Families of symplectic manifolds

The present project investigates the topology of families of symplectic manifolds. Symplectic manifolds arise in many areas of mathematics, for example complex projective varieties (zero-sets of homogeneous polynomial equations) are naturally symplectic, as are moduli spaces of flat connections on surfaces, coadjoint orbits in Lie theory, phase spaces from classical dynamics etc. Studying families of these objects, for example moduli spaces of projective varieties, is an extremely active area. If one remembers only the symplectic structures then one can hope to get a symplectic angle on questions from moduli theory.Families of symplectic manifolds also arise in mirror symmetry. In the same way that there is a line through two points in the plane or a conic through five points, one can try and count curves through points in other symplectic manifolds. Mirror symmetry is a conjecture motivated by string theory, which on a very basic level is a machine for providing surprising formulae for the number of complex curves in symplectic manifolds. For example the number of curves in a quintic threefold is governed by a hypergeometric function. In many interesting cases there are no curves to count, but when one lets the symplectic manifold vary in a family, curves appear at isolated points of the family and can be counted. This was first observed by Kontsevich and Yau-Zaslow who conjectured that similar surprising formulae should hold for families of K3 surfaces (which were proved recently by Maulik and Pandharipande). The aim of this project is to see what these family curve counts can tell us about the topology of the universal family of symplectic manifolds and to find applications of the theory to concrete problems in symplectic geometry. For example, families of symplectic manifolds may arise from surgery (blow-up or cut-and-paste) applied to a family of submanifolds (for example points, Lagrangian tori, symplectic submanifolds) and if the corresponding family of symplectic manifolds is nontrivial (detected by these curve counts) then we should be able to say something nontrivial about the underlying family of objects.