Titles and Abstracts Marie-Claude Arnaud (Université d Avignon, France) Arnol d-liouville theorem in weak regularity. Classical Arnol d-liouville theorem describes precisely the dynamics of Hamiltonian systems that have enough independent C 2 integrals. More precisely, by using a C 1 symplectic change of coordinates, the theorem reduces the Hamiltonian to a Hamiltonian that depends only on the action variables. Here we focus on what happens with low regularity, when we obtain a symplectic homeomorphism instead of a C 1 diffeomorphism. In particular, we will prove that when the integrals are just C 1 and when the Hamiltonian is Tonelli, Arnol d-liouville theorem remains true with a symplectic homeomorphism instead of a C 1 change of coordinates and we will discuss other cases. The motivation for studying low regularity is that when a Tonelli Hamiltonian has no conjugate points, only the existence of continuous integrals can be proved. Dario Bambusi (Università degli Studi di Milano, Italy) Reducibility of the forced quantum oscillator in d dimensions. Consider the Schrödinger equation ψ = ψ + V (x)ψ + ɛw (x, ωt)ψ, x R d, V (x) := d νj 2 x 2 j with W a perturbation which depends quasi-periodically on time and goes to zero at infinity as a function of x. The vector (ν, ω) is assumed to be diophantine. I will present a reducibility of the system, namely existence of a unitary transformation, quasiperiodically dependent on time, which conjugates the system to a time independent one. Boundedness of Sobolev norms and pure point nature of the Floquet spectrum follow. The main difficulty is that the difference between couples of eigenvalues of the harmonic oscillator are dense on the real axis. To overcome this difficulty we first use pseudodifferential calculus in order to conjugate the system to a system with a very regularizing perturbation. Then we develop KAM theory in order to actually get reducibility. The fact the perturbation is very smoothing allows one to control the very bad small denominators that occuring the KAM procedure. Work in collaboration with B. Grbert, A. Maspero, R. Montalto, D. Robert. Massimiliano Berti (S.I.S.S.A. Trieste, Italy) Dynamics of Water waves. We present new recent existence results of small amplitude time quasiperiodic standing waves solutions of the water waves equations, as well as long time existence results for the initial value problem. Rafael de la Llave (Georgia Tech, USA) Instability in Hamiltonian problems in Celestial Mechanics. We consider the problem of whether small perturbations of a Hamiltonian systems can lead to large effects over time. A way of establishing this in concrete systems is to find invariant manifolds and scattering maps which 1 j=1
satisfy certain concrete properties. (Joint work with M. Gidea and T. M- Seara). We will detail a general rigorous mechanism and then present its application in the Jupiter-Sun three body problem. We rigorously show that if some explicit integrals do not vanish, for all non-zero and sufficiently small values of the eccentricity, there are orbits (close to the Lyapunov orbits near a Lagrange point) whose energy increases by a fixed amount (independent of the eccentricity). The integrals are evaluated using standard methods and are found to be not zero (estimating the errors in the conventional way of numerical analysis, we believe that we get at least 5 figures accuracy). (Joint work with M. Gidea and T. M. Seara). Similar methods have been used recently by A. Granados in coupled oscillators applied to energy harvesting. Albert Fathi (Georgia Tech, USA) Recurrence on abelian cover. If h is a homeomorphism on a compact manifold which is chain-recurrent, we will try to understand when the lift of h to an abelian cover is also chainrecurrent. Bassam Fayad (CNRS, IMJ-PRG, France) Some constructions of diffusion in Hamiltonian dynamics. Jacques Féjoz (Université Paris-Dauphine & Observatoire de Paris, France) Linear billiards and Lagrangian relations. Consider as a billiard table the complement of a finite collection of linear subspaces within a Euclidean vector space. A trajectory is a constant speed polygonal curve with vertices on the subspaces and change of direction upon hitting a subspace governed by conservation of momentum (mirror reflection). The itinerary of a trajectory is the list of subspaces it hits, in order. Such non-deterministic billiard process is motivated by the high-energy limit of the N-body problem we construct. Two basic questions are: (A) Are itineraries finite? and (B) What is the structure of the space of all trajectories