Ralph Fox
Encyclopedia
Ralph Hartzler Fox was an American mathematician
. As a professor at Princeton University
, he taught and advised many of the contributors to the Golden Age of differential topology
, and he played an important role in the modernization and main-streaming of knot theory
.
Ralph Fox attended Swarthmore College
for two years, while studying piano at the Leefson Conservatory of Music in Philadelphia. He earned a master's degree from Johns Hopkins University
, and a Ph.D. degree from Princeton University
in 1939. His doctoral dissertation, On the Lusternick-Schnirelmann Category, was directed by Solomon Lefschetz
. (In later years he disclaimed all knowledge of Lyusternik-Schnirelmann category, and certainly never published on the subject again.) He directed 21 doctoral dissertations, including those of John Milnor
, John Stallings
, Francisco González-Acuña
, Guillermo Torres-Diaz and Barry Mazur
.
His mathematical contributions include Fox n-coloring
of knots, the Fox-Artin arc, and the free differential calculus. He also identified the compact-open topology
on function spaces as being particularly appropriate for homotopy theory.
Aside from his strictly mathematical contributions, he was responsible for introducing several basic bits of terminology to knot theory
: the phrases slice knot, ribbon knot
, and Seifert circle all appear in print for the first time under his name, and he also popularized (if he did not introduce) the phrase Seifert surface
.
He popularized the playing of the oriental game of Go at both Princeton and the Institute for Advanced Study
.
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
. As a professor at Princeton University
Princeton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....
, he taught and advised many of the contributors to the Golden Age of differential topology
Differential topology
In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...
, and he played an important role in the modernization and main-streaming of knot theory
Knot theory
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...
.
Ralph Fox attended Swarthmore College
Swarthmore College
Swarthmore College is a private, independent, liberal arts college in the United States with an enrollment of about 1,500 students. The college is located in the borough of Swarthmore, Pennsylvania, 11 miles southwest of Philadelphia....
for two years, while studying piano at the Leefson Conservatory of Music in Philadelphia. He earned a master's degree from Johns Hopkins University
Johns Hopkins University
The Johns Hopkins University, commonly referred to as Johns Hopkins, JHU, or simply Hopkins, is a private research university based in Baltimore, Maryland, United States...
, and a Ph.D. degree from Princeton University
Princeton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....
in 1939. His doctoral dissertation, On the Lusternick-Schnirelmann Category, was directed by Solomon Lefschetz
Solomon Lefschetz
Solomon Lefschetz was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear ordinary differential equations.-Life:...
. (In later years he disclaimed all knowledge of Lyusternik-Schnirelmann category, and certainly never published on the subject again.) He directed 21 doctoral dissertations, including those of John Milnor
John Milnor
John Willard Milnor is an American mathematician known for his work in differential topology, K-theory and dynamical systems. He won the Fields Medal in 1962, the Wolf Prize in 1989, and the Abel Prize in 2011. Milnor is a distinguished professor at Stony Brook University...
, John Stallings
John R. Stallings
John Robert Stallings Jr. was a mathematician known for his seminal contributions to geometric group theory and 3-manifold topology. Stallings was a Professor Emeritus in the Department of Mathematics at the University of California at Berkeley where he had been a faculty member since 1967...
, Francisco González-Acuña
Francisco Javier González-Acuña
Francisco Javier González-Acuña is a mathematician in the UNAM's institute of mathematics and CIMAT, specializing in low-dimensional topology.He did his graduate studies at Princeton University, obtaining his Ph.D. in 1970...
, Guillermo Torres-Diaz and Barry Mazur
Barry Mazur
-Life:Born in New York City, Mazur attended the Bronx High School of Science and MIT, although he did not graduate from the latter on account of failing a then-present ROTC requirement. Regardless, he was accepted for graduate school and received his Ph.D. from Princeton University in 1959,...
.
His mathematical contributions include Fox n-coloring
Fox n-coloring
In the mathematical field of knot theory, Fox n-coloring is a method of specifying a representation of a knot group onto the dihedral group of order n where n is an odd integer by coloring arcs in a link diagram...
of knots, the Fox-Artin arc, and the free differential calculus. He also identified the compact-open topology
Compact-open topology
In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. The compact-open topology is one of the commonly-used topologies on function spaces, and is applied in homotopy theory and functional analysis...
on function spaces as being particularly appropriate for homotopy theory.
Aside from his strictly mathematical contributions, he was responsible for introducing several basic bits of terminology to knot theory
Knot theory
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...
: the phrases slice knot, ribbon knot
Ribbon knot
In the mathematical area of knot theory, a ribbon knot is a knot that bounds a self-intersecting disc with only ribbon singularities. This type of singularity is a self-intersection along an arc; the preimage of this arc consists of two arcs in the disc, one properly embedded in the disc and the...
, and Seifert circle all appear in print for the first time under his name, and he also popularized (if he did not introduce) the phrase Seifert surface
Seifert surface
In mathematics, a Seifert surface is a surface whose boundary is a given knot or link.Such surfaces can be used to study the properties of the associated knot or link. For example, many knot invariants are most easily calculated using a Seifert surface...
.
He popularized the playing of the oriental game of Go at both Princeton and the Institute for Advanced Study
Institute for Advanced Study
The Institute for Advanced Study, located in Princeton, New Jersey, United States, is an independent postgraduate center for theoretical research and intellectual inquiry. It was founded in 1930 by Abraham Flexner...
.
Selected publications
- Introduction to Knot Theory, Richard H. Crowell and Ralph H. Fox, Reprint of the 1963 original, Graduate Texts in MathematicsGraduate Texts in MathematicsGraduate Texts in Mathematics is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard size...
, No. 57, Springer-Verlag, New York-Heidelberg, 1977. ISBN 0-387-90272-4 - A quick trip through knot theory, in: M.K. Fort (Ed.), "Topology of 3-Manifolds and Related Topics", Prentice-Hall, NJ, 1961, pp. 120–167.
- Metacyclic invariants of knots and links, Canadian Journal of Mathematics 22 (1970) 193–201.
- On topologies for function spaces, Bulletin of the American Mathematical Society 51 (1945) 429-432.
External links
- http://arxiv.org/abs/math/0108072 Jozef H. PrzytyckiJózef H. PrzytyckiJózef Henryk Przytycki , is a mathematician specializing in the fields of knot theory and topology.In 1987, he and Pawel Traczyk published "Invariants of links of Conway type" , which included a description of what is now called the HOMFLY polynomial...
, Notes to the early history of the Knot Theory in Japan, 2001.