Sc-Smoothness, Retractions and New Models for Smooth Spaces

TL;DR

This paper surveys the development of polyfolds, a new class of smooth spaces with variable local dimensions, providing the analytical foundation for nonlinear problems in symplectic geometry.

Contribution

It introduces the concept of polyfolds and develops the nonlinear Fredholm theory necessary for analyzing complex geometric problems.

Findings

Polyfolds can handle spaces with varying local dimensions.

The theory applies to Gromov-Witten, Floer, and Symplectic Field Theory.

Provides analytical tools for problems with bubbling-off phenomena.

Abstract

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.