having a fixed itinerary? In a beautiful series of papers Burago-Ferleger-Kononenko answered (A) affirmatively by using non-smooth metric geometry ideas and the notion of a Hadamard space. We answer (B) by proving that this space of trajectories is diffeomorphic to a Lagrangian relation on the space of lines in the Euclidean space. Our methods combine those of BFK with the notion of a generating family for a Lagrangian relation. This is a joint work with Andreas Knauf and Richard Montgomery. Alessio Figalli (ETH Zürich, Switzerland) Tba. Benoît Grébert (Université de Nantes) Growth of Sobolev norms for abstract linear Schrödinger equations. We prove an abstract theorem giving a t ɛ bound ( ɛ > 0) on the growth 2
of the Sobolev norms in linear Schrödinger equations of the form i ψ = H 0 ψ + V (t)ψ when the time t. The abstract theorem is applied to several cases, including the case where H 0 is the (resonant or nonresonant) Harmonic oscillator in R d and V (t) a pseudodifferential operator of order smaller than H 0 depending in a quasiperiodic way on time. The proof is obtained by first conjugating the system to some normal form in which the perturbation is a smoothing operator and then applying the a recent results of Maspero-Robert. This is a joint work with D. Bambusi, A. Maspero and D. Robert. Helmut Hofer (IAS Princeton, USA) Feral J-Pseudohomolomorphic Curves in Dynamics. I will discuss some current joint work with Joel Fish, in which we de fine and establish properties of a new class of pseudoholomorhic curves (feral J-curves) to study certain divergence free flows in dimension three. As an application we show that if H is a smooth, proper, Hamiltonian in R 4, then no energy level of H is minimal, answering a question raised by M. Herman in his ICM98 address. Thomas Kappeler (Universität Zürich, Switzerland) On a version of the Arnold-Liouville theorem in infinite dimension: a case study. It is well known that the focusing nonlinear Schrödinger equation (fnls) is an integrable PDE. When considered on the circle, the periodic eigenvalues of the Zakharov-Shabat (ZS) operator, appearing in the Lax pair formulation of the fnls equation, form an infinite set of integrals of motion. Note that unlike other integrable PDEs such as the KdV equation or the defocusing nonlinear Schrödinger equation, the fnls equation exhibits features of hyperbolic dynamics, in particular homoclinic orbits. In form of a case study for the fnls equation, I present a version of the Arnold-Liouville theorem in infinite dimension. This is joint work with Peter Topalov. Konstantin Khanin (University of Toronto, Canada) Schrödinger operators with quasi-periodic potentials and the Aubry- Mather theory. Michela Procesi (Università Roma Tre, Italy) Small quasi periodic solutions for a class of quasi-linear dispersive PDEs on the circle. I shall discuss existence and linear stability of small quasi-periodic solutions for quasi linear dispersive PDEs on the circle; I shall particularly concentrate on the DP equation which is an integrable Hamiltonian system with asymptotically linear dispersion law. I will give an overview of the general strategies as well as of the difficulties in dealing specifically with the DP equation. This is based on joint work with R. Feola and F. Giuliani. Tere Seara (UPC Barcelona, Catalunya) A General Mechanism of Instability in Hamiltonian Systems. 3
We present a general mechanism to establish the existence of diffusing orbits in a large class Hamiltonian systems, in particular for near integrable ones. Our approach relies on successive applications of the outer dynamics along homoclinic orbits to a normally hyperbolic invariant manifold. The information on the outer dynamics is encoded by a geometrically defined map, referred to as the scattering map. We find pseudo-orbits of the scattering map that keep moving in some privileged direction. Then we use the recurrence property of the inner dynamics, restricted to the normally hyperbolic invariant manifold, to return to those pseudo-orbits. Finally, we apply topological methods to show the existence of true orbits that follow the successive applications of the two dynamics. This method differs, in several crucial aspects, from earlier works. Unlike the well known two-dynamics approach, the method relies heavily on the outer dynamics alone. There are virtually no assumptions on the inner dynamics, as its invariant objects (e.g., primary and secondary tori, lower dimensional hyperbolic tori and their stable/unstable manifolds, Aubry-Mather sets) are not used at all. The method applies to