Smooth Manifolds and Observables

This textbook provides a unified approach to differential calculus, smooth manifolds, and commutative algebra,motivated by the mathematical formalization of observation from classical physics. It demonstrates how these subjects are interconnected and shows how differential calculus can be understood as a generalization of the differential calculus on smooth manifolds. The book is intended for advanced undergraduates, graduate students, and researchers in mathematics and physics and includes new chapters on applications to symplectic manifolds, de Rham cohomology, and Poisson brackets.

Format: Paperback / softback
Length: 433 pages
Publication date: 10 September 2021
Publisher: Springer Nature Switzerland AG

This textbook offers a comprehensive exploration of the interconnectedness of differential calculus, smooth manifolds, and commutative algebra, despite their historical development under different circumstances. Motivated by the mathematical formalization of the process of observation from classical physics, the book aims to demonstrate how these three subjects form a unified and coherent whole. With a broad audience in mind, it provides valuable insights into the underlying connections between geometry, physics, and commutative algebra.

The primary objective of this book is to elucidate the natural relationship between differential calculus and commutative algebra. It achieves this by studying the corresponding algebras of smooth functions, which lead to a general construction of differential calculus on various categories of modules over the given commutative algebra. The authors meticulously show that the ordinary differential calculus and differential geometry on smooth manifolds correspond precisely to the category of geometric modules over smooth algebras. This groundbreaking approach opens up a wide range of applications, spanning from delicate questions of algebraic geometry to the theory of elementary particles.

Smooth Manifolds and Observables is designed for advanced undergraduates, graduate students, and researchers in mathematics and physics. In this second edition, ten new chapters are added to further develop the concept of differential calculus over commutative algebras, showcasing its broader application beyond smooth manifolds. The book explores applications to diverse areas such as symplectic manifolds, de Rham cohomology, and Poisson brackets, providing readers with a comprehensive understanding of these subjects. Additionally, numerous new exercises are included, offering readers ample opportunities to practice and apply the theoretical concepts presented.

In conclusion, Smooth Manifolds and Observables is a groundbreaking textbook that provides a comprehensive and insightful exploration of the interconnectedness of differential calculus, smooth manifolds, and commutative algebra. By presenting a unique approach that bridges these three disciplines, the book offers valuable insights into the fundamental principles governing mathematical structures and their applications in physics and mathematics. Whether you are a student, researcher, or professional in these fields, this book is an essential resource for deepening your understanding and advancing your knowledge.

Weight: 688g
Dimension: 235 x 155 (mm)
ISBN-13: 9783030456528
Edition number: 2nd ed. 2020