2014/07/11 by Hofer, Helmut H., Wysocki, Kris, Zehnder, Eduard
#46 #53 #58 #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA) #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.1407.3185
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.