unperturbed Hamiltonians of arbitrary degrees of freedom that are not necessarily convex. In addition, this mechanism is easy to verify (analytically or numerically) in concrete examples, as well as to establish diffusion in generic systems. This is a joint work with M. Gidea (Yeshiva U.) and R. de la Llave (Georgia Tech). Alfonso Sorrentino (Università degli Studi di Roma Tor Vergata, Italy) On the integrability of mathematical billiards. A mathematical billiard is a system describing the inertial motion of a point mass inside a domain, with elastic reflections at the boundary. This simple model has been first proposed by G.D. Birkhoff as a mathematical playground where the formal side, usually so formidable in dynamics, almost completely disappears and only the interesting qualitative questions need to be considered. Since then billiards have captured much attention in many different contexts, becoming a very popular subject of investigation. Despite their apparently simple (local) dynamics, their qualitative dynamical properties are extremely non-local. This global influence on the dynamics translates into several intriguing rigidity phenomena, which are at the basis of several unanswered questions and conjectures. In this talk I shall focus on several of these questions. In particular, I shall describe some recent results related to the classification of integrable billiards (also known as Birkhoff conjecture). This talk is based on works in collaboration with V. Kaloshin and with G. Huang and V. Kaloshin. Sergei Tabachnikov (PennState University, USA) Introducing symplectic billiards. I shall introduce a simple dynamical system called symplectic billiards. As opposed to the usual (Birkhoff) billiards, where length is the generating function, for symplectic billiards the symplectic area is the generating function. I 4
shall explore basic properties and exhibit several similarities, but also differences, of symplectic billiards to Birkhoff billiards. Symplectic billiards can be defined not only in the plane, but also in linear symplectic spaces. In this multi-dimensional setting, I shall discuss the existence of periodic trajectories and describe the integrable dynamics of symplectic billiards in ellipsoids. Susanna Terracini (Università degli Studi di Torino, Italy) Parabolic trajectories and symbolic dynamics: a survey on the variational approach to the N-body and N-centre problem. In its full generality, the N-body problem of Celestial Mechanics has challenged many generations of mathematicians. It is commonly accepted, since the early works by H. Poincaré, that the periodic problem, through its associated action spectrum, carries precious information on the whole dynamics of a Hamiltonian system. Therefore, the problem of the existence and the qualitative properties of periodic and other selected trajectories for the N- body problem (from the classical celestial mechanics point of view to more recent advances in molecular and quantum models) has been extensively studied over the decades, and, more recently, new tools and approaches have given a significant boost to the field. We shall review some old and new results on the existence and classification of selected trajectories of the classical N-centre and N-body problem, with an emphasis on new analytical and geometrical techniques. Dmitry Treschev (Steklov Mathematical Institute, Russia) Titchmarsh convolution theorem. Generalizations and dynamical applications.. Let f 1 and f 2 be two Lebesgue integrable functions on a line. According to the Titchmarsh convolution theorem sup supp f 1 f 2 = sup supp f 1 + sup supp f 2 provided the right-hand side is finite. I plan to discuss the so-called relative version of this theorem and applications in dynamics of infinite-dimensional Hamiltonian systems. Claude Viterbo (ENS de Paris, France) Barcodes and area preserving maps of surfaces. Ke Zhang (University of Toronto, Canada) Arnold diffusion via Aubry-Mather type. We describe a recent effort to better conceptualize the proof for Arnold diffusion in two and a half degrees of freedom. We say a cohomology class is of Aubry-Mather type, if its discrete Aubry set is contained in a twodimensional normally hyperbolic invariant cylinder, and the unstable bundle satisfy a geometrical condition. We show a cohomology class of Aubry- Mather type is always connected to nearby cohomologies for a residue perturbation of the Hamiltonian. The question of Arnold diffusion then reduces to the question of finding large connected components of such cohomologies. This is a joint work with Vadim Kaloshin. 5