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Proc Natl Acad Sci U S A. 2019 Apr 30;116(18):8787-8797. doi: 10.1073/pnas.1811701116. Epub 2019 Apr 12.

Counterexamples in scale calculus.

Author information

1
Polyfold Laboratory, University of California, Berkeley, CA 94720-3840.
2
Polyfold Laboratory, University of California, Berkeley, CA 94720-3840 wehrheim@berkeley.edu.

Abstract

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.

KEYWORDS:

implicit function theorem; inverse function theorem; polyfold theory; scale calculus

PMID:
30979800
PMCID:
PMC6500174
[Available on 2019-10-12]
DOI:
10.1073/pnas.1811701116

Conflict of interest statement

The authors declare no conflict of interest.

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