Dario Bambusi
Birkhov normal form for tame Hamiltnian PDEs
I will present an abstract Birkhoff normal form theorem for Hamiltonian Partial Differential Equations. The theorem applies to semilinear equations with nonlinearity belonging to a large class. As a corollary one gets that any small amplitude solution remains very close to a torus for very long times. Applications to some concrete models (NLW,NLS) will also be presented. From the general theorem one also deduces a result of almost global existence of solution.
Joint work with Benoit Grebert

Leonid Bunimovich
Some Billiard News
I'll describe some examples of billiards with divided phase space (containing integrable islands and chaotic components) in 2D and 3D. Other examples are concerned with convex chaotic billiards in 3D and with positive measure ergodic components consisting of rays propagating in one direction ("tracks").

Dolgopyat, Dmitry
Averaging in systems with periodic fast motion
We consider the averaging problem in the case when the fast motion is the simplest possible, namely, periodic. We show that the methods of partially hyperbolic dynamics can be used to establish the convergence of adiabatic invariants to a Markov process. We conclude with some open questions especially emphasizing quasiperiodic case.

Eliasson, Hakan
KAM for the non-linear Schr\"odinger equation
We shall discuss the non-linear Schr\"odinger equation with perodic boundary conditions in dimension $d$. This is an infinite-dimensional Hamiltonian system and one central problem is the perturbation theory for lower-dimensional tori = quasi-periodic solutions, usually known as KAM. The difficulties in applying KAM in infinite dimensions are substantial and become larger with increasing $d$. For NLS the case $d=1$ was solved in the late 80's by Kuksin and, later, Bourgain. The case $d=2$ was solved in the early 90's by Bourgain, but by an approach (known as the Craig-Wayne scheme) that provides less information than KAM. We shall discuss these issues and, if times admits, report on a recent work (with Kuksin) that aims to solve the problem for any $d$.

Fathi, Albert
Sard, Whitney, Assouad and Mather
For a Lagrangian $L:TM\to \bf R$ on a compact manifold, John Mather has introduced a function $h:M\times M\to \bf R$, called the Peierls barrier. The set ${\cal A}=\{x\in M\mid h(x,x)=0\}$ is the Aubry set. The function $\delta(x,y)=h(x,y)+h(y,x)$ is a semi-metric on ${\cal A}$. Identifying points $x, y$ such that $\delta(x,y)=$ we obtain a quotient, called the Mather quotient, which is a genuine metric space. John Mather has asked the following question (related to his work on Arnold diffusion): If $L $ is C$^\infty$ is the Mather quotient totally disconnected? For any $r\geq 2$, Mather has given an example of a C$^r$ Lagrangian on some high-dimensional torus whose Mather quotient is isometric to the interval. In this lecture we will explain the relationship of this problem with Whitney's extension theorem, Whitney's counterexamples to Sard's theorem, and the work of Assouad on bi-Lipschitz embeddabilty in Euclidean space of metric spaces with the doubling property. We will explain how to obtain a Lagrangian whose Mather quotient is Lipschitz equivalent to any given metric space with the doubling property.

Gentile, Guido
Stability for quasi-periodically perturbed Hill's equations
I will describe a result, obtained in a joint work with J. Barata and D. Cortez, for quasi-periodically perturbed Hill's equations. Assuming Diophantine conditions on the frequencies of the decoupled system, but without making any assumptions on the perturbing potential, we prove that quasi-periodic solutions of the unperturbed equation can be continued for values of the perturbation parameter in a Cantor set of relatively large measure around the origin. The method is based on a resummation procedure of a formal Lindstedt series for a generalized Riccati equation associated to Hill's equation.

Diogo Aguiar Gomes
A variational formulation for the Navier-Stokes equation
In this talk we discuss a new variational principle for the Navier-Stokes equation which asserts that its solutions are critical points of a stochastic control problem in the group of area-preserving diffeomorphisms. This principle is a natural extension of the results by Arnold, Ebin, and Marsden concerning the Euler equation.

Ursula Hamenstaedt
Dynamical properties of the Teichmueller geodesic flow
We show that the dynamical properties of the Teichmueller geodesic flow on moduli space are very similar to the properties of the geodesic flow on a non-compact hyperbolic manifold of finite volume: The Lebesgue measure is mixing, and the set of initial directions of flow lines remaining in some compact subset of moduli space has full Hausdorff dimension.

Konstantin Khanin
Renormalizations, multi-dimensional continued fractions and KAM theory
We use a new multi-dimensional continued fractions algorithm to prove several results of the KAM type. The proofs is very simple conceptually and based on renormalizations.

Kuksin, Sergei
On damped-driven Hamiltonian PDE
Consider a nonlinear PDE of the following form:
[hamiltonian PDE] + \nu [Laplacian] = C(\nu) [random force],
where \nu>0 is a small parameter. I consider behaviour of solutions of this equation when \nu goes to zero and the scaling constant C(\nu) is such that the solutions remain of order one. In particular, I shall discuss this problem for the case when the Hamiltonian PDE is integrable, and show that the limiting behaviour is described by an infinite-dimensional SDE with a non-additive noice. I believe that these results are related to the misterious phenomenon of weak turbulence. They are obtained jointly with Andrei Piatnitski in a work under preparation.

