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Title: Morse Theory. (AM-51) by John Willard Milnor ISBN: 0-691-08008-9 Publisher: Princeton Univ Pr Pub. Date: 01 May, 1963 Format: Paperback Volumes: 1 List Price(USD): $49.95 |
Average Customer Rating: 5 (4 reviews)
Rating: 5
Summary: A Gem of Exposition
Comment: This book needs no review, as it is so well-known. But reading the first critique about the need for a more intuitive development of the subject, perhaps one viewpoint is worth restating. This book is a model of concise exposition. All of Milnor's works are written in a fashion that "makes it clear", except that sometimes years of thought are really needed to see the essence of the arguments. The writing sets up the framework for understanding, and the reader must work to fill it in. This is what makes it great writing. The reader accepts the truth of the statements, but Milnor does not bludgeon the reader with details which can be filled in by a professional mathematician. Or as many of us understand it, when you can fill in all the details, you have developed geometric intuition and are on the way to a deeper understanding of the subject.
Each chapter of the book is a classic. Chapter 2 on Riemannian geometry gives an overview of the subject which can be used as a basis for teaching a course on the same. When the students can fill in the details, they understand the core of the subject.
Rating: 5
Summary: An important book
Comment: Morse theory is about finding geometric information about manifolds using analysis of functions(smooth functions) having these manifolds as their domain. The important fact is that Morse theory can be generalized to infinite dimensional manifolds (i.e. understanding the "geometric" structure of these manifolds using analysis of functions defined on the manifolds). Milnor's book motivate the "infinite" Morse theory in the first chapter explaining finite Morse theory very clearly and in a beautifull manner. The second chapter is preliminaries on Riemannian geometry which leads to the second part of the book about infinite Morse theory. To be more clear:the spaces/manifolds are path spaces on lie groups (compact lie groups mostly ) and the functions on these spaces are "energy" functions of these paths.
Once the reader understand the first chapter, the infinite dimentional Morse theory is much more easy to understand.
And Milnor does a very good job explaining the similarity.
The final chapter shows some implications of Morse theoy in particular Bott periodicity theorem.
One remark is in order for someone who doesnt know(Riemannian) geometry the book can be very hard to read, so I recommend as a starting point the beautiful book of Matsumoto "Morse theory" this is avery good book in a basic level which can serve as a very good introduction to milnor's book.
Rating: 5
Summary: The best
Comment: When I was just becoming a mathematician, my teacher gave me this book, saying "You're not ready for this yet, but you should have it --- it's the best piece of mathematical exposition there is." Maybe that claim's exaggerated, but I've yet to find one I prefer. Along with Milnor's Lectures on the h-Cobordism Theorem, and his Characteristic Classes, this book is a lesson not only in topology (and wonderful topology, too!), but in clear writing as well.
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Title: Topology from the Differentiable Viewpoint by John Willard Milnor ISBN: 0691048339 Publisher: Princeton Univ Pr Pub. Date: 24 November, 1997 List Price(USD): $17.95 |
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Title: Characteristic Classes. (AM-76) by John Willard Milnor, James D. Stasheff ISBN: 0691081220 Publisher: Princeton Univ Pr Pub. Date: 01 August, 1974 List Price(USD): $65.00 |
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Title: The Topology of Fibre Bundles. (PMS-14) by Norman Steenrod ISBN: 0691005486 Publisher: Princeton Univ Pr Pub. Date: 05 April, 1999 List Price(USD): $29.95 |
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Title: Differential Forms in Algebraic Topology (Graduate Texts in Mathematics ; 82) by Raoul Bott, Loring W. Tu ISBN: 0387906134 Publisher: Springer Verlag Pub. Date: July, 1997 List Price(USD): $69.95 |
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Title: The Geometry of Schemes by David Eisenbud, Joe Harris ISBN: 0387986375 Publisher: Springer Verlag Pub. Date: 22 December, 1999 List Price(USD): $34.95 |
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