Smooth Manifolds and Observables

Description

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. Foreword Book_BookNotesTitle_1 Preface Contents 1 Introduction 2 Cutoff and Other Special Smooth Functions on mathbbRn 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 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 Coming from the Spencer Sequence 19 Differential Forms: Classical and Algebraic Approach 20 Cohomology 21 Differential Operators over Graded Algebras Afterword Appendix A.M. Vinogradov Observability Principle, Set Theory and the ``Foundations of Mathematics'' References Index