Smooth Manifolds and Observables

• Cutoff and Other Special Smooth Functions on ?n.- Algebras and Points.- Smooth Manifolds (Algebraic Definition).- Charts and Atlases.- Smooth Maps.- Equivalence of Coordinate and Algebraic Definitions.- Spectra and Ghosts.- The Differential Calculus as a Part of Commutative Algebra.- Smooth Bundles.- Vector Bundles and Projective Modules.

From the reviews: "Main themes of the book are manifolds, fibre bundles and differential operators acting on sections of vector bundles. ... A classical treatment of these topics starts with a coordinate description of a manifold M ... . The present book is based on an alternative point of view, where calculus on manifolds is treated as a part of commutative algebra. ... The book contains quite a few exercises and many useful illustrations." (EMS, September, 2004) "The book provides a self-contained introduction to the theory of smooth manifolds and fibre bundles, oriented towards graduate students in mathematics and physics. The approach followed here, however, substantially differs from most textbooks on manifold theory. ... This book is certainly quite interesting and may appeal even to people who merely want to study algebraic geometry, in the sense that they will gain extra insight here by the attention which is paid to making certain constructions in algebraic geometry physically or intuitively acceptable." (Willy Sarlet, Zentralblatt MATH, Vol. 1021, 2003)