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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.
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Oliver Fabert, Joel W. Fish, Roman Golovko, Katrin Wehrheim, Polyfolds: A first and second look. EMS Surv. Math. Sci. 3 (2016), no. 2 pp. 131–208