Polyfolds: A First and Second Look

TL;DR

Polyfold theory provides a novel framework for analyzing complex geometric PDEs, addressing issues of compactification and transversality through new smoothness concepts and local models, with applications demonstrated in Morse theory.

Contribution

This paper offers a meta-mathematical overview and a streamlined exposition of polyfold theory, highlighting its core ideas and applications in geometric analysis.

Findings

Unified framework for elliptic PDE moduli spaces

New notions of smoothness and local models in Banach spaces

Application to Morse theory demonstrated

Abstract

Polyfold theory was developed by Hofer-Wysocki-Zehnder by finding commonalities in the analytic framework for a variety of geometric elliptic PDEs, in particular moduli spaces of pseudoholomorphic curves. It aims to systematically address the common difficulties of compactification and transversality with a new notion of smoothness on Banach spaces, new local models for differential geometry, and a nonlinear Fredholm theory in the new context. We shine meta-mathematical light on the bigger picture and core ideas of this theory. In addition, we compiled and condensed the core definitions and theorems of polyfold theory into a streamlined exposition, and outline their application at the example of Morse theory.