For complete Riemannian manifolds M the condition to have no conjugate points is equivalent to the following strong minimality property of all of its geodesics: If c : R → M is a geodesic and if s < t ∈ R, then c|[s, t] is the (up to parametrization) unique shortest curve connecting c(s) and c(t) in the homotopy class of c|[s,t]. In particular, if the fundamental group of M is free abelian, this is the minimality property of orbits studied by J. Mather for much more general variational principles. Starting with E. Hopf’s result (1948) that every Riemannian 2-torus without conjugate points is flat, there is long list of rigidity results for Riemannian manifolds without conjugate points. We treat complete, 2-dimensional, Riemannian planes and cylinders without conjugate points. These were first studied by K. Burns and G. Knieper (1991). Since there are complete Riemannian metrics of negative Gaussian curvature on these manifolds one has to impose additional conditions in order to have a chance to prove flatness of metrics without conjugate points. A natural and weak invariant that distinguishes between flat and negatively curved metrics is the (asymptotic) area growth. 
We prove flatness of complete Riemannian planes and cylinders without conjugate points under optimal conditions on the area growth.
This is a joint work with P. Emmerich.

We consider a  Hamiltonian system with a nondegenerate normally hyperbolic symplectic critical manifold $M\subset H^{-1}(0)$.

We show that under certain conditions a chain of heteroclinic solutions corresponding to a hyperbolic orbit of the scattering map of $M$ can be shadowed by a solution on $H^{-1}(\eps)$ which passes $O(\sqrt\eps)$ close to $M$.  An applications to the Poincaré second species solutions of the 3 body problem will be given.

Homogenization of convex hamiltonians on compact manifolds
We describe a setting for homogenization of the Hamilton-Jacobi equation on free abelian covers of compact manifolds. In this context we also provide a new simple variational proof of standard homogenization results.

It is over forty years since John Mather, following earlier work of Rene Thom, completed his pioneering series of papers on the Stability of $C^\infty$--mappings. He followed this with a number of other important results in singularity theory concerning stratifications and topological stability, the theory of versal unfoldings, and generic projections, each of which further expanded the reach of singularity theory. It would require a whole series of lectures to even begin to survey the many applications and further developments that have grown out of his work.

Instead this lecture will concern several recent illustrative results in a variety of different directions, involving both theoretical results and applications, which use the methods and ideas introduced in John Mather's work and show its continuing impact. This will include the topology of highly nonisolated singularities such as matrix singularities, the local features of objects in natural images allowing viewer movement, the geometry of shape as it applies to 2D and 3D medical images, and how scale deals with problems of noise in medical images.

This is joint work with A. Delshams, T. M-Seara. 

We consider a system which is a perturbation of an integrable system (with any number of degrees of freedom) and an pendulum (or, more generally a a system with a hyperbolic point and a homoclinic intersection) and subject it to a periodic perturbation. We find explicit conditions which show that the system has orbits whose actions experience a chance which is independent of the size of the perturbation. 
The main new phenomena when increasing the number of degrees of freedom is resonances of high multiplicity, but we note that they happen in sets of large codimension,  so that they can be contoured. We show that given any path in action space which avoids resonances of order 2 or higher there is a trajectory that follows it. 

Combining two scattering maps to obtain diffusion in the elliptic restricted three body problem
The goal of the talk is to show the existence of global instability in the elliptic restricted three body problem. The main tool is to combine two different scattering maps associated to the normally parabolic manifold of infinity to build trajectories whose angular momentum increases arbitrarily. This is a joint work with V. Kaloshin, A. de la Rosa and T.M. Seara.

A generic property of exact magnetic Lagrangians
In the article "A Generic Property of Families of Lagrangian Systems" G. Contreras and P. Bernard proved that generically, in the sense of Mañe (i.e, by adding a potential), for every cohomology class c there is only a finite number of minimizing measures [Ann. of Math. (2008)]. This theorem is a consequence of an abstract result which is useful in different situations.

In this talk we show the genericity of finitely many minimizing measures for Exact Magnetic Lagrangians and apply it to the dynamics of the Aubry set.

In general, when we are dealing with a specific class of Lagrangians, perturbations by adding a potential are not allowed. However, due to the abstract nature of Bernard-Contreras proof, it may be adapted to the case treated here.

This is a joint work with Alexandre Rocha from Universidade Federal de Vicosa, Brazil.

Some recent progress on the theory of billiards in polygons
We will discuss some recent results and some open questions on the dynamics of billiards in polygons from the point of view of Teichmuller dynamics.

On Space-Time Stationary Random Weak KAM Solutions in Non-Compact Setting
I shall discuss a recent joint paper with Y. Bakhtin and E. Cator in which we construct global space-time stationary solutions for the random forced Hamilton-Jacobi equation in the 1D case. The equation is considered on the whole real line and we do not assume space or time periodicity.

Prime ends rotation numbers and periodic points
In a joint work with Andres Koropecki and Meysam Nassiri, we study the problem of existence of a periodic point in the boundary of an invariant domain for an area preserving surface homeomorphism. In particular, we prove the converse of a classic result of Cartwright and Littlewood related to Caratheodory's prime ends theory. This has a number of consequences for generic area preserving surface diffeomorphisms. As an application, we extend previous results of J. Mather on the boundary of invariant open sets for C^r -generic area preserving diffeomorphisms.

