for a paper of fundamental and lasting importance, 'On manifolds homeomorphic to the 7-sphere', Annals of Mathematics 64 (1956), 399- 405. Milnor's current interest is dynamics, especially holomorphic dynamics. His work in dynamics is summarised by Peter Makienko in his review of [ 9 ]:- For a good overview of Milnor's mathematics, see the citations for the various prizes which he has won at THIS LINK. It is evident now that low-dimensional dynamics, to a large extent initiated by Milnor's work, is a fundamental part of general dynamical systems theory. Milnor cast his eye on dynamical systems theory in the mid- 1970s. By that time the Smale program in dynamics had been completed. Milnor's approach was to start over from the very beginning, looking at the simplest nontrivial families of maps. The first choice, one-dimensional dynamics, became the subject of his joint paper with Thurston. Even the case of a unimodal map, that is, one with a single critical point, turns out to be extremely rich. This work may be compared with Poincaré's work on circle diffeomorphisms, which 100 years before had inaugurated the qualitative theory of dynamical systems. Milnor's work has opened several new directions in this field, and has given us many basic concepts, challenging problems and nice theorems.

The Algebraist - Iain M. Banks - Google Books The Algebraist - Iain M. Banks - Google Books

The Algebraist marks a return to the happy hunting grounds of Banks's early SF, replete with all the whizzy boys' toys, wildly improbable extreme sports, damning character assassinations and good-humoured condemnation of all that's wearying about humanity. The Culture, the great civilisation of many of his previous SF novels, is absent, but it's been replaced by a baroque sweep of aliens in capitalist overdrive, providing more than adequate fuel for the author's twin obsessions of sociopolitics and having fun, the two always riding hand in glove, switching with enviable effortlessness between the intimate and the cosmic. E H Brown, Review: Topology from the differentiable viewpoint, by John Willard Milnor, Amer. Math. Monthly 74 (4) (1967), 461. Algebra began with computations similar to those of arithmetic, with letters standing for numbers. [7] This allowed proofs of properties that are true no matter which numbers are involved. For example, in the quadratic equation a x 2 + b x + c = 0 , {\displaystyle axJ Milnor, Differential Topology Forty-six Years Later, Notices Amer. Math. Soc. 58 (6) (2011), 804- 809. He began research at Princeton after graduating with his B.A. and, in 1953, before completing his doctoral studies, he was appointed to the faculty in Princeton. While undertaking research he enjoyed playing games in the common room. In particular he played Kriegspiel (a game of blindfold chess ), Go and Nash (a game invented by John Nash and now called Hex ). In fact John Nash was at Princeton during these years and Milnor and Nash often talked about game theory. Milnor's next paper, written while he was undertaking research, was Sums of positional games (1953). Milnor writes in the Introduction:-

The Algebraist - Iain M. Banks - Google Books

By a link homotopy is meant a deformation of one link onto another, during which each component of the link is allowed to cross itself, but such that no two components are allowed to intersect. The purpose of this paper is to study links under the relation of homotopy. The fundamental tool in this study will be the link group. The link group of a link is a factor group of the fundamental group of its complement, which is invariant under homotopy. ... I am indebted to R H Fox for assistance in the preparation of this paper. The word algebra is not only used for naming an area of mathematics and some subareas; it is also used for naming some sorts of algebraic structures, such as an algebra over a field, commonly called an algebra. Sometimes, the same phrase is used for a subarea and its main algebraic structures; for example, Boolean algebra and a Boolean algebra. A mathematician specialized in algebra is called an algebraist. The quadratic formula expresses the solution of the equation ax 2 + bx + c = 0, where a is not zero, in terms of its coefficients a, b and c.With an article, it means an instance of some algebraic structure, like a Lie algebra, an associative algebra, or a vertex operator algebra. Milnor has received many awards and honours for his extraordinarily important contributions. He received the National Medal of Science in 1967 and was elected a member of the National Academy of Sciences, the American Academy of Arts and Science. He is a member of the American Philosophy Society and has played a major role in the American Mathematical Society. In August 1982 Milnor received the Leroy P Steele Prize:- Without an article, it means a part of algebra, such as linear algebra, elementary algebra (the symbol-manipulation rules taught in elementary courses of mathematics as part of primary and secondary education), or abstract algebra (the study of the algebraic structures for themselves). Algebra (from Arabic ‏ الجبر‎ ( al-jabr)'reunion of broken parts, [1] bonesetting' [2]) [ʔldʒbr] ( listen ⓘ) is the study of variables and the rules for manipulating these variables in formulas; [3] it is a unifying thread of almost all of mathematics. [4] For those not acquainted with large-scale SF, The Algebraist is a perfect place to have your mind blown to smithereens with all that its vast canvas delivers. In particular, if you're used to the less ambitious and necessarily less physically astonishing pleasures of contemporary fiction, you might want to take out insurance on the integrity of your skull.

