Joint IAS/Princeton/Montreal/Paris/Tel-Aviv Symplectic Geometry Zoominar
On symplectic capacities and their blind spots
In this talk I will discuss a joint project with Yuanpu Liang in which we establish several properties of the sequence of symplectic capacities defined by Gutt and Hutchings for star-shaped domains using $S^1$-equivariant symplectic homology. Among the results discussed will be the fact that, unlike the first of these capacities, the others all fail to satisfy the symplectic version of the Brunn Minkowski established by Artstein-Avidan and Ostrover. We also show that the Gutt-Hutchings capacities, together with the volume, do not constitute a complete set of symplectic invariants even for convex bodies with smooth boundary. The examples constructed to prove these results are not exotic. They are convex and concave toric domains. The main new tool used is a significant simplification of the formulae of Gutt and Hutchings for the capacities of such domains, that holds under an additional symmetry assumption. This allows us to compute the capacities in new examples and to identify and exploit blind spots that they sometimes share.