John horton conway

John Horton Conway (Liverpool, December 26, 1937 - Princeton, New Jersey, April 11, 2020) was a prolific British mathematician, a specialist in group theory (theory of finite groups), knot theory, number theory, game theory, and code theory.

Born and raised in Liverpool, Conway spent the first half of his career at Cambridge University before moving to the United States, where he held the John von Neumann Professorship at Princeton University for the remainder of his career. On April 11, 2020, at age 82, he died of complications from COVID-19.

Biography

Conway was born on 26 December 1937 in Liverpool, the son of Cyril Horton Conway and Agnes Boyce. He became interested in mathematics at a very early age. When he was 11, his ambition was to become a mathematician. After leaving the sixth form, he studied mathematics at Gonville and Caius College, Cambridge. A "terribly introverted teenager"; At school, he took his admission to Cambridge as an opportunity to transform himself into an extrovert, a change that later earned him the nickname "the world's most charismatic mathematician."

Conway earned a bachelor's degree in 1959, and under the supervision of Harold Davenport, began to conduct research in number theory. Having solved Davenport's open problem about writing numbers as sums of fifth powers, Conway became interested in infinite ordinals. His interest in games seems to have begun during his years of study at the Cambridge Mathematical Tripos , where he became an avid backgammon player, spending hours playing in the common room. He obtained his PhD in 1964 and was appointed a University Fellow and Professor of Mathematics at Sidney Sussex College, Cambridge.After leaving Cambridge in 1986, he took up the position of John von Neumann Chair in Mathematics at Princeton University.

Conway's Game of Life

A Gosper Canyon creating "gliders" in Conway's Life Game

Conway was best known for inventing the Game of Life, one of the earliest examples of a cellular automaton. His initial experiments in this field were done with pencil and paper, long before personal computers existed.

Since Martin Gardner introduced the game to Scientific American in 1970, it has spawned hundreds of computer programs, websites, and articles. It is a staple of recreational mathematics. There is an extensive wiki dedicated to curating and cataloging the various aspects of the game. From the earliest days, it has been a favorite in computer labs, both for its theoretical interest and as a practical exercise in programming and data visualization. Conway used to hate the Game of Life, in large part because it had come to eclipse some of the other deeper and more important things he had done, but the game helped launch a new branch of mathematics, the field of cellular automata.

The game of life is known to be Turing complete.

Conway and Martin Gardner

Conway's career was intertwined with that of mathematics popularizer and Scientific American columnist Martin Gardner. When Gardner included Conway's Game of Life in his Mathematical Games column in October 1970, it became the most widely read of all his columns and made Conway an instant celebrity. Gardner and Conway had first corresponded in the late 1950s, and over the years Gardner had frequently written about recreational aspects of Conway's work. For example, he discussed the game of Conway Shoots (July 1967), The Hackenbush (January 1972) and His Angel Trouble (February 1974). In the September 1976 column, he reviewed Conway's book On Numbers and Games and even managed to explain Conway's surreal numbers.

Conway was a leading member of Martin Gardner's Mathematical Grapevine. He regularly visited Gardner and wrote him long letters summarizing his recreational research. On a 1976 visit, Gardner kept him for a week, pressing him for information on the newly announced Penrose tilings. Conway had discovered many (if not most) of the main properties of tilings. Gardner used these results when introducing the world to Penrose tilings in his January 1977 column. The cover of that issue of Scientific American features the Penrose mosaics and is based on a sketch by Conway.

Conferences called Gathering 4 Gardner are held every two years to celebrate the legacy of Martin Gardner, and Conway himself was often a featured speaker at these events, discussing various aspects of the recreational mathematics.

Main research areas

Combinatorial Game Theory

Conway was widely known for his contributions to combinatorial game theory (CGT), a theory of partisan games. He developed this with Elwyn Berlekamp and Richard Guy, and with them he also co-authored the book Winning Ways for your Mathematical Plays . He also wrote the book On Numbers and Games (ONAG) which lays out the mathematical foundations of combinatorial game theory.

He was also one of the inventors of the game of shoots, as well as Phutball. He developed detailed analyzes of many other games and puzzles, such as Soma Cube, Peg Solitaire, and Conway's Soldiers. He came up with the angel problem, which was solved in 2006.

He invented a new number system, 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 extremely large numbers, arrow notation Conway Chains. Much of this is discussed in part 0 of On Numbers and Games.

Geometry

In the mid-1960s with Michael Guy, Conway established that there are sixty-four convex uniform polychores that exclude two infinite sets of prismatic shapes. They discovered the great antiprism in the process, the only non-Wythoffian uniform construction. Conway has also suggested a system of notation dedicated to describing polyhedra called the Conway polyhedra notation.

In tiling theory, he devised the Conway criterion, which is a quick way to identify many prototypes that tile the plane.

He investigated lattices in higher dimensions and was the first to determine the symmetry group of the Leech lattice.

Geometric topology

In knot theory, Conway formulated a new variation of the Alexander polynomial and produced a new invariant now called the Conway polynomial. After lying dormant for over a decade, this The concept became central to work in the 1980s on the novel knot polynomials. Conway further developed knot theory and invented a notational system for tabulating knots, today known as Conway's notation, while correcting a number of errors in the knot tables of the 19th century and extending them to include all but four of non-alternating primes with 11 crosses. In knot theory, the Conway knot is named after him.

