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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 text serves as an introduction to the theory of smooth manifolds, but it's quite unlike any other text on the subject. The classic approach to manifold theory is heavy on the (point-set) topology and analysis; this one is heavy on the algebra. In this book, the author shows that a manifold (in the traditional sense) is completely characterized by its ring of smooth functions. The question then becomes: which rings can be constructed in this way? The author answers this question and, in the process, ends up building a completely algebraic theory of manifolds--it's totally awesome.

What's more, he uses the idea of physical observables to motivate the construction. This is a delightful surprise, since algebraists don't tend to focus much on applications of a theory. Indeed, the concept of physical observation is brought up in the first chapter and used to justify many of the algebraic constructions throughout the book. A traditional manifold book would probably leave the entire discussion of mechanics (and more generally, symplectic structures)until the end (if it's included at all).

Though the technical prerequisites are modest, I suggest you study some other texts first (or concurrently). The traditional approach to manifolds is still quite important and (as of yet) irreplaceable. I suggest you study this either before or along with this text. Indeed, you might get more out of this book if you are already familiar with the traditional theory. I suggest John Lee's "An Introduction to Smooth Manifolds"--the best out there in my opinion. Furthermore, you should make sure your algebra is sharp before you undertake this book. A graduate level understanding should suffice (something on the order of Dummit/Foote's Abstract Algebra text should do quite nicely).

Some of the material is quite similar to commutative algebra and algebraic geometry. This book actually makes a good spring board for studying those subjects. You should pick up an algebraic geometry book or, preferably, Eisenbud's "Commutative Algebra: with a view towards algebraic geometry" either after or along with this book.

Overall, it's a spectacular text with lots of exercises. I recommend this to anyone who hasn't learned this material.

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