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Counterexamples in scale calculus.

Benjamin Filippenko¹, Zhengyi Zhou¹, Katrin Wehrheim²

¹Polyfold Laboratory, University of California, Berkeley, CA 94720-3840.

Proceedings of the National Academy of Sciences of the United States of America

|April 14, 2019

Summary

This summary is machine-generated.

Researchers created counterexamples to calculus theorems in scale calculus, a new infinite-dimensional calculus. This work clarifies the complex foundations of polyfold theory, crucial for regularizing mathematical spaces.

Keywords:

implicit function theorem inverse function theorem polyfold theory scale calculus

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Area of Science:

Differential Geometry
Symplectic Geometry
Infinite-Dimensional Analysis

Background:

Classical calculus theorems like the inverse and implicit function theorems are fundamental in mathematics.
Polyfold theory, a tool for regularizing moduli spaces, relies on scale calculus, a generalization of multivariable calculus.
The application of polyfold theory often implicitly assumes properties that lack formal justification.

Purpose of the Study:

To construct counterexamples to classical calculus facts within the framework of scale calculus.
To formally establish and investigate the continuity of differentials of basic germs in polyfold theory.
To justify the technical complexity inherent in the foundations of polyfold theory.

Main Methods:

Development of counterexamples to the inverse and implicit function theorems in scale calculus.
Introduction of the nonlinear scale-Fredholm notion within polyfold theory.
Construction of a scale-diffeomorphism and scale-Fredholm map exhibiting discontinuous differentials.

Main Results:

Counterexamples demonstrate that classical calculus theorems do not directly translate to scale calculus.
The continuity of differentials of basic germs is shown to hold only in specific coordinate systems.
A scale-Fredholm map with discontinuous differentials is constructed, highlighting coordinate-dependent properties.

Conclusions:

The study formally addresses the lack of implicit function theorems in scale calculus by establishing continuity properties of differentials.
The findings underscore the necessity of specific coordinate choices in polyfold theory, explaining its technical demands.
This work provides crucial foundational insights into scale calculus and polyfold theory, impacting areas like symplectic geometry.