, Benoît David, and Bas Lemmens.

3. Name: Marie-Claude Arnaud Institution: Université d'Avignon

Title: Green bundles along weak KAM solutions.

Abstract: For Tonelli Hamiltonians, we will explain the link between the Green bundles and the weak KAM solutions. In particular, we will explain in which sense the nullity of the Lyapunov exponents for the minimizing measures implies some regularity.

4. Name: Martino Bardi

Title: Critical value of some non-convex Hamiltonians

Abstract: We consider Hamiltonians H convex in the first n moment variables and concave in the other m, e.g., the difference of two usual convex and coercive Hamiltonians, the former in n and the latter in m variables. At least formally, one can associate to it a variational problem where one seeks trajectories whose first n components minimize and the last m maximize a convex-concave Lagrangian. This link can be made rigorous by the theory of differential games, at least in part.

We define as critical value of H the infimum of the constants c such that the stationary Hamilton-Jacobi equation with right hand side c has a (viscosity) subsolution, and show that some of the properties of the critical value in the convex case still hold for a large class of non-convex Hamiltonians.

Using ideas from differential games we prove the existence and a formula for the critical value of some convex-concave Hamiltonians whose corresponding Lagrangian satisfies a saddle condition. Finally we discuss some examples, studied with Gabriele Terrone, where such saddle condition does not hold.

5. Name: Ugo Bessi
Institution: Università Roma 3, Italy

Title: Some remarks on Aubry-Mather theory and the Vlasov equation.

Abstract: We shall recall some results of W. Gangbo and A. Tudorascu on the connections between Aubry-Mather theory and the Vlasov equation; we shall see that the theorems of J. Mather and P. Bernard on the existence of chaotic orbits survive in this setting, together with the notion of minimal measures, Aubry sets, etc... We shall show that the viscous Mather theory developed by D. Gomes can be adapted to this situation.

6. Name: Michael Bialy
Institution: Tel Aviv University, Israel

Title: Maximizing orbits of convex billiards

Abstract: In this talk I shall discuss properties of locally maximizing orbits inside strictly convex billiards. This is well understood in dimension two, for example all orbits corresponding to caustics are locally maximal. However in higher dimension this changes. I shall prove that there are no maximizing orbits sufficiently close to the boundary. Using this result it is easy to prove the existence of conjugate points in the multidimensional case.

7. Name: Rodrigo Bissacot
Institution: Universidade de São Paulo, Bresil

Title: A dynamical proof of the existence of maximizing measures for irreducible countable Markov shifts. Part A.

Abstract: We prove that if Sigma_A(N) is an irreducible Markov shift space over N and f:Sigma_A (N) \rightarrow R is coercive with bounded variation then there exists a maximizing probability measure for f, whose support lies on a Markov subshift over a finite alphabet. Furthermore, the support of any maximizing measure is contained in this same compact subshift. To the best of our knowledge, this is the first proof beyond the finitely primitive case on the general irreducible non-compact setting. It's also noteworthy that our technique works for the full shift over positive real sequences.

8. Name : Bernard Bonnard
Institution: Université de Bourgogne, France

Title: Lieu conjuges et de coupure pour les metriques riemanniennes sur les 26spheres de revolution

Abstract: Dans ce travail en collaboration avec J.B. Caillau on donne un critére pour qu'une métrique Riemannienne sur une 2-sphére de révolution admette un lieu de coupure qui soit un segment et que le lieu conjugé ait exactement 4 plis. On présente ensuite une application au probléme de transport optimal sur l'ellipsoide de révolution dans le cas oblat en montrant que le domaine d'injectivité tangent est convexe si le rapport entre les deux axes est plus petit que 1/31/2 . Des extensions de ces résultats sont discutés pour des problémes de contrôle optimal sur les 2 sphéres.

