We show how the functorial properties of contact homology can be used to detect nontrivial elements in the homotopy groups of the space of contact structures. This technique will then be illustrated with various examples.
In this talk I will review shortly the construction of the cluster homology of a Lagrangian submanifold and describe some applications to the detection of pseudoholomorphic disks (this part of the talk is based on joint work with Francois Lalonde). I will also indicate how this construction is related to a previous one-introduced jointly with Jean-Francois Barraud-which serves to detect algebraically pseudoholomorphic strips.
Every toric manifold M has a natural automorphism group A that is a maximal compact subgroup of the symplectomorphism group S of M. I will discuss recent work due to Susan Tolman and myself concerning the relation between these groups. In particular I will discuss the question of when the induced map p_{1} (A) ® p_{1} (S) is injective. It turns out that toric manifolds for which injectivity fails have a very special structure.
I will define invariants under deformation of real symplectic 4-manifolds. These invariants are obtained via an algebraic count of real rational J-holomorphic curves which pass through a given configuration of points, for a generic almost complex structure J. These invariants provide lower bounds in real enumerative geometry, namely for the total number of such real rational J-holomorphic curves.