In mathematics , specifically in differential topology , Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse , a typical differentiable function on a manifold will reflect the topology quite directly. Morse theory allows one to find CW structures and handle decompositions on manifolds and to obtain substantial information about their homology. Morse originally applied his theory to geodesics critical points of the energy functional on paths. These techniques were used in Raoul Bott 's proof of his periodicity theorem. The analogue of Morse theory for complex manifolds is Picard—Lefschetz theory.
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Would you like to tell us about a lower price? If you are a seller for this product, would you like to suggest updates through seller support? One of the most cited books in mathematics, John Milnor's exposition of Morse theory has been the most important book on the subject for more than forty years.
Morse theory was developed in the s by mathematician Marston Morse. One classical application of Morse theory includes the attempt to understand, with only limited information, the large-scale structure of an object. This kind of problem occurs in mathematical physics, dynamic systems, and mechanical engineering. Morse theory has received much attention in the last two decades as a result of a famous paper in which theoretical physicist Edward Witten relates Morse theory to quantum field theory.
Milnor was awarded the Fields Medal the mathematical equivalent of a Nobel Prize in for his work in differential topology. He has since received the National Medal of Science and the Steele Prize from the American Mathematical Society twice and in recognition of his explanations of mathematical concepts across a wide range of scienti. The citation reads, "The phrase sublime elegance is rarely associated with mathematical exposition, but it applies to all of Milnor's writings.
Reading his books, one is struck with the ease with which the subject is unfolding and it only becomes apparent after re. Read more Read less. Frequently bought together. Add all three to Cart. These items are shipped from and sold by different sellers. Show details.
Buy the selected items together This item: Morse Theory. Customers who viewed this item also viewed. Page 1 of 1 Start over Page 1 of 1. Previous page. Differential Forms in Algebraic Topology. Heat Kernels and Dirac Operators. Mathematical Methods of Classical Mechanics. Introduction to Smooth Manifolds. Next page. Customers who bought this item also bought.
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Write a customer review. Most helpful customer reviews on Amazon. Verified Purchase. 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.
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.
An example is a double planar pendulum. Since each pendulum is free to rotate 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. Not easy going but rewarding. I'm still working on it. The classic work on the subject, written in the incomparable style of John Milnor.
The difficult subject is explained with a simplicity and clarity which rivals how Richard Feynman was able to write about Physics. 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.
This is a good supplement for more information on Morse Theory but be warned that it is a set of lecture notes. Great quality, book came with handwritten note! Go to Amazon.
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Morse Theory. (AM-51), Volume 51