Counterexamples in scale calculus

We construct counterexamples to classical calculus facts such as the inverse and implicit function theorems in scale calculus-a generalization of multivariable calculus to infinite-dimensional vector spaces, in which the reparameterization maps relevant to symplectic geometry are smooth. Scale calcu...

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Veröffentlicht in:Proceedings of the National Academy of Sciences - PNAS Jg. 116; H. 18; S. 8787
Hauptverfasser: Filippenko, Benjamin, Zhou, Zhengyi, Wehrheim, Katrin
Format: Journal Article
Sprache:Englisch
Veröffentlicht: United States 30.04.2019
Schlagworte:
scale calculus
polyfold theory
inverse function theorem
implicit function theorem
ISSN:1091-6490, 1091-6490
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Zusammenfassung:We construct counterexamples to classical calculus facts such as the inverse and implicit function theorems in scale calculus-a generalization of multivariable calculus to infinite-dimensional vector spaces, in which the reparameterization maps relevant to symplectic geometry are smooth. Scale calculus is a corner stone of polyfold theory, which was introduced by Hofer, Wysocki, and Zehnder as a broadly applicable tool for regularizing moduli spaces of pseudoholomorphic curves. We show how the novel nonlinear scale-Fredholm notion in polyfold theory overcomes the lack of implicit function theorems, by formally establishing an often implicitly used fact: The differentials of basic germs-the local models for scale-Fredholm maps-vary continuously in the space of bounded operators when the base point changes. We moreover demonstrate that this continuity holds only in specific coordinates, by constructing an example of a scale-diffeomorphism and scale-Fredholm map with discontinuous differentials. This justifies the high technical complexity in the foundations of polyfold theory.
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ISSN:1091-6490
1091-6490
DOI:10.1073/pnas.1811701116