From Gromov´s non-squeezing to the Weinstein conjecture and back

In this talk I will report on a joint work with Hansjörg  Geiges. We study holomorphic spheres in
certain symplectic cobordisms  and derive information about periodic Reeb orbits in the concave
end of these cobordisms from the non-compactness of the relevant moduli spaces. We use this
to confirm the strong Weinstein conjecture (predicting the existence of null-homologous Reeb
links) for various higher-dimensional contact manifolds, including  contact type hypersurfaces in
subcritical Stein manifolds and in cotangent bundles with a base manifold, which is a product
with the circle. The quantitative character of this result leads to the definition of a symplectic
capacity, and  in turn to a proof of symplectic non-squeesing.