EMS Surveys in Mathematical Sciences

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Volume 3, Issue 2, 2016, pp. 131–208
DOI: 10.4171/EMSS/16

Polyfolds: A first and second look

Oliver Fabert[1], Joel W. Fish[2], Roman Golovko[3] and Katrin Wehrheim[4]

(1) Vrije Universiteit Amsterdam, Netherlands
(2) University of Massachusetts Boston, USA
(3) Université Libre de Bruxelles, Belgium
(4) University of California Berkeley, United States

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.

Keywords: Non-linear functional analysis, Fredholm theory, transversality, polyfolds

Fabert Oliver, Fish Joel, Golovko Roman, Wehrheim Katrin: Polyfolds: A first and second look. EMS Surv. Math. Sci. 3 (2016), 131-208. doi: 10.4171/EMSS/16