New geometric and functional analytic ideas arising from problems in symplectic geometry

The study of moduli spaces of holomorphic curves in symplectic geometry is the key ingredient for the construction of symplectic invariants. These moduli spaces are suitable compactifications of solution spaces of a first order nonlinear Cauchy-Riemann type operator. The solution spaces are usually not compact due to bubbling-off phenomena and other analytical difficulties. Moreover, there are serious transversality issues. As it turns out one can develop an abstract nonlinear Fredholm theory (applicable to a variety of other problems as well) for which the solution sets would be the compactified spaces. Such a general Fredholm theory takes place in spaces of varying dimensions.

However, in the case of transversality the solution spaces would still be smooth (finite-dimensional) manifolds, orbifolds or branched manifolds depending on the generality of the situation. In fact one has the usual Fredholm package of transversality and perturbation theory. In order to develop such a theory one needs to have a fresh look at the ways how finite-dimensional calculus should be generalized to infinite dimensions. Using a somewhat more exotic generalization one even can build a differential geometry containing new kind of spaces which can be used to build the ambient spaces for the generalized Fredholm theories. These spaces are called polyfolds. In this talk we describe some of the ideas and goals of the theory.