Sc-Smoothness, Retractions and New Models for Smooth Spaces

We survey a (nonlinear) Fredholm theory for a new class of ambient spaces called polyfolds, and develop the analytical foundations for some of the applications of the theory. The basic feature of these new spaces, which can be finite and infinite dimensional, is that in general they may have locally varying dimensions. These new spaces are needed for a functional analytic treatment of nonlinear problems involving analytic limiting behavior like bubbling-off. The theory is applicable to Gromov-Witten and Floer Theory as well as Symplectic Field Theory.