Journal Article

Counterexamples in scale calculus

Wehrheim, Katrin, Filippenko, Benjamin, Zhou, Zhengyi

2019

Overview

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.

• Publisher: National Academy of Sciences
• Subject: Calculus / Isomorphism / Mathematics / Operators (mathematics) / Physical Sciences / PNAS Plus / Theorems / Vector spaces