Polyfold and Fredholm Theory

This book pioneers a nonlinear Fredholm theory in polyfolds, generalizing aspects of nonlinear analysis and differential geometry with a pinch of category theory. It introduces smooth spaces and bundles with locally varying dimensions, allowing for implicit function theorems and transversality and perturbation theory. The theory is applied to symplectic geometry and has potential applications in other areas.

Format: Hardback
Length: 1001 pages
Publication date: 22 July 2021
Publisher: Springer Nature Switzerland AG

This groundbreaking book pioneers a nonlinear Fredholm theory in a broad class of spaces known as polyfolds, revolutionizing the field of differential geometry and analysis. By seamlessly integrating certain aspects of nonlinear analysis and differential geometry with a touch of category theory, it introduces a novel framework that encompasses local symmetries.

On the differential geometrical side, the book unveils a vast array of "smooth spaces and bundles" that possess locally varying dimensions, ranging from finite to infinite-dimensional. These bundles come equipped with a significant class of sections, exhibiting properties reminiscent of classical nonlinear Fredholm theory and enabling implicit function theorems. Within this nonlinear analysis framework, a versatile transversality and perturbation theory is developed to also cover equivariant settings.

The genesis of this theory can be traced back to the authors' work between 2007 and 2010, driven by nonlinear moduli problems in symplectic geometry. These problems are typically described locally as nonlinear elliptic systems, necessitating the study up to a notion of isomorphism. This introduces symmetries, as such a system can be isomorphic to itself in different ways.

Bubbling-off phenomena are a common occurrence in these systems, and a thorough understanding of them is essential to yield algebraic invariants. Consequently, a transversality theory for bubbling-off phenomena in the presence of symmetries becomes crucial. However, geometric perturbations may not always be sufficient to achieve transversality, necessitating the consideration of abstract perturbations.

Fortunately, the theory is already demonstrating remarkable success in its intended applications in symplectic geometry, and it is expected to find applications in numerous other domains where partial differential equations, geometry, and functional analysis intersect.

Written by the originators themselves, Polyfold and Fredholm Theory is a comprehensive and authoritative resource that delves into the intricate details of this groundbreaking field. It is a must-read for researchers and scholars seeking to expand their knowledge in differential geometry and analysis.

Weight: 1694g
Dimension: 235 x 155 (mm)
ISBN-13: 9783030780067
Edition number: 1st ed. 2021