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).

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.

For a Hamiltonian diffeomorphism ϕ, consider the cyclic subgroup {ϕⁿ : n ∈ ℤ}⊂ Ham(M,ω) equipped with the Hofer metric d_H. We show that on certain closed symplectic manifolds, there exists a C¹-open and dense subset of Ham(M,ω) for which the growth of the Hofer length of ϕⁿ will be linear, i.e. lim (d_H(Id, ϕⁿ) / n) > 0. The proof involves a mixture of known results in symplectic topology and in dynamics of volume preserving diffeomorphisms.

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 a² + b² + c² = 3abc - we get a monotone Lagrangian torus T(a²,b²,c²) in ℂP² . Using neck-stretching techniques we are able to get enough information on the count of Maslov index 2 pseudo-holomorphic disks that allow us to show that for (d,e,f) a Markov triple distinct from (a,b,c), T(d²,e²,f²) is not Hamiltonian isotopic to T(a²,b²,c²).

In this talk we will describe how to get almost toric fibrations for all del Pezzo surfaces, in particular for ℂP²#kℂP² 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.

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.