Smooth Manifolds and Observables

Jet Nestruev9780387955438, 0387955437, 9780387227399

Smooth Manifolds and Observables is about the differential calculus, smooth manifolds, and commutative algebra. While these theories arose at different times and under completely different circumstances, this book demonstrates how they constitute a unified whole. The motivation behind this synthesis is the mathematical formalization of the process of observation in classical physics.
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.
This unique textbook contains a large number of exercises and is intended for advanced undergraduates, graduate students, and research mathematicians and physicists.

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Table of contents :
Title……Page 1
Preface to the English Edition……Page 3
Preface……Page 4
Contents……Page 9
1 Introduction……Page 11
2 Cutoff and Other Special Smooth Functions on R^n……Page 22
3 Algebras and Points……Page 29
4 Smooth Manifolds (Algebraic Definition)……Page 45
5 Charts and Atlases……Page 60
6 Smooth Maps……Page 71
7 Equivalence of Coordinate and Algebraic Definitions……Page 83
8 Spectra and Ghosts……Page 91
9 The Differential Calculus as a Part of Commutative Algebra……Page 100
10 Smooth Bundles……Page 147
11 Vector Bundles and Projective Modules……Page 166
Afterword……Page 210
Appendix A.M.Vinogradov Observability Principle, Set Theory and the “Foundations of Mathematics”……Page 212
References……Page 219
Index……Page 221