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This digital document, covering the life and work of John (Willard) Milnor, is an entry from Contemporary Authors, a reference volume published by Thompson Gale. The length of the entry is 1274 words. The page length listed above is based on a typical 300-word page. Although the exact content of each entry from this volume can vary, typical entries include the following information:

• Place and date of birth and death (if deceased)
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Chapters: John Forbes Nash, Jr., Richard Feynman, Roald Hoffmann, Murray Gell-Mann, John Milnor, William Thurston, Richard Smalley, Robert Curl, Stephen Smale, Heisuke Hironaka, Vladimir Voevodsky, Ahmed Zewail, Frederick Reines, Stanley B. Prusiner, Carl Wieman, Shing-Tung Yau, John R. Stallings, Steven Weinberg, Jack Steinberger, Alan Macdiarmid, David Mumford, Frank Wilczek, Terence Tao, Yuan T. Lee, Markus Brunnermeier, Elias James Corey, Richard R. Schrock, Mario J. Molina, Dudley R. Herschbach, Sheldon Lee Glashow, John G. Thompson, Theodor W. Hänsch, David Gross, Leon Cooper, Mark G. Raizen, Kenneth G. Wilson, Robert H. Grubbs, Sloan Fellowship, Linda B. Buck, Charles Fefferman, Michael Freedman, Hugh David Politzer, John Charles Polanyi, Daniel Quillen, Vaughan Jones, Melvin Schwartz, Alan J. Heeger, Emmanuel Candès, Val Logsdon Fitch, Curtis T. Mcmullen, Karl Barry Sharpless, Rudolph A. Marcus, James Cronin, Thomas M. Liggett, Avner Friedman, Harry Kesten, Andrew Ng, Boris Aronov. Source: Wikipedia. Pages: 277. Not illustrated. Free updates online. Purchase includes a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: Richard Phillips Feynman (pronounced ; May 11, 1918 February 15, 1988) was an American physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics (he proposed the parton model). For his contributions to the development of quantum electrodynamics, Feynman, jointly with Julian Schwinger and Sin-Itiro Tomonaga, received the Nobel Prize in Physics in 1965. He developed a widely used pictorial representation scheme for the mathematical expressions governing the behavior of subatomic particles, which later became known as Feynman diagrams. During his lifetime, Feynman became ...More: http://booksllc.net/?id=25523 ... Read more

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High Quality Content by WIKIPEDIA articles! High Quality Content by WIKIPEDIA articles!John Willard Milnor (born February 20, 1931, in Orange, New Jersey) is an American mathematician known for his work in differential topology, K-theory and dynamical systems, and for his influential books. He won the Fields Medal in 1962 and Wolf Prize in 1989. As of 2005, Milnor is a distinguished professor at the State University of New York at Stony Brook. His wife, Dusa McDuff, is a professor of mathematics at Barnard College. ... Read more

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This scarce antiquarian book is a selection from Kessinger Publishing's Legacy Reprint Series. Due to its age, it may contain imperfections such as marks, notations, marginalia and flawed pages. Because we believe this work is culturally important, we have made it available as part of our commitment to protecting, preserving, and promoting the world's literature. Kessinger Publishing is the place to find hundreds of thousands of rare and hard-to-find books with something of interest for everyone! ... Read more

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standard introduction to differential topology, included in Milnor's "Collected Papers"
These brief notes from Milnor's 1958 lectures at Princeton University have circulated for years and were frequently referenced in earlier works in differential topology before other texts, such as Hirsch's Differential Topology, Guillemin & Pollack's Differential Topology, and Milnor's own Topology from the Differentiable Viewpoint became available. This 78-page typescript, created by Munkres, has at long last been incorporated into the third book of Milnor's collected papers published by the AMS (Collected Papers of John Milnor. Volume III: Differential Topology), where after being typeset it only occupies about half as many pages. This valuable collection also includes a number of Milnor's papers on exotic spheres, cobordism, and differential topology, including not only these lecture notes but also his complementary 1961 Princeton notes on differential structures and his chapter on differential topology that appeared in Lectures on Modern Mathematics in 1964, so there really is no reason to buy these notes separately, unless you are a collector.

