  The main topic is the development of a Fredholm theory in a new class of
spaces called M-polyfolds. In the subsequent Volume II the theory will be
generalized to an even larger class of spaces called polyfolds, which can also
incorporate local symmetries. The whole package provides a functional analytic
framework to deal with compactness and transversality issues as they occur in
moduli problems of symplectic geometry. Applications of the theory cover
Floer-type theories as they occur in symplectic geometry. M-polyfolds and the
more general polyfolds are smooth spaces which can be finite-dimensional as
well as infinite-dimensional. In applications of interest they in general have
locally varying dimensions. Despite the fact that the spaces are much more
general than Banach manifolds a nonlinear Fredholm theory with the usual
features is possible (Sard-Smale type perturbation theory). This generalized
Fredholm theory can be applied to classes of nonlinear elliptic problems which
show bubbling-off phenomena but allow for certain kind of compactifications.