9. Name: Julien Brémont
Institution: Université Paris-Est, France

Title: Maximizing points and coboundaries for a rotation

Abstract: Consider an irrational rotation of the unit circle and a real continuous function. A point is declared "maximizing" if the growth of the ergodic sums at this point is maximal up to an additive constant. In case of two-sided ergodic sums the existence of a maximizing point for a continuous function implies that it is the coboundary of a continuous function. In contrast, we build for the "usual" one-sided ergodic sums examples in Holder or smooth classes indicating that all kinds of behaviour of the function with respect to the dynamical system are possible. We also show that generic continuous functions are without maximizing points, not only for rotations, but for the transformation 2x mod 1 as well. For this latter transformation it is known that any Holder continuous function has a maximizing point.

10. Name: Andrea Davini
Institution: Università La Sapienza Roma, Italy

Title: Weak KAM Theory topics for stationary ergodic Hamiltonians

Abstract: In this talk I will present some recent results concerning the qualitative analysis of the critical equation associated with a stationary ergodic Hamiltonian. This is performed through a stochastic version of the metric methods of H-J equations and of the Aubry-Mather theory. A fundamental role is played by the notion of closed random stationary set, issued from stochastic geometry. This is a joint research with A. Siconolfi (University of Roma "La Sapienza").

11. Name: Alessio Figalli
Institution: University of Texas at Austin, USA

Title: Closing Aubry sets

Abstract: Given a Hamiltonian H on a compact manifold, the Mañ:é conjecture in C^k topology states that, for a generic potential V (generic w.r.t. the C^k topology), the Aubry set associated to H+V is either a fixed point or a periodic orbit. In this talk we will see how, given a Hamiltonian which possesses a sufficiently smooth viscosity (sub)solution to the Hamilton-Jacobi equation, for any \epsilon >0 there exists a potential V_\epsilon, whose C^2-norm is bounded by \epsilon, such that the Aubry set associated to H+V_\epsilon is either a fixed point of a periodic orbit. This represents a first step through the solution of the Mane Conjecture in C^2 topology. Moreover, we will see how these techniques allow to solve the Mañ:é conjecture in C^1 topology. This is a joint work with Ludovic Rifford.

12. Name: Ricardo Freire
Institution: Universidade de São Paulo, Bresil

Title: A dynamical proof of the existence of maximizing measures for irreducible countable Markov shifts. Part B.

Abstract: Abstract (common): We prove that if $\Sigma_{\mathbf A}(\mathbb N)$ is an irreducible Markov shift space over $\mathbb N$ and $f:\Sigma_{\mathbf A}(\mathbb N) \rightarrow \mathbb R$ is coercive with bounded variation then there exists a maximizing probability measure for f, whose support lies on a Markov subshift over a finite alphabet. Furthermore, the support of any maximizing measure is contained in this same compact subshift. To the best of our knowledge, this is the first proof beyond the finitely primitive case on the general irreducible non-compact setting. It's also noteworthy that our technique works for the full shift over positive real sequences.

13. Name: Eduardo Garibaldi
Institution: Universidade Estatual de Campinas, Bresil

Title: Effective potentials and the zero temperature

Abstract: There are several interesting connections between Aubry-Mather theory and ergodic optimization. In the weak KAM theory, one may find, for instance, the construction of viscosity solutions of Hamilton-Jacobi equations by entropy penalization methods. In a joint work with A. Lopes (UFRGS), we study similar techniques on a transitive two-sided shift in order to obtain effective potentials. We also study how to describe an equivalent version of this problem at zero temperature.

14. Name: Nicola Gigli
Institution: Université de Nice, France

Title: On the interplay between horizontal and vertical derivation

Abstract: I will talk about some recent developments on the interplay between the L^2 and W_2 geometries. Results include the possibility of studying the continuity equation in a genuine metric framework, and the identification of the heat flow as gradient flow in spaces with Ricci curvature bounded from below.

