Attempts to two new theories in Symplectic topology that could eventually be useful
We will discuss informally with the audience two potential new theories in Symplectic topology. We call the first one Weighted Gromov-Witten invariants (WGW), and the second one is a hyperellipticity theory on 4-dimensional symplectic manifolds. The idea of the first one is to compute the capacity, computed in the moduli space, of all J-curves passing throught k subsets (say open subsets or Lagrangian submanifolds) of a symplectic manifold of any dimension. It must be normalised so that we recover the GW-invariants when the open subsets collapse to points. We then take the infimum over all J-structures. We have computed this invariant on an example in CP^2 that requires a new projective Non-Squeezing theorem. The second one, hyperellipticity, is more limited. It requires two homology classes A and B in M^4 such that B \cdot B = 0, and A \cdot B is a stricly positive number. We assume that the corresponding J-cuves in class A and B are of expected dimension and we generalize to the symplectic case the concept of branched coverings from a Riemann surface to another. Here, in the simplest case, the graph of the branched covering is seen in class A, while the class B is a fibration.