Group Theory

Examples of dodecágonos according to their symmetry
r24
d12	g12	p12		i8
d6	g6	p6		d4	g4	p4
g3				d2	g2	p2
a1
Conway classified these symmetries using a letter and the order of symmetry below. She assigned the letter. r the symmetry group of the regular figure; and in the case of the subgroups used the letter d (diagonal) for figures with axes of symmetry only through their vertices; p for figures with axles of symmetry only through perpendicular axles on their sides; i for figures with axles of symmetry both through vertices and through centers of sides; and g for those figures only with rotational symmetry. With a1 are labeled those figures with absence of symmetry. The lower types of symmetries allow one or more degrees of freedom to define different irregular figures. Only the subgroup g12 He has no degrees of freedom, but he can look like a directed graph.

He was the main author of the Atlas of Finite Groups which provides properties of many finite simple groups. Working with his colleagues Robert Curtis and Simon P. Norton he built the first concrete representations of some of the sporadic groups. More specifically, he discovered three sporadic groups based on the symmetry of the Leech lattice, which have been designated Conway groups.This work made him a key player in the successful classification of finitely simple groups.

Based on a 1978 observation by mathematician John McKay, Conway and Norton formulated the complex conjecture known as Monstrous moonshine. This course, named after Conway, relates the monster group to elliptic modular functions, thus uniting two previously distinct areas of mathematics: finite groups and the theory of complex functions. Now it has been revealed that the Monstrous moonshine also has deep connections to string theory.

Conway introduced the Mathieu groupoid, an extension of the Mathieu M₁₂ group to 13 points.

Number Theory

As a graduate student, he proved a case for an Edward Waring conjecture, that each integer could be written as the sum of 37 numbers each raised to the fifth power, although Chen Jingrun solved the problem independently before the Conway's work could be published.

Algebra

Conway has written textbooks and done original work in algebra, concentrating particularly on quaternions and octonions. Together with Neil Sloane, he invented the icosians.

Analysis

Invented a base 13 function as a counterexample to the Converse of the Intermediate Value Theorem: The function takes all real values in every interval of the real line, so it has a Darboux property but is not continuous.

Algorithms

To calculate the day of the week, he invented the Doomsday algorithm. The algorithm is simple enough that anyone with basic arithmetic ability can do the math in their heads. Conway could usually give the correct answer in less than two seconds. To improve his speed, he practiced his calendar calculations on his computer, which was programmed to ask him questions with random dates each time he logged on. One of his first books was on finite state machines.

Theoretical Physics

In 2004, Conway and Simon B. Kochen, another Princeton mathematician, proved the free will theorem, a stunning refutation of the hidden variables theory of quantum mechanics. He claims that, given certain conditions, if an experimenter can freely decide which quantities to measure in a particular experiment, then elementary particles must be free to choose their spins for measurements to be consistent with physical law. In Conway's provocative wording: "if experimenters have free will, so do elementary particles".

Awards and distinctions

Conway received the Berwick Prize (1971), was elected a Fellow of the Royal Society (1981), became a Fellow of the American Academy of Arts and Sciences in 1992, was the first recipient of the Pólya Prize (LMS), he won the Nemmers Prize in Mathematics (1998) and received the Leroy Steele Prize for Mathematical Exposition (2000) from the American Mathematical Society. In 2001 he was awarded an honorary degree from the University of Liverpool, and in 2014 one from the Alexandru Ioan Cuza University of Iași.

His FRS nomination, in 1981, reads:

A versatile mathematician that combines a deep, combined vision with algebraic virtuosity, particularly in the construction and manipulation of "out of rhythm" algebraic structures that illuminate a wide variety of problems of completely unexpected forms. He has made outstanding contributions to the theory of finite groups, to the theory of knots, to mathematical logic (both the theory of sets and the theory of automatons) and the theory of games (as well as to their practice).

In 2017, Conway was awarded an Honorary Membership of the British Mathematical Association.

Death

On April 8, 2020, Conway developed symptoms of COVID-19. On April 11, he died in New Brunswick, New Jersey at the age of 82.

Books and publications

• 1971 - Regular algebra and finite machines. Chapman and Hall, London, 1971, Series: Chapman and Hall mathematics series.
• 1976 - On numbers and games. Academic Press, New York, 1976, Series: L.M.S. monographs, 6.
• 1979 - On the Distribution of Values of Angles Determined by Coplanar Points (with Paul Erdős, Michael Guy, and H. T. Croft). Journal of the London Mathematical Society, vol. II, series 19, pp. 137–143.
• 1979 - Monstrous Moonshine (with Simon P. Norton). Bulletin of the London Mathematical Society, vol. 11, issue 2, pp. 308-339.
• 1982 - Winning Ways for your Mathematical Plays (with Richard K. Guy and Elwyn Berlekamp). Academic Press.
• 1985 - Atlas of finite groups (with Robert Turner Curtis, Simon Phillips Norton, Richard A. Parker, and Robert Arnott Wilson). Clarendon Press, New York, Oxford University Press, 1985.
• 1988 - Sphere Packings, Lattices, and Groups(with Neil Sloane). Springer-Verlag, New York, Series: Grundlehren der mathematischen Wissenschaften, 290.
• 1995 - Minimal-Energy Clusters of Hard Spheres (with Neil Sloane, R. H. Hardin, and Tom Duff). Discrete & Computational Geometry, vol. 14, no. 3, pp. 237–259.
• 1996 - The Book of Numbers (with Richard K. Guy). Copernicus, New York, 1996.
• 1997 - The Sensual (quadratic) Form (with Francis Yein Chei Fung). Mathematical Association of America, Washington, DC, 1997, Series: Carus mathematical monographs, no. 26.
• 2002 - On Quaternions and Octonions (with Derek A. Smith). A. K. Peters, Natick, MA, 2002.
• 2008 - The Symmetries of Things (with Heidi Burgiel and Chaim Goodman-Strauss). A. K. Peters, Wellesley, MA, 2008.