Smooth Manifolds and Observables / by Jet Nestruev.

Format:

Book

Author/Creator:

Nestruev, Jet, author.

Contributor:

SpringerLink (Online service)

Series:

Mathematics and Statistics (SpringerNature-11649)

Graduate texts in mathematics 0072-5285 ; 220.

Graduate Texts in Mathematics, 0072-5285 ; 220

Language:

English

Subjects (All):

Manifolds (Mathematics).

Complex manifolds.

Algebra.

Quantum theory.

Quantum computers.

Spintronics.

Manifolds and Cell Complexes (incl. Diff.Topology).

Quantum Physics.

Quantum Information Technology, Spintronics.

Local Subjects:

Manifolds and Cell Complexes (incl. Diff.Topology).

Algebra.

Quantum Physics.

Quantum Information Technology, Spintronics.

Physical Description:

1 online resource (XVIII, 433 pages) : 88 illustrations.

Edition:

Second edition 2020.

Contained In:

Springer Nature eBook

Place of Publication:

Cham : Springer International Publishing : Imprint: Springer, 2020.

System Details:

text file PDF

Summary:

This textbook demonstrates how differential calculus, smooth manifolds, and commutative algebra constitute a unified whole, despite having arisen at different times and under different circumstances. Motivating this synthesis is the mathematical formalization of the process of observation from classical physics. A broad audience will appreciate this unique approach for the insight it gives into the underlying connections between geometry, physics, and commutative algebra. 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.

Contents:

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.

Other Format:

Printed edition:

ISBN:

978-3-030-45650-4

9783030456504

Access Restriction:

Restricted for use by site license.