Attempts to two new theories in Symplectic topology that could eventually be useful

François Lalonde (Université de Montréal)
Mon, Nov 28 2022, 2:30pm

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.