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Topology from the Differentiable Viewpoint
This elegant book by distinguished mathematician John Milnor, provides a clear and succinct introduction to one of the most important subjects in modern mathematics. Beginning with basic concepts such as diffeomorphisms and smooth manifolds, he goes on to examine tangent spaces, oriented manifolds, and vector fields. Key concepts such as homotopy, the index number of a map, and the Pontryagin construction are discussed. The author presents proofs of Sard's theorem and the Hopf theorem.
Characteristic Classes. (AM-76), Volume 76
The theory of characteristic classes provides a meeting ground for the various disciplines of differential topology, differential and algebraic geometry, cohomology, and fiber bundle theory. As such, it is a fundamental and an essential tool in the study of differentiable manifolds. In this volume, the authors provide a thorough introduction to characteristic classes, with detailed studies of Stiefel-Whitney classes, Chern classes, Pontrjagin classes, and the Euler class. Three appendices cover the basics of cohomology theory and the differential forms approach to characteristic classes, and provide an account of Bernoulli numbers. Based on lecture notes of John Milnor, which first appeared at Princeton University in 1957 and have been widely studied by graduate students of topology ever since, this published version has been completely revised and corrected.
Morse Theory. (AM-51), Volume 51
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 t...
Lectures on the h-Cobordism Theorem
Important lectures on differential topology by acclaimed mathematician John Milnor These are notes from lectures that John Milnor delivered as a seminar on differential topology in 1963 at Princeton University. These lectures give a new proof of the h-cobordism theorem that is different from the original proof presented by Stephen Smale. Milnor's goal was to provide a fully rigorous proof in terms of Morse functions. This book remains an important resource in the application of Morse theory.
Dynamics in One Complex Variable
This volume studies the dynamics of iterated holomorphic mappings from a Riemann surface to itself, concentrating on the classical case of rational maps of the Riemann sphere. This subject is large and rapidly growing. These lectures are intended to introduce some key ideas in the field, and to form a basis for further study. The reader is assumed to be familiar with the rudiments of complex variable theory and of two-dimensional differential geometry, as well as some basic topics from topology. This third edition contains a number of minor additions and improvements: A historical survey has been added, the definition of Lattés map has been made more inclusive, and the écalle-Voronin theory of parabolic points is described. The résidu itératif is studied, and the material on two complex variables has been expanded. Recent results on effective computability have been added, and the references have been expanded and updated. Written in his usual brilliant style, the author makes difficult mathematics look easy. This book is a very accessible source for much of what has been accomplished in the field.
Singular Points of Complex Hypersurfaces. (AM-61), Volume 61
The description for this book, Singular Points of Complex Hypersurfaces. (AM-61), Volume 61, will be forthcoming.
Introduction to Algebraic K-Theory
Algebraic K-theory describes a branch of algebra that centers about two functors. K0 and K1, which assign to each associative ring ∧ an abelian group K0∧ or K1∧ respectively. Professor Milnor sets out, in the present work, to define and study an analogous functor K2, also from associative rings to abelian groups. Just as functors K0 and K1 are important to geometric topologists, K2 is now considered to have similar topological applications. The exposition includes, besides K-theory, a considerable amount of related arithmetic.
Frontiers in Complex Dynamics
John Milnor, best known for his work in differential topology, K-theory, and dynamical systems, is one of only three mathematicians to have won the Fields medal, the Abel prize, and the Wolf prize, and is the only one to have received all three of the Leroy P. Steele prizes. In honor of his eightieth birthday, this book gathers together surveys and papers inspired by Milnor's work, from distinguished experts examining not only holomorphic dynamics in one and several variables, but also differential geometry, entropy theory, and combinatorial group theory. The book contains the last paper written by William Thurston, as well as a short paper by John Milnor himself. Introductory sections put t...
Prospects in Mathematics
Five papers by distinguished American and European mathematicians describe some current trends in mathematics in the perspective of the recent past and in terms of expectations for the future. Among the subjects discussed are algebraic groups, quadratic forms, topological aspects of global analysis, variants of the index theorem, and partial differential equations.