15. Name: Diogo Gomes
Institution: Instituto Superior Técnico de Lisbon, Portugal

Title: Calculus of variations methods in mean-field game

Abstract: In this talk we present several applications of calculus of variations methods to mean-field games. As a motivation we start we by a discussion of the stochastic Evans-Aronsson's problem. We show that this problem gives rise to class of mean-field games for which we establish several new a-priori estimates. In the low dimensional setting these prove the existence of smooth classical solutions. These existence results complement the existence theory by Lions and Lasry

16. Name: Naoyuki Ichihara
Institution: Hiroshima University, Japan

Title: Large time behavior of solutions of Hamilton-Jacobi-Bellman equations with superlinear nonlinearity in gradients

Abstract: We discuss the large time behavior of solutions to the Cauchy problem for semilinear parabolic equations having superlinear nonlinearity in gradients. Equations of this kind often appear in the stochastic control theory. We prove that, as time tends to infinity, the solution converges to a function of variable separation type which is characterized by an ergodic stochastic control problem.

17. Name: Hitoshi Ishii
Institution: Waseda University, Tokyo, Japan

Title: Large time behavior of solutions of Hamilton-Jacobi equations with Neumann type BC.

Abstract: I will discuss Hamilton-Jacobi equations u_t +H(x,Du) = 0 in \Omega \times (0,\infty) with the nonlinear Neumann type boundary condition B(x,Du) = 0 on \partial\Omega \times 0,\infty), where is a bounded domain of Rn. Based on a recent joint work with Guy Barles and Hiroyoshi Mitake, I explain the large time asymptotic behavior of solutions of the above problem as well as the corresponding the Skorokhod problem and Aubry-Mather sets.

18. Name: Renato Iturriaga
Institution: Centro de Investigación en Matemática, Mexico

Title: On the convergence of solutions of discounted Hamilton Jacobi equation

Abstract: We study the convergence of solutions of \lamda u + H(x,Du) = c, as \lambda tends to zero. We show that the limit exist when the Mather quotient, has null 1-dimensional Haussdorf measure.
19. Name: Oliver Jenkinson
Institution: Queen Mary, University of London, United Kingdom

Title: Ergodic Dominance and Ergodic Optimization

Abstract: For various dynamical systems, I will describe certain partial orders imposed on the set of all invariant probability measures. Typically, these orders are induced by the cone of all increasing functions, or the cone of all convex functions, and are related to the notion of stochastic dominance. The identification of maximal and minimal elements of these posets has consequences for ergodic optimization.

20. Name: Renaud Leplaideur
Institution: Université de Brest, France

Title: Some advances in the selection problem.

Abstract: I will present the problem of selection at temperature zero. Given a Dynamical System X and f : X ->X and a potential phi we look for the f-invariant measures mu which maximize the integral of phi with respect to mu. Such a measure is called a phi-maximizing measure.

On the other hand, the Thermodynamical formalism produces an Equilibrium State mu_beta for the potential beta phi with beta > 0. A ground state for phi is a phi-maximizing measure which is an accumulation point for mu_beta as beta goes to infty.

The selection problem is to determine if mu_beta converges, and also how it chooses the limit between all the phi-maximizing measures. The convergence does not always occur, but it turns out that fore some potentials, it is possible to prove convergence and to determine the limit.

The talk will recall part of the background needed to understand and present the problem and will show some results. In particular, we emphasize that the Max-Plus formalism seems to be a powerful tool to solve the problem.

21. Name: Olivier Ley
Institution: Université de Rennes, France

Title: Long time behavior for some systems of first-order Hamilton-Jacobi equations.

Abstract: We show the large time behavior of solutions of a weakly coupled system of Hamilton-Jacobi equations. This result extend to the case of systems the well-known theorem of Namah-Roquejoffre (1999) Some relations with control will be discussed. This is a joint work with F. Camilli, P. Loreti from Roma and V. Nguyen from Rennes.