These notes contain standard introductory material on smooth manifolds, including a proof of the (easy) Whitney embedding theorem and vector bundles, as well as more advanced material, such as the Pontrjagin-Thom construction. Munkres, in his own Elementary Differential Topology, uses these notes as a reference, so his book has only a small overlap with them. As always with Milnor's writings, the explanations are clear and the definitions are precise, but the level is a little higher than, say, his TFTDV or G&P, with manifolds being defined abstractly rather than as subsets of a Euclidean space, and practically no space is spent motivating the subject matter (there aren't even any figures). The style is most similar to that of Munkres, with virtually all the material being contained within Hirsch's book. ... Read more

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Kapitel: Chen Ning Yang, John Willard Milnor, Michail Leonidowitsch Gromow, Shing-Tung Yau, Axel Meyer, Dennis Sullivan, Melba Phillips, Detlef Gromoll, Michael Kimmel, Oleg Janowitsch Wiro, Rose Laub Coser, Massimo Pigliucci, Paul Christian Lauterbur, Jeff Cheeger, Anthony Knapp, Robert Schatten, Barry Mccoy, Bernard Maskit, Mark Aronoff, Douglas J. Futuyma. Aus Wikipedia. Nicht dargestellt. Auszug: Chen-Ning Franklin Yang (simplified Chinese: ; traditional Chinese: ; pinyin: Yáng Zhènníng) (born October 1, 1922) is a Chinese-American physicist who worked on statistical mechanics and particle physics. He, together with Tsung-dao Lee, received the 1957 Nobel prize in physics for their work on parity nonconservation of weak interaction. Yang naturalized as a United States citizen in 1964. Yang was born in Hefei, Anhui, China, his father Yang Ko-Chuen (simplified Chinese: ; pinyin: Yáng Wǔzhī) (1896-1973) was a mathematician and his mother Luó Mènghuà (罗孟华) was a housewife. Yang attended elementary school and high school in Beijing, and in the autumn of 1937 his family moved to Hefei after Japanese invaded China. In 1938 they moved to Kunming, Yunnan, where the National Southwestern Associated University was located. In the same year, as a second year student, Yang passed the entrance examination and studied at the National Southwestern Associated University. He received his bachelor's degree in 1942, the thesis being about the application of group theory to molecular spectra, under the supervision of Wu Ta-you (吴大猷) (1907-2000). He continued to study graduate courses there for two years under the supervision of Wang Zhuxi (王竹溪) (1911-1983), working on statistical mechanics. In 1944 he received his master's degree was awarded a scholarship known as the Boxer Indemnity (simplified Chinese: ; traditional Chinese: ; pinyin: Gēngzǐ péikuǎn), a scholarship set up by the United States government using the funds raised from the money C...http://booksllc.net/?l=de ... Read more

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classic proof of perhaps the most important theorem in higher-dimensional (n>4) topology
Milnor's "Lectures on the h-cobordism theorem" consists of a proof and applications of the h-cobordism theorem, an important technical result that, among other things, leads immediately to a proof of the Poincare conjecture for smooth manifolds of dimension >= 5. The theorem was originally proved by Smale in 1962 (part of the basis for his Fields Medal) using handlebody techniques, but in this book Milnor presents a (partially) different proof using Morse theoretic lemmas due to Morse and Barden. Along the way, a number of theorems and techniques in differential topology are used or derived (including the Whitney trick, finger moves, bicollarings, surgery, extensions of embeddings and isotopies, smooth structures on unions, Poincare duality), which makes this valuable as much for the methods as for the final result. Definitely it is one of the best books for learning how to actually prove things in topology, mixing in Morse theory, Riemannian geometry, and algebraic topology, too.

