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...
Saved in:
| Published in: | Proceedings of the National Academy of Sciences - PNAS Vol. 116; no. 18; p. 8787 |
|---|---|
| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
United States
30.04.2019
|
| Subjects: | |
| ISSN: | 1091-6490, 1091-6490 |
| Online Access: | Get more information |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | 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. |
|---|---|
| Bibliography: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 23 |
| ISSN: | 1091-6490 1091-6490 |
| DOI: | 10.1073/pnas.1811701116 |