Smooth Manifolds and Observables
Springer International Publishing (Verlag)
978-3-030-45649-8 (ISBN)
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.
Jet Nestruev is a collective of authors, who originally convened for a seminar run by Alexandre Vinogradov at the Mechanics and Mathematics Department of Moscow State University in 1969. In the present edition, Jet Nestruev consists of Alexander Astashov (Senior Researcher at the State Research Institute of Aviation Systems), Alexandre Vinogradov (Professor of Mathematics at Salerno University), Mikhail Vinogradov (Diffiety Institute), and Alexey Sossinsky (Professor at the Independent University of Moscow).
Foreword.- Preface.- 1. Introduction.- 2. Cutoff and Other Special Smooth Functions on R^n.- 3. Algebras and Points.- 4. Smooth Manifolds (Algebraic Definition).- 5. Charts and Atlases.- 6. Smooth Maps.- 7. Equivalence of Coordinate and Algebraic Definitions.- 8. Points, Spectra and Ghosts.- 9. The Differential Calculus as Part of Commutative Algebra.- 10. Symbols and the Hamiltonian Formalism.- 11. Smooth Bundles.- 12. Vector Bundles and Projective Modules.- 13. Localization.- 14. Differential 1-forms and Jets.- 15. Functors of the differential calculus and their representations.- 16. Cosymbols, Tensors, and Smoothness.- 17. Spencer Complexes and Differential Forms.- 18. The (co)chain complexes that come from the Spencer Sequence.- 19. Differential forms: classical and algebraic approach.- 20. Cohomology.- 21. Differential operators over graded algebras.- Afterword.- Appendix.- References.- Index.
| Erscheinungsdatum | 12.09.2020 |
|---|---|
| Reihe/Serie | Graduate Texts in Mathematics |
| Zusatzinfo | XVIII, 433 p. 88 illus. |
| Verlagsort | Cham |
| Sprache | englisch |
| Maße | 155 x 235 mm |
| Gewicht | 856 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Algebra |
| Mathematik / Informatik ► Mathematik ► Geometrie / Topologie | |
| Naturwissenschaften ► Physik / Astronomie ► Quantenphysik | |
| Schlagworte | Alexandre Vinogradov book • Algebraic geometry smooth manifolds • Differential calculus commutative algebras • differential forms • differential geometry smooth manifolds • Diffieties • Diffiety • Hamiltonian Formalism • observability • observables • Projective modules • Smooth bundles • Smooth Functions • Smooth manifolds • Smooth manifold theory • smooth maps • Spencer complexes • Vector Bundles |
| ISBN-10 | 3-030-45649-8 / 3030456498 |
| ISBN-13 | 978-3-030-45649-8 / 9783030456498 |
| Zustand | Neuware |
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