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**Marcelo Alves (Orsay): Legendrian contact homology and forcing of topological entropy**

In this talk I will explain how Legendrian contact homology can be used to produce positive lower bounds for the topological entropy of Reeb flows on contact 3-manifolds. As an application I will explain a contact topological generalization of a result of Denvir and Mackay which says that if a Riemannian metric on the two-dimensional torus has a simple contractible closed geodesic then its geodesic flow has positive topological entropy. This is partly joint work with Pedro A. S. Salomão.

**Edward Belbruno (Princton): ****Ballistic Capture Transfer in the Three and Four-Body Problems with Applications**

Ballistic capture transfer occurs when a particle becomes captured into orbit about one of the primary mass points of the three or four body problem. Capture in this case is defined when the Kepler two-body energy of particle with respect to the primary is negative. This was first numerically demonstrated by E. Belbruno(EB) in 1986. This capture occurs in a region about the primary called a weak stability boundary(WSB). It was proven by EB (2005) that the WSB contains within it a hyperbolic invariant set. Although crudely numerically estimated in 1986, numerical work by Garcia and G. Gomez(2007) gave deeper insights into its makeup giving surprising results as to its complexity. Further insights in the makeup of this region were made in a series of studies from 2010-2012 by M. Gidea, F. Topputo, EB. The relationship of ballistic capture to permanent capture and the work of K. Sitnikov in the 1960s is discussed.

Ballistic capture transfer has important applications. One such transfer was used in 1991 by J. Miller and EB to rescue an errand Japanese lunar probe and successfully bring it to the Moon. This same transfer class was used in 2011 for NASAs lunar Grail mission, and another type was used by ESAs lunar SMART 1 mission in 2004. A new result that is described is the existence of a ballistic capture transfer to Mars that was developed by F. Topputo and EB (2014). This promises to have interesting applications. If there is time, the use of ballistic capture transfers in open star clusters is described which shed light on the Lithopanspermia Hypothesis and the origin of life on Earth(EB, A. Moro-Martin, R. Malhotra, D. Savransky, 2012).

**Alvaro del Pino Gomez (Madrid): An existence h-principle for Engel structures **

It has been shown recently, in joint work with Casals, Pérez, and Presas, that any flag in a 4-manifold can be homotoped to the flag induced by an Engel structure. This raises many questions regarding the nature of Engel structures, the main one being whether they present some rigidity.

The aim of the talk will be to sketch the proof of the result, and to give a feeling for these open questions.

**Marco Golla (Uppsala): Symplectic hats**

A symplectic cap of a contact 3-manifold Y is a compact 4-manifold with concave boundary -Y; a symplectic hat of a transverse knot K in Y is a compact symplectic surface in a cap, whose boundary is -K. We study the existence problem for hats and their topology, with a particular emphasis on the case of transverse knots in the standard 3-sphere; we will also discuss applications to fillings of contact 3-manifolds.

This is joint work (in progress) with John Etnyre.

**Liu Guogang (Nantes): Positive Legendrian isotopy**

The aim of the talk is to explain the following theorem still on working:

There exists positive loop of Legendrian isotopy for every loose Legendrian submanifold.

First of all, I will talk about the basic facts about positive Legendrian isotopy; then explain why we need the loose condition in dimension one and two cases. Finally, I will explain the ideas of h-principle in general case.

**Asaf Kislev (Tel Aviv): Hofer growth of C ^{1}-generic Hamiltonian flows **

For a Hamiltonian diffeomorphism ϕ, consider the cyclic subgroup {ϕ^{n} : n ∈ ℤ}⊂ Ham(M,ω) equipped with the Hofer metric d_{H}. We show that on certain closed symplectic manifolds, there exists a C^{1}-open and dense subset of Ham(M,ω) for which the growth of the Hofer length of ϕ^{n} will be linear, i.e. lim (d_{H}(Id, ϕ^{n}) / n) > 0. The proof involves a mixture of known results in symplectic topology and in dynamics of volume preserving diffeomorphisms.

**Susanna Terracini (Turin): Scattering parabolic solutions for the spatial N-centre problem**

For the N-centre problem in the three dimensional space,

where N ≥ 2, m_{i} > 0 and α ∈ [1, 2), we prove the existence of entire parabolic trajectories having prescribed asymptotic directions. The proof relies on a variational argument of min-max type. Morse index estimates and regularization techniques are used in order to rule out the possible occurrence of collisions. This is a joint work with Alberto Boscaggin and Walter Dambrosio.

**Renato Vianna (Cambridge): Infinitely many monotone Lagrangian tori in del Pezzo surfaces**In 2014, we showed how the Chekanov torus arises as a fiber of an almost toric fibration and how this perspective enable us to describe an infinite range of monotone Lagrangian tori. More precisely, for any Markov triple of integers (a,b,c) - satisfying a

In this talk we will describe how to get almost toric fibrations for all del Pezzo surfaces, in particular for ℂP^{2}#kℂP^{2} for 4 ≤ k ≤ 8, where there is no toric fibrations (with monotone symplectic form). From there, we will be able to construct infinitely many monotone Lagrangian tori. Some Markov like equations appear. They are the same as the ones appearing in the work of Haking-Porokhorov regarding degeneration of surfaces to weighted projective spaces.

**Evgeny Volkov (Uppsala): Cyclic homologies and equivariant cohomologies**

Cyclic homology of the de Rham algebra of a simply connected manifold maps to equivariant cohomology of the loop space of the manifold. The map is provided by Chen's iterated integrals. More precisely, using the language of mixed complexes one can define several versions of cyclic homology and correspondingly several versions of equivariant cohomology. For some versions Chen's integrals give us an isomorphism, for some not. We explain some peculiarities entering the discussion.

**Kai Zehmisch (Münster): Diffeomorphism type of symplectically aspherical fillings**

Symplectically aspherical fillings of simply connected contact manifolds that are subcritically Stein fillable and of dimension at least five are unique up to diffeomorphism. In my talk I will present a proof and indicate generalisations to classes of contact manifolds with non-trivial fundamental groups. This is joint work with Kilian Barth und Hansjörg Geiges.

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- 11.02.2016