TY - JOUR
T1 - Counterexamples in scale calculus
JF - Proceedings of the National Academy of Sciences
JO - Proc Natl Acad Sci USA
SP - 8787
LP - 8797
DO - 10.1073/pnas.1811701116
VL - 116
IS - 18
AU - Filippenko, Benjamin
AU - Zhou, Zhengyi
AU - Wehrheim, Katrin
Y1 - 2019/04/30
UR - http://www.pnas.org/content/116/18/8787.abstract
N2 - The counterexamples presented here are the first of this kind for notions of differentiability that satisfy a chain rule. Their context arises naturally from requiring differentiability of crucial maps in the theory of pseudoholomorphic curves in symplectic geometry. This serves to illuminate some of the key reasons and major technical obstacles for the extensive recent development of analytic foundations in this field. We moreover reestablish some key calculus facts in polyfold theory under additional assumptions that are imposed by the theory and satisfied in practice. This illuminates how the obstacles are overcome and allows users of polyfold theory to make new arguments more akin to those in standard calculus using general properties rather than working from definitions.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.
ER -