Counterexamples in Scale Calculus

TL;DR

This paper constructs counterexamples in Scale Calculus, highlighting the limitations of classical calculus theorems in infinite-dimensional settings, which impacts the foundations of Polyfold Theory used in symplectic geometry.

Contribution

It provides explicit counterexamples demonstrating the failure of inverse and implicit function theorems in Scale Calculus, clarifying the technical challenges in Polyfold Theory.

Findings

Counterexamples show classical calculus theorems fail in Scale Calculus.

Continuity of differentials depends on specific coordinates.

Justifies the technical complexity in Polyfold Theory foundations.

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 cornerstone of Polyfold Theory, which was introduced by Hofer-Wysocki-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.