Ralph Hartzler Fox (March 24, 1913 – December 23, 1973) was an American

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

. As a professor at

Princeton University Princeton University is a private university, private Ivy League research university in Princeton, New Jersey, United States. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial ...

, he taught and advised many of the contributors to the ''Golden Age of

differential topology In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...

'', and he played an important role in the modernization of

knot theory In topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot be und ...

and of bringing it into the mainstream.

Biography

Ralph Fox attended

Swarthmore College Swarthmore College ( , ) is a Private college, private Liberal arts colleges in the United States, liberal arts college in Swarthmore, Pennsylvania, United States. Founded in 1864, with its first classes held in 1869, Swarthmore is one of the e ...

for two years, while studying piano at the Leefson Conservatory of Music in

Philadelphia Philadelphia ( ), colloquially referred to as Philly, is the List of municipalities in Pennsylvania, most populous city in the U.S. state of Pennsylvania and the List of United States cities by population, sixth-most populous city in the Unit ...

. He earned a master's degree from

Johns Hopkins University The Johns Hopkins University (often abbreviated as Johns Hopkins, Hopkins, or JHU) is a private university, private research university in Baltimore, Maryland, United States. Founded in 1876 based on the European research institution model, J ...

, and a

PhD A Doctor of Philosophy (PhD, DPhil; or ) is a terminal degree that usually denotes the highest level of academic achievement in a given discipline and is awarded following a course of graduate study and original research. The name of the deg ...

degree from

in 1939. His doctoral dissertation, ''On the Lusternick–Schnirelmann Category'', was directed by

Solomon Lefschetz Solomon Lefschetz (; 3 September 1884 – 5 October 1972) was a Russian-born American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear ordinary differential equatio ...

. (In later years he disclaimed all knowledge of the Lusternik–Schnirelmann category, and certainly never published on the subject again.) He directed 21 doctoral dissertations, including those of

John Milnor John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook Uni ...

, John Stallings, Francisco González-Acuña, Guillermo Torres-Diaz and

Barry Mazur Barry Charles Mazur (; born December 19, 1937) is an American mathematician and the Gerhard Gade University Professor at Harvard University. His contributions to mathematics include his contributions to Wiles's proof of Fermat's Last Theorem in ...

, and supervised Ken Perko's undergraduate thesis. He was an Invited Speaker at the

International Congress of Mathematicians The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the IMU Abacus Medal (known before ...

held in 1950 in

Cambridge, Massachusetts Cambridge ( ) is a city in Middlesex County, Massachusetts, United States. It is a suburb in the Greater Boston metropolitan area, located directly across the Charles River from Boston. The city's population as of the 2020 United States census, ...

. 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 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 ...

function spaces In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a ...

as being particularly appropriate for

homotopy theory In mathematics, homotopy theory is a systematic study of situations in which Map (mathematics), maps can come with homotopy, homotopies between them. It originated as a topic in algebraic topology, but nowadays is learned as an independent discipli ...

. Aside from his strictly mathematical contributions, he was responsible for introducing several basic phrases to

: the phrases ''

slice knot A slice knot is a knot (mathematics), mathematical knot in 3-dimensional space that bounds an embedded disk in 4-dimensional space. Definition A knot K \subset S^3 is said to be a topologically slice knot or a smoothly slice knot, if it is the ...

'', '' 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 In mathematics, a Seifert surface (named after German mathematician Herbert Seifert) is an orientable 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 exampl ...

''. He popularized the playing of the game of Go at both Princeton and the

Institute for Advanced Study The Institute for Advanced Study (IAS) is an independent center for theoretical research and intellectual inquiry located in Princeton, New Jersey. It has served as the academic home of internationally preeminent scholars, including Albert Ein ...

Selected publications

* ''Introduction to Knot Theory'', Richard H. Crowell and Ralph H. Fox, Reprint of the 1963 original,

Graduate Texts in Mathematics Graduate Texts in Mathematics (GTM) () 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 (with va ...

, No. 57, Springer-Verlag, New York-Heidelberg, 1977. * "A quick trip through knot theory", in: M. K. Fort (Ed.), ''Topology of 3-Manifolds and Related Topics'', Prentice-Hall, New Jersey, 1961, pp. 120–167. * * * * * * * * * * * *

References

External links

*

Jozef H. Przytycki, ''Notes to the early history of the Knot Theory in Japan'', 2001. {{DEFAULTSORT:Fox, Ralph 1913 births 1973 deaths 20th-century American mathematicians Institute for Advanced Study visiting scholars American topologists Princeton University faculty Swarthmore College alumni People from Morrisville, Pennsylvania Mathematicians from Pennsylvania