Authors:
- Motivates the algebraic study of smooth manifolds by looking at them from the point of view of physics, in particular by using the observability principle
- Bridges between ideas from physics, geometry, and algebra, offering new perspectives to students and researchers in each field
- Incorporates a wide array of exercises and detailed illustrations, allowing readers to better familiarize themselves with the material
- Includes ten new chapters in the second edition, as well as additional exercises, examples, and illustrations
- Includes supplementary material: sn.pub/extras
Part of the book series: Graduate Texts in Mathematics (GTM, volume 220)
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Table of contents (21 chapters)
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Front Matter
About this book
The main objective of this book is to explain how differential calculus is a natural part of commutative algebra. This is achieved by studying the corresponding algebras of smooth functions that result in a general construction of the differential calculus on various categories of modules over the given commutative algebra. It is shown in detail that the ordinary differential calculus and differential geometry on smooth manifolds turns out to be precisely the particular case that corresponds to the category of geometric modules over smooth algebras. This approach opens the way to numerous applications, ranging from delicate questions of algebraic geometry to the theory of elementary particles.
Smooth Manifolds and Observables is intended for advanced undergraduates, graduate students, and researchers in mathematics and physics. This second edition adds ten new chapters to further develop the notion of differential calculus over commutative algebras, showing it to be a generalization of the differential calculus on smooth manifolds. Applications to diverse areas, such as symplectic manifolds, de Rham cohomology, and Poisson brackets are explored. Additional examples of the basic functors of the theory are presented alongside numerous new exercises, providing readers with many more opportunities to practice these concepts.
Keywords
- Smooth manifolds
- Smooth manifold theory
- Observables
- Observability
- differential geometry smooth manifolds
- Algebraic geometry smooth manifolds
- Differential calculus commutative algebras
- Diffiety
- Diffieties
- Alexandre Vinogradov book
- Smooth functions
- Smooth maps
- Smooth bundles
- Vector bundles
- Projective modules
- Spencer complexes
- Differential forms
- Hamiltonian formalism
Authors and Affiliations
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Lizzano in Belvedere, Italy, Moscow, Russia
Jet Nestruev
About the author
Bibliographic Information
Book Title: Smooth Manifolds and Observables
Authors: Jet Nestruev
Series Title: Graduate Texts in Mathematics
DOI: https://doi.org/10.1007/978-3-030-45650-4
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Nature Switzerland AG 2020
Hardcover ISBN: 978-3-030-45649-8Published: 10 September 2020
Softcover ISBN: 978-3-030-45652-8Published: 10 September 2021
eBook ISBN: 978-3-030-45650-4Published: 10 September 2020
Series ISSN: 0072-5285
Series E-ISSN: 2197-5612
Edition Number: 2
Number of Pages: XVIII, 433
Number of Illustrations: 88 b/w illustrations
Topics: Manifolds and Cell Complexes (incl. Diff.Topology), Algebra, Quantum Physics, Quantum Information Technology, Spintronics