22. Name: Grigory L. Litvinov
Institution:Independent University of Moscow(IMU), Russia

Title: Dequantization,idempotent/tropical mathematics and the hamilton-jacobi equation

Abstract: This is a talk on heuristic aspects of tropical and idempotent mathematics. Tropical mathematics can be treated as a result of a dequantization of the traditional mathematics as the Planck constant tends to zero taking purely imaginary values. For example, the field of real or complex numbers can be treated as a quantum object whereas tropical algebras and idempotent semirings can be examined as "classical" objects (a semiring is called idempotent if the semiring addition is idempotent,i.e. x + x = x). The Hamilton-Jacobi equation is a result of this dequantization applied to the Schroedinger equation.

In the spirit of N. Bohr's correspondence principle there is a (heuristic) correspondence between important, useful, and interes- ting constructions and results over fields and similar results over idempotent semirings. For example, the superposition principle in quantum mechanics (i.e. the linearity of the Schroedinger equation) corresponds to a linearity of the Hamilton-Jacobi and Bellman equa- tions over idempotent semirings. A systematic application of this correspondence principle leads to a variety of results including such exotic applications as a methodology to construct computer devices (processors) for numerical calculations and to get the cor- responing patents.

Dequantization of mathematical structures and relations between dequantization and the so-called tropical geometry are discussed.

23. Name: Artur Lopes
Institution: Universidade Federal do Rio Grande do Sul, Bresil
Title: Ergodic Transport
Abstract: We present several results where techniques of Transport Theory and Transshipment are used in order to analyze problems in Ergodic Theory. In particular one can get that Ergodic Optimization can be seen as a particular case of Ergodic Transport.
These results are joint works with several collaborators.

24. Name: William M. McEneaney
Institution: University of California San Diego, USA

Title: Computational-Cost Optimality and Idempotent Approaches for HJ PDEs

Abstract: Idempotent (e.g., max-plus, min-max) algebra based methods for solution of Hamilton-Jacobi PDEs have had tremendous success for some classes of problems. This is particularly the case for high-dimensional problems (four to fifteen space dimensions), where many orders of magnitude speed improvement over traditional approaches can sometimes be seen. On the other hand, these methods are more difficult to apply and code than classical approaches. Hence, critical questions regard how, and under what conditions, these methods may be expected to perform well. A definition of epsilon-complexity for value functions (HJ PDE viscosity solutions) will be introduced, where the definition will cover multiple semiring and associated basis representations. We will discuss conditions under which idempotent algebra based methods can obtain an approximate solution with computational complexity proportional to the epsilon complexity of the value function. A priori estimation of the epsilon complexity remains an open question.

25. Name: Toshio Mikami
Institution: Hiroshima University, Japan

Title: Stochastic optimal transportation and marginal problem for stochastic processes

Abstract: In this talk we introduce a stochastic optimal transportation as a stochastic analogue of optimal mass transportation problem and its application to marginal problems for stochastic processes. If time permits, I'd like to introduce recent progress.
Part of my talk is based on the joint work with Prof. Michèle Thieullen, Univ. Paris 6.

26. Name: Hiroyoshi Mitake Institution: Hiroshima University, Japan

Title: Large-time Asymptotics for noncoercive Hamilton-Jacobi equations appearing in crystal growth Abstract: We investigate the large-time behavior of viscosity solutions of Hamilton-Jacobi equations with noncoercive Hamiltonian in a multidimensional Euclidean space. Our motivation comes from a model describing growing faceted crystals recently discussed by E. Yokoyama, Y. Giga and P. Rybka (Phys. D 237 (2008), no. 22, 2845--2855). We prove that the average growth rate of a solution is constant only in a subset, which will be called effective domain, of the whole domain and give the asymptotic profile in the subset. This means that the large-time behavior for noncoercive problems may depend on the space variable in general, which is different from the usual results under the coercivity condition. Moreover, on the boundary of the effective domain, the gradient with respect to the x-variable of solutions blows up as time goes to infinity. Therefore, we are naturally led to study singular Neumann problems for stationary Hamilton-Jacobi equations. We establish the existence and comparison results for singular Neumann problems and apply the results for a large-time asymptotic profile on the effective domain. This is a joint work with Prof. Giga and Dr. Liu.

