The Algebraist

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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] Imagine that he is hugely enthusiastic and charming, and that his thoughtful analyses of contemporary human politics range from the individual to the mass, from theory to action, from ideology to consequence. The word "algebra" has several related meanings in mathematics, as a single word or with qualifiers.

The Algebraist by Iain M. Banks | Hachette Book Group

Sometimes both meanings exist for the same qualifier, as in the sentence: Commutative algebra is the study of commutative rings, which are commutative algebras over the integers. 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. The first time that I developed a particular interest in mathematics was as a freshman at Princeton University. I had been rather socially maladjusted and did not have too many friends, but when I came to Princeton, I found myself very much at home in the atmosphere of the mathematics common room. People were chatting about mathematics, playing games, and one could come by at any time and just relax. I found the lectures very interesting. I felt more at home there than I ever had before and I have stayed with mathematics ever since. As a single word with an article or in the plural, "an algebra" or "algebras" denotes a specific mathematical structure, whose precise definition depends on the context. Usually, the structure has an addition, multiplication, and scalar multiplication (see Algebra over a field). When some authors use the term "algebra", they make a subset of the following additional assumptions: associative, commutative, unital, and/or finite-dimensional. In universal algebra, the word "algebra" refers to a generalization of the above concept, which allows for n-ary operations. H Bass, John Milnor, the algebraist, in Topological methods in modern mathematics (Houston, TX, 1993), 45- 84.

al·ge·bra

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 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. for a paper of fundamental and lasting importance, 'On manifolds homeomorphic to the 7-sphere', Annals of Mathematics 64 (1956), 399- 405. 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. Milnor's current interest is dynamics, especially holomorphic dynamics. His work in dynamics is summarised by Peter Makienko in his review of [ 9 ]:-

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

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). Biography John Milnor's parents were Joseph Willard Milnor (1889- 1949) and Emily Cox (1891- 1973). Joseph Milnor, born in Williamsport, Pennsylvania, graduated from Lehigh University in 1912 with first class honours in mathematics. After serving for a year with the General Electric Company in Pittsfield, he entered the engineering department of the Western Union Telegraph Company in 1913. Nine years later he was promoted to research engineer and, in 1936, became a transmission engineer. He was appointed consulting engineer in 1943 and retired in the following year. 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. Imagine that the storyteller has a well-educated and thoughtful mind with which he fills you in on all the details of these new worlds and peculiar personalities, and that he has the skill to paint in words the most breathtaking portraits of our universe on levels from the chemical to the personal.

For the kind of algebraic structure, see Algebra over a field. For other uses, see Algebra (disambiguation).