  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.