Gallavotti Giovanni
Lindstedt series and their divergent subseries

Marian Gidea
Topological methods in the instability problem of Hamiltonian systems
We use topological methods to investigate some recently proposed mechanisms of instability (Arnol'd diffusion) in Hamiltonian systems. In these mechanisms, chains of heteroclinic connections between whiskered tori are constructed, based on the existence of a normally hyperbolic manifold $\Lambda$, so that: (a) the manifold $\Lambda$ is covered rather densely by transitive tori (possibly of different topology), (b) the manifolds $W^s_\Lambda$, $W^u_\Lambda$ intersect transversally, (c) the systems satisfies some explicit non-degeneracy assumptions, which hold generically. In this paper we use the method of correctly aligned windows to show that, under the assumptions (a), (b), (c), there are orbits that move a significant amount. As a matter of fact, the method presented here does not require that the tori are exactly invariant, only that they are approximately invariant. Hence, compared with the previous papers, we do not need to use KAM theory. This lowers the assumptions on differentiability. Also, the method presented here allows to produce concrete estimates on the time to move, which were not considered in the previous papers.

Roberto Livi
Anomalous heat transport in low-dimensional systems
Statistical fluctuations are strongly dependent on the space dimension and may yield ill-defined transport coefficients in stationary out--of--equilibrium conditions. Specifically, numerical simulations indicate that heat conductivity diverges as a power--law of the system size in several 1d models of coupled anharmonic oscillators and hard--sphere gases. In this talk we aim at summarizing the recent theoretical developments providing an explanation of these anomalous hydrodynamic properties.

Robert MacKay
Towards an ergodic space-time explanation of Hubble's law
Hubble's law relates redshift with intensity for the radiation received from distant objects. Assuming existence of a preferred time coordinate such that each constant-time section is nearly homogeneous, it is interpreted as implying an expanding universe. Very similar relations follow, however, for space-times which have no net expansion but instead have significant inhomogeneity, albeit statistically homogeneous. For example, for an emitter and receiver which remain on average a constant distance apart, the average redshift with respect to receiver time equals its variance with respect to emission time, hence positive. It is plausible that the variance increases with distance between the emitter and receiver. An additional factor is that intensity is determined by not just an inverse square law but also focusing effects. There is a correlation between focusing and redshift. Combining these two factors may lead to predictions compatible with Hubble's law.

Renato Iturriaga
On the Stochastic Aubry Mather
We prove the differentiability of the stochastic analogues of $\alpha $ and $\beta $ Mather's functions. We also prove that the solution to the viscous Hamilton Jacobi equation associated to $\af$ is differentiable in the parameter.

Jürgen Pöschel
On the Well-Posedness of KdV in Weighted Sobolev Spaces
We consider the KdV equation \[ u_t = -u_{xxx}+6uu_x \] with periodic boundary conditions. We will show that the associated initial value problem is well posed in a large family of weighted Sobolev spaces, which includes families of spaces of analytic functions.
The proof is based on the global system of Birkhoff coordinates -- the cartesian counterpart of action-angle coordinates -- for the integrable KdV equation on $H^1(S^1)$ developed in \cite{KP}. They allow us to write the KdV equation as a classical integrable system in infinitely many coordinates. The main point is to show that this coordinate system also respects weighted Sobolev spaces. That is, weighted subspaces of $H^1$ are mapped into the corresponding spaces of weighted $\ell^2$-sequences. Then it is obvious that the evolution of this system stays within such spaces. By applying a simple extension of the KAM theory of KdV, also described in \cite{KP}, this result applies also to small Hamiltonian perturbations of the KdV equation, provided the perturbation has the corresponding regularity.
(Work in collaboration with Thomas Kappeler)

Harald Posch
Tangent-space dynamics of many-body systems
We characterize the phase-space instability of many-body systems in terms of their sets of Lyapunov exponents. Both soft and hard interaction potentials are considered. The computer simulations are also extended to rough hard disks, which provides the possibility to consider qualitatively-different degrees of freedom, translation and rotation. We study the spatial localization of the perturbation vectors associated with the large exponents, and the delocalized Lyapunov modes connected with the small (in the absolute sense) exponents. For rough hard disks, translation-rotation coupling has a pronounced effect on the Lyapunov spectrum. Corresponding spectra are compared to our previous results for hard dumbbells. The relaxation times for tangent-vector rotation is also examined.

Szasz Domokos
SLIGHTLY SUPERDIFFUSIVE DYNAMICAL MOTIONS
For the planar, infinite horizon Lorentz process with a periodic configuration of scatterers, denote by $S_n$ the displacement of the Lorentz particle in the moment of the $n$th collision. P. Bleher, in 1992, gave a heuristic proof for the existence of a Gaussian limit law of $\frac{S_n}{\sqrt{n \log n}}$. This is what we call a slightly superdiffusive behaviour. (N. B. in the case of a finite horizon, the scaling is $\sqrt{n}$, i. e. diffusive.) Last year, with T. Varj\'u, we presented a sketch of a rigorous proof of Bleher's statement. Recently, P. B\'alint and S. Gou\"ezel discovered that, for the stadium billiard and typical functions, the same slightly superdiffusive limit law holds. By also using their arguments, now we can strongly simplify our approach for establishing Bleher's statement. First this proof will be presented. Moreover, by extending our earlier methods to this case, we can also obtain a local version of the aforementioned global limit law and use it for proving the recurrence of the planar Lorentz process in general. It is interesting to mention that, by results of N. Sim\'anyi and F. P\`ene, this recurrence also implies the ergodicity of the infinite, invariant (Liouville-)measure of the Lorentz process. The results are joint with T. Varj\'u.

Jean-Christophe YOCCOZ
Diophantine conditions for interval exchange maps
We will discuss conditions related to the Rauzy-Veech-Zorich algorithm for interval exchange maps which allow to solve the associated cohomological equations. This is a joint work with Stefano Marmi and Pierre Moussa

Wojtkowski, Maciej
Jacobi fields, symmetries and hyperbolic billiards

Anton Zorich
Closed trajectories on flat surfaces corresponding to quadratic differentials
(joint work with H.Masur)