27. Name: Régis Monneau
Institution: Ecole des Ponts ParisTech, France

Title: Hamilton-Jacobi equations on a jonction and application to traffic flow.
Abstract: This work is concerned with the study of a model case of first order Hamilton-Jacobi equations posed on a junction'', that is to say the union of a finite number of half-lines with a unique common point. The main result is a comparison principle. We also prove existence and stability of solutions. The two challenging difficulties are the singular geometry of the domain and the discontinuity of the Hamiltonian. As far as discontinuous Hamiltonians are concerned, these results seem to be new. They are applied to the study of some models arising in traffic flows. The techniques developed in the present work provide new powerful tools for the analysis of such problems.

28. Name: Blaz Mramor
Institution: VU University Amsterdam, The Netherlands

Title: Dichotomy for minimizers of monotone recurrence relations

Abstract: Variational monotone recurrence relations arise in solid state physics as generalizations of the Frenkel-Kontorova crystal model. For such problems, Aubry-Mather theory establishes the existence of Birkhoff global minimizers of arbitrary rotation number. Global minimizers of a nearest neighbor crystal model, can cross only once. As a consequence, every one of them is Birkhoff. In crystals with a larger range of interaction the single crossing property does not hold and there could exist global minimizers that are not Birkhoff. Under a natural twist condition, the following dichotomy holds: global minimizers are either Birkhoff, and thus very regular, or extremely irregular and nonphysical: they then grow exponentially and oscillate.

29. Name: Levon Nurbekian Institution: University of Texas at Austin, USA

Title: Lagrangian Dynamics and a Weak KAM theorem

Abstract: The space L^2[0,1] has a natural Riemannian structure on the basis of which in their recent work W. Gangbo and A. Tudorascu introduced an infinite dimensional torus T. For a certain class of Hamiltonians they prove an existence of a viscosity solution to the cell problem on T. Furthermore they exploit the solution to prove the existence of the solution to the so called nonlinear 1-dimensional Vlasov system and obtain asymptotics for the solution.

In the current work we try to generalize results obtained by Gangbo and Tudorascu to the so called higher dimensional case, where the ambient space is the L^2([0,1]^d; R ^d) instead of the L^2[0,1]. More precisely, for a certain class of Hamiltonians H, defined on the cotangent bundle of the infinite dimensional Hilbert space L^2([0,1]^d; R ^d), and for any c \in R ^d we prove the existence of the periodic continuous viscosity solution U to the cell problem H(M, Du+c)=\lambda, where \lambda in R is a constant depending on c.

U is a value function of a certain variational problem. We study the existence of the minimizing trajectories of that problem. It turns out that U is differentiable in a Frechet sense on a dense G_{\delta} subset of L^2([0,1]^d; R ^d) and for every point of differentiability there exists a unique trajectory starting at that point. Furthermore, we prove the existence of the so called two sided minimizers.

Once we prove the existence of the solution to that problem we are able to prove the existence of the solution to the nonlinear d-dimensional Vlasov system and obtain an asymptotics for the solution.
30. Name: Yaron Ostrover
Institution: Tel-Aviv University, Israel

Title:Bounds for Minkowski Billiard Trajectories.

Abstract: In this talk we shall discuss how a certain symplectic invariant on the classical phase space can be used to obtain bounds and inequalities on the length of the shortest periodic billiard trajectory in a convex domain. Moreover, we shall explain how the above approach applies both for the classical (Euclidean) case and the more general case of Minkowski billiards. This talk is based on a joint work with Shiri Artstein-Avidan from Tel-Aviv University.

31. Name: Anthony Quas

Title: Rates of approximation in ergodic optimization

Abstract: A well-known conjecture states that for a residual set of Holder continuous potentials, the invariant measure optimizing the integral is supported on a periodic orbit. Simple examples show that this cannot hold for all Holder continuous potentials. In the case where the optimizing measure is not supported on a periodic orbit, we consider the rate of approximation of the measure by periodic orbit measure.