Algebraist - definition of algebraist by The Free Dictionary Algebraist - definition of algebraist by The Free Dictionary

E H Spanier, Review: Characteristic classes, by John Willard Milnor and James D Stasheff, Bull. Amer. Math. Soc. 81 (5) (1975), 862- 866. He received the Wolf Prize (1989), the Leroy P Steele Prize for Mathematical Exposition (2004), the Leroy P Steele Prize for Lifetime Achievement (2011), the Abel Prize (2011) and in 2014 was made a Fellow of the American Mathematical Society. The references [ 4 ] to [ 18 ] give a good indication of the wide influence of Milnor's work up to 1992 (when these articles were written ). The article [ 4 ] is a survey of Milnor's work in algebra, particularly in algebraic K K K-theory, where his work continues to have important influences. The article [ 17 ] looks at nine papers which Milnor had written on differential geometry. It discusses Milnor's theorem, which shows that the total curvature of a knot is at least 4. Among other results discussed are Milnor's result showing that we cannot necessarily "hear the shape" of a 16-dimensional torus, and another result giving upper and lower bounds on the number of distinct words of a given length in a finitely generated subgroup of the fundamental group. M Raussen and J Milnor, Interview with John Milnor, Notices Amer. Math. Soc. 59 (3) (2012), 400- 408.. In the 1950s Milnor did a substantial amount of work on algebraic topology which is discussed in [ 18 ]. He constructed the classifying space of a topological group and gave a geometric realisation of a semi-simplicial complex. He also studied the Steenrod algebra and its dual, investigated the structure of Hopf algebras, and studied characteristic classes and their relation to mathematical physics.

This was only one of several papers that Milnor published in 1953. The others were: The characteristics of a vector field on the two-sphere; On total curvatures of closed space curves; and (with Israel Herstein ) An axiomatic approach to measurable utility. Another paper, Link groups, was published in 1954 but it had been submitted for publication in March 1952, over a year before the first of the 1953 papers just mentioned. Milnor writes in the Introduction to Link groups:-

Milnor (1931 - ) - Biography - MacTutor History of John Milnor (1931 - ) - Biography - MacTutor History of

M Spivak, A brief report of John Milnor's brief excursions into differential geometry, in Topological methods in modern mathematics (Houston, TX, 1993), 31- 43. N H Kuiper, Review: Morse theory, by John Willard Milnor, Bull. Amer. Math. Soc. 71 (1) (1965), 136- 137. Milnor has written eight important books: Morse theory (1963); Lectures on the h-cobordism theorem (1965); Topology from the differentiable viewpoint (1965); Singular points of complex hypersurfaces (1968); Introduction to algebraic K-theory (1971); (with Dale Husemoller ) Symmetric bilinear forms (1973); (with James D Stasheff ) Characteristic classes (1974); and Dynamics in one complex variable (1999). J Sondow, An aroma of paradox and audacity : Milnor's work in differential topology, in Topological methods in modern mathematics (Houston, TX, 1993), 23- 30.J Hubbard, Review: Dynamics in one complex variable, by John Willard Milnor, Bull. Amer. Math. Soc. (N.S. ) 38 (4) (2001), 495- 498. For the kind of algebraic structure, see Algebra over a field. For other uses, see Algebra (disambiguation). In low dimensions manifolds are things that are easily visualized. A curve in space is an example of a one-dimensional manifold; the surfaces of a sphere and of a doughnut are examples of two-dimensional manifolds. But for mathematicians the dimensions one and two are just the beginning; things get more interesting in higher dimensions. Also, for physicists manifolds are very important, and it is essential for them to look at higher-dimensional examples. For example, suppose you study the motion of an airplane. To describe just the position takes three coordinates, but then you want to describe what direction it is going in, the angle of its wings, and so on. It takes three coordinates to describe the point in space where the plane is centred and three more coordinates to describe its orientation, so already you are in a six-dimensional space. As the plane is moving, you have a path in six-dimensional space, and this is only the beginning of the theory. If you study the motion of the particles in a gas, there are enormously many particles bouncing around, and each one has three coordinates describing its position and three coordinates describing its velocity, so a system of a thousand particles will have six thousand coordinates. Of course, much larger numbers occur, so mathematicians and physicists are used to working in large-dimensional spaces.