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.
journal_name
Proc Natl Acad Sci U S Aauthors
Filippenko B,Zhou Z,Wehrheim Kdoi
10.1073/pnas.1811701116subject
Has Abstractpub_date
2019-04-30 00:00:00pages
8787-8797issue
18eissn
0027-8424issn
1091-6490pii
1811701116journal_volume
116pub_type
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