32. Name: Bob Rink
Institution: VU University Amsterdam, The Netherlands

Title: A destruction theorem for generalized Frenkel-Kontorova crystals

Abstract: The atoms of a generalized Frenkel-Kontorova crystal interact beyond their nearest neighbors. This implies that the ground states of these crystals are not characterized as orbits of a twist map.
In this talk, I will present the following converse KAM theorem for such crystals: if the crystal model admits a continuous family of ground states with an average particle spacing that is "easy to approximate by rational numbers", then this family can be destroyed by an arbitrarily small smooth perturbation of the crystal model. This means that a "typical" crystal will display "forbidden regions" for its atoms.
I obtained this result with my PhD student Blaz Mramor. It generalizes a theorem of Mather for the destruction of Liouville invariant circles of twist maps. Our proof is quite different though and may allow for generalizations to lattice Aubry-Mather theory and elliptic PDEs.

33. Name: Marco Rorro
Institution: CASPUR, Università La Sapienza Roma, Italy

Title: Numerical methods for weak KAM (sub-)solutions and Aubry-Mather sets

Abstract: We present two numerical methods for the computation respectively of viscosity solutions and viscosity sub-solutions of Hamilton-Jacobi equations. The first method is based on direct discretization of a cell problem via a semi-Lagrangian approximation scheme. Then the Aubry-Mather set is obtained following backward the characteristic lines. The second method is based on a variational formulation due to Evans and allows to obtain directly an approximation of the Aubry-Mather set.
We discuss the properties of the two approaches and present some numerical tests in 1D and 2D.

34. Name: Rafael Ruggiero

Title: Generic properties of closed orbits of Tonelli Hamiltonians from Mane's viewpoint

Abstract: Given a Tonelli Hamiltonian H in a compact manifold and a real number E we show that there exist arbitrarilly small C-infinity functions f:M-->R such that the Hamiltonian Hf = H+f satisfies the following property: in the energy level Hf =E the spectra of the differentials of Poincaré maps of periodic orbits are disjoint from the set of roots of unity.

Combining this result with the work of E. Oliveira we get a version of the Kupka-Smale theorem for Tonelli Hamiltonians from Mañé's viewpoint. This is a joint work with Ludovic Rifford from the University of Nice, Sophia-Antipolis.

35. Name: Antonio Siconolfi
Institution: Università La Sapienza Roma, Italy

Title: Homogenization of noncoercive Hamilton-Jacobi equations

Abstract: We study the homogenization problem for convex Hamilton-Jacobi equations without assuming any coercivity condition. We show that homogenization takes place if a controllability-type condition is fulfilled. More precisely, we demonstrate that it is sufficient that a multivalued dynamics, suitably related to the Hamiltonian, does not admit any periodic invariant subset. As an application, we consider the G-equation, appearing in the turbulent combustion models, and recover recent results of Xin-Yu and Cardialiguet-Nolen-Souganidis, effecting considerable simplication of the proof.

36. Name: Kohei Soga
Institution: Waseda University, Japan

Title: Stochastic and variational approach to the Lax-Friedrichs scheme

Abstract: I present a stochastic and variational approach to the Lax-Friedrichs scheme applied to scalar conservation laws. This is a finite difference version of Fleming's results ('69), where the vanishing viscosity method is characterized by stochastic processes and calculus of variations. We convert the difference equations into these of Hamilton-Jacobi types and introduce corresponding calculus of variations which consists of random walks. The convergence of approximation is derived from asymptotics of continuous limit of random walks under hyperbolic scaling. The main advantages of our approach is the following: Our framework is basically pointwise convergence, not $L^1$ as usual, which yields uniform convergence except "small" neighborhoods of shocks; The convergence proof is verified for arbitrarily large time interval, which is hard to obtain in the case of flux functions of general types depending on both space and time; The approximation of characteristics curves is available as well. As Bessi ('03) nicely applies Fleming's results to the weak KAM theory, the stochastic and variational approach to the Lax-Friedrichs scheme can be a useful tool for the numerical analysis of the weak KAM theory.

