Albouy: The most classical way to be Hamiltonian in two different ways The relation between bi-hamiltonian systems and classical integrable systems was the object of many discussions. Here we recall (with Hans Lundmark and Stefan Rauch-Wojciechowski) that the most classical systems in mechanics are defined by Newton equations, the right-hand side of which is a force field depending only on position. We observe that such Newton equations are defined on an affine space. They define a Hamiltonian system only if there is a Euclidean structure such that the force field is a gradient vector field. This raises the question: May the affine space be a Eucidean space in two different ways, in such a way that the given force field is a gradient? If this occurs, the Newton equations are bi-hamiltonian. This indeed occurs, but only rather trivially. If we observe, with Appell, that the Newton equations are already defined at the level of the locally projective space, and ask the corresponding question about the force field being a gradient in two different ways, then we reach interesting classical integrable examples such as Kepler problem, two fixed center problem, Neumann problem, Braden problem. This leads to new observations about these problems and their relations. We will present the general theory as developed in a join work with Hans Lundmark. Bartsch: Point vortex dynamics Chekanov: The homotopy type of the space of contact structures on certain 3-manifolds Chenciner: Non-avoided crossings for n-body balanced configurations in R^3 near a central configuration The balanced configurations are those n-body configurations which admit a relative equilibrium motion in a Euclidean space E of high enough dimension 2p. They are characterized by the commutation of two symmetric endomorphisms of the (n-1)-dimensional Euclidean space of codispositions, the intrinsic inertia endomorphism B which encodes the shape and the Wintner-Conley endomorphism A which encodes the forces. In general, p is the dimension d of the configuration, which is also the rank of B. Lowering to 2(d-1) the dimension of E occurs when the restriction of A to the (invariant) image of B possesses a double eigenvalue. It is shown that, while in the space of all dxd-symmetric endomorphisms, having a double eigenvalue is a condition of codimension 2 (the avoided crossings of physicists), here it becomes of codimension 1 provided some condition (H) is satisfied. As the condition is always satisfied for configurations of the maximal dimension (i.e. if d=n-1), this implies in particular the existence, in the neighborhood of the regular tetrahedron configuration of 4 bodies with no three of the masses equal, of exactly 3 families of balanced configurations which admit relative equilibrium motion in a four dimensional space. de la Llave: Automatic reducibiliy in KAM Theory: A posteriori theorems, fast and validated algorithms, analyticity properties We present an approach to KAM theory based on studying the embedding and applying corrections that improve the conditions of invariance. These corrections are obtained in a systematic way using symplectic properties. In particular, it has been implemented in symplectic systems and in conformally symplectic systems. This leads to "a-posteriori" theorems that show if there is an approximate solution with a good condition number there is a true solution nearby. It also leads to very efficient algorithms that are guaranteed to converge up to the domain of validity of the results in some cases. Besides numerical applications, the a-posteriori approach leads to detailed results on analyticity domain in singular perturbations and to very quick proofs of other well known results such as monogenicity and Whiteney regularity in the frequency. This is joint work with R. Calleja and A. Celletti Fejoz: On bounded motions in the N-body problem In the spatial N-body problem, for any given masses, there is a set of positive Lebesgue measure of initial conditions leading to quasiperiodic motions. This is a variant of Arnold's theorem, where one mass is supposed much larger than the others. Godinho: Polygons, hyperpolygons and beyond Knauf: New techniques for the N-body problem Onaran: Nonloose transverse knots A knot which is everywhere transverse to the contact planes is called a transverse knot. In this talk, we will discuss recently defined depth, tension and order invariants for transverse knots in overtwisted contact structures. In particular, we will discuss the invariants of a connected binding of an open book decomposition which is a transverse knot in the contact structure supported by the open book. This work is joint with K. Baker. Ortega: Dynamics of a ping-pong model Ozbagci: Contact open books with exotic pages We consider a fixed contact 3-manifold that admits infinitely many compact Stein fillings which are all homeomorphic but pairwise non-diffeomorphic. Each of these fillings gives rise to a closed contact 5-manifold described as a contact open book whose page is the filling at hand and whose monodromy is the identity symplectomorphism. We show that the resulting infinitely many contact 5-manifolds are all diffeomorphic but pairwise non-contactomorphic. Moreover, we explicitly determine these contact 5-manifolds. (This is a joint work with Otto van-Koert) Pelayo: Integrable systems in the semiclassical limit I will describe recent work on the spectral theory of semiclassical systems given by collections of symplectic operators emphasizing the symplectic geometry of their classical counterparts. Ratiu: The Flaschka transformation as a momentum map In this talk I will show how the celebrated Flaschka transformation, which is crucial in the study of the integrable finite Toda system, is a momentum map. This is based on reduction theory and certain conditions introduced by Pukanszky, that will be discussed. It turns out that there is a second sent of Pukanszky conditions that ultimately imply that simply connected coadjoint orbits of connected, simply connected, solvable Lie groups are symplectomorphic to the standard symplectic vector space. A sketch of the proof will be given. Schlenk: Non-existence of intermediate symplectic capacities via symplectic folding, following Richard Hind Symplectic capacities formalize Gromov's Nonsqueezing Theorem, a paradigm of symplectic rigidity, that can be phrased as follows: For any symplectic embedding $\phi$ of the unit ball $B^2n$ into $R^{2n}$, the area of the projection of$\phi (B^{2n})$ to the first coordinate plane $R^^2 (x_1,y_1)$ is at least $\pi$. This is a 2-dimensional rigidity statement, and of course also the 2n-dimensional volume of a ball cannot be reduced by symplectic embeddings. Hofer asked long ago whether there are "intermediate" versions of this (to be made precise in the talk). An ingenious construction by Guth showed that this is not the case, that is: The only symplectic non-squeezing theorem is 2-dimensional. Guth' rather envolved construction was made sharp by Pelayo-Vu Ngoc and by Hind-Kerman, and Abbondandolo used it to prove a beautiful variant of the non-existence of intermediate capacities. In this talk we follow Richard Hind and show how the old-fashioned method of symplectic folding can be used to obtain these symplectic embedding results in a very transparent way. Terracini: A variational approach to colliding solutions in celestial mechanics Tolman: Non-Hamiltonian actions with isolated fixed points I explain how to construct a non-Hamiltonian symplectic circle action on a closed, connected,six-dimensional symplectic manifold with exactly 32 fixed points. Based in part on joint work with J. Watts. Zehmisch: Magnetic two-spheres In my talk I will explain the following result: For any Riemannian two-sphere provided with a magnetic field there exists a closed magnetic geodesic for almost all kinetic energy levels. This is joint work with Gabriele Benedetti.