Abstract



Symplectic Field Theory is a very general theory of algebraic invariants for symplectic and contact manifolds. It generalizes Floer-theory, Gromov-Witten theory, and Contact Homology and is constructed by measuring moduli spaces of pseudoholomorphic curves. The richness of its structure comes from the fact that the moduli spaces have boundaries and singularities and that infinitely many moduli spaces interact with each other. These structures can be captured by a novel nonlinear Fredholm theory which is distinguished by two facts. The ambient spaces do not carry smooth structures in the usual sense and even have locally varying dimensions. However there is a notion of transversality and the solution sets are at points of transversality smooth orbifolds with boundaries with corners. Moreover the Fredholm theory makes precise what it means that infinitely many Fredholm operators interact with each other leading to a so called "Fredholm Theory with Operations". This abstract Fredholm Theory is developed and illustrated by its application to Symplectic Field Theory.