Hyperbolic cylinders and KAM tori for generic nearly integrable systems in ${\mathbb A}^3$
This talk presents the construction of a geometrical framework enabling one to deduce the existence of global Arnold diffusion in {\em a priori} stable systems in ${\mathbb A}^3$ from the diffusion properties of {\em a priori} unstable systems.
We consider perturbed systems of the form $H(\theta,r)=h(r)+\varepsilon f(\theta,r)$, where $h$ is a $C^k$ strictly convex and superlinear  function in ${\mathbb R}^3$ and $f$ an element of the unit ball $B$ of $C^k({\mathbb T}^3\times{\mathbb R}^3)$.
Given an energy $e>{\rm Min\,} h$, our main result is the existence of a residual subset of $B$ such that for each $f\in B$, there exists $\varepsilon_0>0$ such that for $0<\varepsilon<\varepsilon_0$, the system $H$ admits in $H^{-1}(e)$ a ``net'' of compact normally hyperbolic invariant cylinders (diffeomorphic to ${\mathbb T}^2\times[0,1]$), which becomes ``asymptotically dense'' in its energy level when $\varepsilon$ tends to~$0$. These cylinders admit torsion properties and homoclinic connections similar to those usually assumed in the context of {\em a priori} unstable diffusion.

Geometry of the reduced and regularized three-body problem
For the planar three-body problem, it is possible to combine a global symplectic reduction with Levi-Civita type regularization of all three binary collisions and blow-up of the triple collision singularity to obtain a complete flow.    Most of the ingredients in this procedure have been known for a long time, but I will describe a nice geometrical approach to the complete result, emphasizing the size-shape splitting of the configuration space.   The regularized and unregularized shape spaces are both two-spheres and are related by a beautiful four-to-one branched covering map first discovered by Lemaitre.

Symplectic topology of the Poisson bracket
I shall discuss "hard" symplectic invariants coming from the Poisson bracket. Applications include function theory, Hamiltonian dynamics and quantum mechanics. In parts the talk is based on a joint work with Michael Entov and Lev Buhovsky.

Oscilatory orbits in the circular restricted planar three body problem for big values of the Jacobi constant and any mass ratio.
In this work we consider the circular restricted three body problem. In this problem the Jacobi constant is a first integral. On the other hand, the restricted three body problem can be considered as a perturbation of the two body problem for small mass ratio, where the total energy is also a first integral. 
In 1980 J. Llibre and C. Sim\'o \cite{LlibreS80} proved the existence of oscillatory motions for the restricted planar circular three body problem, that is, of orbits  which leave every bounded region but which return infinitely often to some fixed bounded region. To prove their existence they  had to assume that  the ratio  between the masses of the two primaries was exponentially small with respect to the angular momentum (or the Jacobi constant). In the present work, we generalize their work proving the existence of oscillatory motions for any value of the mass ratio.
In fact, we will show that, for big values of the Jacobi constant and for any mass ratio, there exist transversal intersections between the stable and unstable manifolds of infinity which guarantee the existence of a symbolic dynamics that  creates the so called oscillatory orbits. The main achievement is to rigorously prove the existence of these orbits without assuming the mass ratio small and therefore, this transversality  can not be checked by using classical perturbation theory respect to the mass ratio.The main point in our approach  is that, for big values of the Jacobi constant, the restricted three body problem is a fast and small perturbation of a two body problem. This makes the transversality of the invariant manifolds to be exponentially small, but non zero, for big Jacobi constant and any mass ratio. 
This is a joint work with M. Guardia and P. Martin.

Polymorphisms and adiabatic chaos
Polymorphisms are multivalued (Lebesgue) measure preserving self-maps of the interval [0,1]. This class of dynamical systems was introduced by Vershik. Later polymorphisms appeared in the problem of destruction of an adiabatic invariant after crossing through a separatrix. We plan to discuss ergodic properties and typical singularities of "adiabatic" polymorphisms.

Linearization of nonlinear perturbations of interval exchange maps
We present a smooth linearization result, joint work with S.Marmi and P.Moussa, for nonlinear perturbations of interval exchange maps or nonlinear perturbations of linear flows on translation surfaces. We discuss in particular the diophantine condition on the unperturbed map or flow, as well as some related open questions.

Time-dependent billiards with slowly moving scatterers
We propose a model of Sinai billiards with moving scatterers, in which the locations of the scatterers may be shifted by small amounts between collisions. Our main result is the exponential loss of memory of initial data, and our proof consists of a coupling argument for non-stationary compositions of maps similar to classical billiard maps. This can be seen as a prototypical result on convergence to quasi-stationary states for time-dependent dynamical systems with slowly varying coefficients.

Arnold Diffusion via Invariant Cylinders and Mather Variational Method
The quasi-ergodic hypothesis, proposed by Ehrenfest and Birkhoff, says that a typical Hamiltonian dynamics on a typical energy surface has a dense orbit. This question is wide open. In early 60th Arnold constructed an example of instabilities for a nearly integrable Hamiltonian with degree of freedom n>2 and conjectured that this is a generic phenomenon, nowadays, called Arnold diffusion. A proof of Arnold's conjecture in two and half degrees of freedom has been announced by J. Mather in 2003. Based on two works: one joint with V. Kaloshin and P. Bernard, another with V. Kaloshin, we provide a proof for two and half degrees of freedom using an alternative approach. Our approach is based on constructing a net of normally hyperbolic invariant cylinders and a version of Mather variational method.