John Horton Conway

John Horton Conway (born December 26, 1937, Liverpool, England) is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He has also contributed to many branches of recreational mathematics.

Conway is currently professor of mathematics at Princeton University. He studied at University of Cambridge, where he started research under Harold Davenport. In 1981 he was elected a Fellow of the Royal Society.

Game theory

Among amateur mathematicians, he is perhaps most widely known for his contributions to combinatorial game theory, a theory of partizan games. This he developed with Elwyn Berlekamp and Richard Guy.

He is also one of the inventors of sprouts, as well as philosopher's football. He developed detailed analyses of many other games and puzzles, such as the Soma cube, and peg solitaire. He came up with the still unsolved Angel problem.

He invented a new system of numbers, the surreal numbers, which are closely related to certain games and have been the subject of a mathematical novel by Donald Knuth. He also invented a nomenclature for exceedingly large numbers, the Conway chained arrow notation.

He is also known for the invention of the game of life. This is one of the early and still celebrated examples of a cellular automaton.

Geometry

In the mid-1960s with Michael Guy, son of Richard Guy, he established that there are sixty-four convex nonprismatic uniform polychora.

Geometric topology

Conway's approach to computing the Alexander polynomial of knot theory, in a variant now called the Alexander-Conway polynomial, involved a skein relation. After lying dormant for more than a decade, this concept became central to work in the 1980s on the novel knot polynomials. Conway further developed tangle theory. Conway has also suggested a system of notation dedicated to describing polyhedra called Conway polyhedron notation.

Group theory

He worked on the classification of finite simple groups and discovered the Conway groups. He was the primary author of the Atlas of Finite Groups giving properties of many finite simple groups. He with collaborators constructed the first concrete representations of some of the sporadic groups.

With Simon Norton he formulated the complex of conjectures relating the monster group with modular functions, christened by them monstrous moonshine.

Algorithmics

For calculating the day of the week, he invented the Doomsday algorithm. One of his early books was on finite state machines.

Theoretical physics

In 2004, Conway and Simon Kochen, another Princeton mathematician, proved the Free will theorem, a startling version of the No Hidden Variables principle of Quantum Mechanics. It states that given certain conditions (that almost every physicist agrees are true), if an experimenter can freely decide what quantities to measure in a particular experiment, then elementary particles must be free to choose their spins in order to make the measurements consistent with physical law. Or, in Conway's provocative wording, if experimenters have free will, then so do elementary particles.

Books

He has (co-)written several books including the Atlas of Finite Groups, Sphere Packings, Lattices and Groups, The Sensual (Quadratic) Form, On Numbers and Games, Winning Ways for your Mathematical Plays, The Book of Numbers, and On Quaternions and Octonions.

External links and references

by O'Connor and Robertson
Charles Seife, "Impressions of Conway", The Sciences
Mark Alpert, "Not Just Fun and Games", Scientific American April 1999. (official online version; registration-free online version)
Jasvir Nagra, "Conway's Proof Of The Free Will Theorem" ^*
Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A.: "Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups." Oxford, England 1985.

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