Morse theory

by Milnor, John W., (John Willard)

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Title Statements

Full title :Morse theory / by J. Milnor ; based on lecture notes by M. Spivak and R. Wells.

Summary

"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 1920s by mathematician Marston Morse. (Morse was on the faculty of the Institute for Advanced Study, and Princeton published his Topological Methods in the Theory of Functions of a Complex Variable in the Annals of Mathematics Studies series in 1947.) 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 1962 for his work in differential topology. He has since received the National Medal of Science (1967) and the Steele Prize from the American Mathematical Society twice (1982 and 2004) in recognition of his explanations of mathematical concepts across a wide range of scientific disciplines. 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 reflection that this ease is the mark of a master.'"--Publisher

Edition Statements

Publication Date:©1969.

Publication Statements

Publisher(s):Princeton University Press
Place of Publication:Princeton, N.J
Language:eng
Publication Date:©1969.

The Physical Object

Description:viii, 153 pages : : illustrations ; : 24 cm

Edition Identifiers

ISBN:0691080089, 9780691080086
oclc:ocm11597574
LC Classification:QA611
Dewey Decimal:514.24

Part I. Non-Degenerate Smooth Functions on a Manfold -- Introduction -- Definitions and Lemmas -- Homotopy Type in Terms of Critical Values -- Examples -- The Morse Inequalities -- Manifolds in Euclidean Space: The Existence of Non-degenerate Functions -- The Lefschetz Theorem on Hyperplane Sections -- Part II. A Rapid Course in Riemannian Geometry -- Covariant Differentiation -- The Curvature Tensor -- Geodesics and Completeness -- Part III. The Calculus of Variations Applied to Geodesics -- The Path Space of a Smooth Manifold -- The Energy of a Path -- The Hessian of the Energy Function at a Critical Path -- Jacobi Fields: The Null-space of E** -- The Index Theorem -- A Finite Dimensional Approximation to ^c -- The Topology of the Full Path Space -- Existence of Non-conjugate Points -- Some Relations Between Topology and Curvature -- Part IV. Applications to Lie Groups and Symmetric Spaces -- Symmetric Spaces -- Lie Groups as Symmetric Spaces -- Whole Manifolds of Minimal Geodesics -- The Bott Periodicity Theorem for the Unitary Group -- The Periodicity Theorem for the Orthogonal Group.