The h-cobordism theorem states that an n-dim simply connected cobordism W between 2 simply connected (n-1)-dim manifolds V,V' that are each homotopy equivalent to W (which is the definition of an h-cobordism) is isomorphic to a product, with the isomorphism depending on the category of manifolds with which one is working. In particular, if such a W exists, then V and V' must be diffeomorphic. This book, as well as Smale's original proof, only deals with the smooth case, where the theorem is shown to be true for all n except possibly 4 or 5 (with the n=4 case being equivalent to the original Poincare conjecture that has now been proved using different techniques).PL and topological versions of the theorem also has been proved, as well as one with somewhat relaxed requirements (the s-cobordism theorem), but none of these extensions are explored in the book, although there is some mention of them. The proof basically proceeds by finding a Morse function on the cobordism for which it can be decomposed into a union of "elementary cobordisms," each with a single critical value and index. The Morse function is then perturbed in a series of steps so that the critical points can be cancelled, eventually leaving a cobordism with no critical points, which is diffeomorphic to a product. In the handlebody version of the proof (not presented here, but see Kosinski's Differential Manifolds), this decomposition is equivalent to a handlebody decomposition, which is then manipulated by sliding and cancelling handles and until they are all removed. The dimensional restriction (n>=6) primarily comes from the application of the Whitney trick - in lower dimensions there is not enough room to remove algebraically cancelling intersections between embedded spheres - and it is this limitation that accounts for the difficulties in 4-dimensional smooth topology.

The book is accessible to graduate students who have taken 1st-year courses in differential and algebraic topology. He cites such texts as Munkres' Elementary Differential Topology, his own lectures notes (which appear in Collected Papers of John Milnor. Volume III: Differential Topology), and de Rham's Varietes differentiables for the differential topology, but the reader may prefer more modern treatments, such as Guillemin & Pollack's Differential Topology or Hirsch's Differential Topology. For the algebraic topology he often mentions results (such as Alexander duality) without citing any reference, so you'll need a good background in homology theory (I'd recommend Dold's Lectures on Algebraic Topology or Greenberg & Harris's Algebraic Topology: A First Course). One definite prerequisite is the first part of Milnor's own Morse Theory, which provides the foundation for the proof. In fact, this book is good companion to "Morse Theory," in the sense that many important and basic results in the subject that were surprisingly left out of "Morse Theory" are proved here.

Overall the book is relatively clean, with only some typos, chiefly in the form of missing or mistaken subscripts and indices. This only really becomes a problem at the end of Chapter 5, where a key subscript 1 is omitted in the last equation of page 65 (should be E_{1} instead of E) and then repeatedly for the constant k on the next page (i.e., k_{1} instead of K). On pg. 101 he states that he has proved the case index = 1 when he means index = 0 (but this should be obvious), and in a number of places a theorem is cited by the wrong number, in particular, 3.14 and 3.4 are switched a lot.

Despite its age this remains one of the most significant books in the theory of manifolds. The aforementioned book by Kosinski, which you should certainly read as well, is the only other one that I can think of that contains the complete proof of the h-cobordism theorem, but the method is different. More recently, Matsumoto's An Introduction to Morse Theory copies much of the material from this book, including identical proofs and diagrams for Theorem's 4.1 and 5.6, and mixes handlebody arguments with the analytic ones, but stops well short of even stating the h-cobordism theorem. The final few pages contain applications of the theorem to prove various results about smooth manifolds that are contractible or homotopy spheres; cf. Kosinski for more extensive results of this type.
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This volume contains papers of one of the best modern geometers and topologists, John Milnor, on various topics related to the notion of the fundamental group. It is excellent reading for any mathematician with an interest in geometry and topology and for any person with an interest in mathematics.The volume contains sixteen papers and is partitioned into four parts: Knot theory, Free action on spheres, Torsion, and Three-dimensional manifolds. Each part is preceded by an introduction containing the author's comments on further development of the subject. Although some of the papers were written quite a while ago, they appear more modern than many of today's publications. Milnor's excellent, clear, and laconic style makes the book a real treat.This volume is highly recommended to a broad mathematical audience, and, in particular, to young mathematicians who will certainly benefit from their acquaintance with Milnor's mode of thinking and writing. ... Read more