37. Name: Xifeng Su
Institution: Chinese Academy of Sciences, China

Title: KAM theory for quasi-periodic equilibria in 1-D quasiperiodic media.

Abstract: We consider Frenkel-Kontorova models corresponding to 1 dimensional quasicrystals.

We present a KAM theory for quasi-periodic equilibria. The theorem presented has an a-posteriori format. We show that, given an approximate solution of the equilibrium equation, which satisfies some appropriate non-degeneracy conditions, then, there is a true solution nearby. This solution is locally unique. Such a-posteriori theorems can be used to validate numerical computations and also lead immediately to several consequences a) Existence to all orders of perturbative expansion and their convergence b) Bootstrap for regularity c) An efficient method to compute the breakdown of analyticity.

Since the system does not admit an easy dynamical formulation, the method of proof is based on developing several identities. These identities also lead to very efficient algorithms.

38. Names: Enrico Valdinoci
Institution: Università di Roma Tor Vergata, Italy

Title: Semilinear equations and (non)local minimal surfaces

Abstract: I would like to present some differential equations arising in phase cohexisence models. In the classical case, phase changes are regulated by a local interaction, which leads to an elliptic partial differential equations and which produces interfaces close to surfaces of minimal perimeter.
In order to take into account long range effects, a new model has been recently considered, in which the interaction is non-local, for instance, modeled on the Gagliardo seminorm on fractional Sobolev spaces. The associated equation is driven by a fractional Laplace operator and the interfaces are now either local or non-local, according to the fractional parameter.

39. Name: Andrea Venturelli
Institution: Université d'Avigon, France Title: Brake to Syzygy map in the zero angular momentum planar three body problem.

Abstract: We consider the zero angular momentum planar three body problem, and we fix the energy to a negative value. A syzygy is a three-body configuration where the three body are collinear. A result of R. Montgomery says that every solution with negative energy and zero angular momentum has a syzygy in forward time. For this reason, if we take a three body configuration on the zero velocity curve and the corresponding solution starting with zero velocity, at some time we will fond a syzygy. In this way, we define a map called "brake to syzygy" map. We will describe some interesting properties of the "brake to syzygy" map, and in particular, we construct an interesting brake periodic solution. It is a joint work with Rick Moeckel and Richard Montgomery.

40. Name: Claude Viterbo
Institution: Ecole Polytechnique, France

Title: Symplectic Homogenization and applications (?)

41. Name: Lilian Wong
Institution: Georgia Tech School of Mathematics, USA

Title: On the behavior at infinity of solutions to difference equations in Schroedinger form

Abstract: We offer several perspectives on the behavior at infinity of solutions of discrete Schroedinger equations. First we study pairs of discrete Schroedinger equations whose potential functions differ by a quantity that can be considered small in a suitable sense as the index n \rightarrow \infty. With simple assumptions on the growth rate of the solutions of the original system, we show that the perturbed system has a fundamental set of solutions with the same behavior at infinity, employing a variation-of-constants scheme to produce a convergent iteration for the solutions of the second equation in terms of those of the original one. We use the relations between the solution sets to derive exponential dichotomy of solutions and elucidate the structure of transfer matrices.

Later, we present a sharp discrete analogue of the Liouville-Green (WKB) transformation, making it possible to derive exponential behavior at infinity of a single difference equation, by explicitly constructing a comparison equation to which our perturbation results apply. In addition, we point out an exact relationship connecting the diagonal part of the Green matrix to the asymptotic behavior of solutions. With both of these tools it is possible to identify an Agmon metric, in terms of which, in some situations, any decreasing solution must decrease exponentially. This is joint work with Evans Harrell.