Polyfolds And A General Fredholm Theory

TL;DR

This paper introduces a comprehensive Fredholm theory for polyfolds, a new class of ambient spaces with variable local dimensions, facilitating the analysis of complex nonlinear problems in symplectic geometry.

Contribution

It develops a general Fredholm framework for polyfolds, enabling the treatment of nonlinear problems with analytic limits in symplectic topology.

Findings

Polyfolds can have locally varying dimensions.

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

Provides a functional analytic approach to nonlinear problems with limits.

Abstract

We survey a very general (nonlinear) Fredholm theory for a new class of ambient spaces, called polyfolds. This theory is being currently developed jointly with K. Wysocki and E. Zehnder. The basic feature of these new spaces 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. This theory is applicable to Gromov-Witten and Floer Theory as well as Symplectic Field Theory.