Customer Reviews

Morse Theory (Annals of Mathematic Studies AM-51)

5 star:		(5)
4 star:		(1)
3 star:		(0)
2 star:		(0)
1 star:		(0)

 

 

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.

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.

Perhaps everyone who has had a bite in topology feels that this book is too famous to be given any kind of reference. The above having been said, this book is really a gem, elegantly explaining the Bott periodicity in the spirit of the original article by Bott. Of course, a simpler proof using K-theory has been available since the sixties, but that does not deteriorate the value of this book.

This is a great book, a classic. Sure, the fonts and diagrams could use some modernization. But if you really want to learn a subject in Math, learn from the giants.
This is very easy to read (assuming you are a first year in grad school in Math, or an advanced undergrad.)

I gave this book a 5star because it does succeed in presenting the main ideas,theorems,and proofs in Morse theory. This book was given as a reference in V.I.Arnold's"Mathematical Methods Of Classical Mechanics" for further information on Riemannian curvature(AppendixI). It turns out that Arnold's book contains much material which is prerequisite to Milnor. Euler-Lagrange equations,vector fields on manifolds,Poisson Brackets,and more which you will encounter in Milnor are explained in Arnold. Another book which helps is Bishop and Crittendon's"Geometry Of Manifolds." The configuration space of a mechanical system can and does in many cases translate to a Riemannian manifold with a motion of the system necessarily translating to a geodesic of the manifold. An example is a double planar pendulum. Since each pendulum is free to rotate 360 degrees,its configuration space is a torus or donut. Geodesics are extremal paths,hopefully minimal. One possible path is a closed spiral on the donut. Is it a geodesic? What's the motion? This book answers a great many questions as to how the shape or curvature and topology of the manifold influences and determines its geodesics. A knowledge of homotopy theory,deformation retracts,cw complexes,etc. is needed. Not easy going but rewarding. I'm still working on it.

This book is a classic. If your grandpa did math, he probably did it out of this book. I think maybe Gauss picked up a few of his tricks here.

The age of this book was an issue. Some of Milnor's words are not in common use any more. He doesn't phrase his results in the modern language of tensors, which was troubling for me. With that said, some of Milnor's proofs are so clear and readable that they can't be improved upon. This is a great place to start learning Morse theory, though I'd look at a more modern treatment too if you are serious.