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, structure, space, models, and change. History On ...

. As a professor at

Princeton University Princeton University is a private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the ...

, 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.

Biography

Ralph Fox attended Swarthmore College for two years, while studying piano at the Leefson Conservatory of Music in

Philadelphia Philadelphia, often called Philly, is the List of municipalities in Pennsylvania#Municipalities, largest city in the Commonwealth (U.S. state), Commonwealth of Pennsylvania, the List of United States cities by population, sixth-largest city i ...

. He earned a master's degree from

Johns Hopkins University Johns Hopkins University (Johns Hopkins, Hopkins, or JHU) is a private research university in Baltimore, Maryland. Founded in 1876, Johns Hopkins is the oldest research university in the United States and in the western hemisphere. It consi ...

, and a PhD degree from

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

Solomon Lefschetz Solomon Lefschetz (russian: Соломо́н Ле́фшец; 3 September 1884 – 5 October 1972) was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear o ...

. (In later years he disclaimed all knowledge of the

Lusternik–Schnirelmann category In mathematics, the Lyusternik–Schnirelmann category (or, Lusternik–Schnirelmann category, LS-category) of a topological space X is the homotopy invariant defined to be the smallest integer number k such that there is an open covering \_ of X ...

, 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 Univ ...

John Stallings John Robert Stallings Jr. (July 22, 1935 – November 24, 2008) 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 ...

, Francisco González-Acuña, Guillermo Torres-Diaz and Barry Mazur, and supervised Ken Perko's undergraduate thesis. He was an Invited Speaker at the International Congress of Mathematicians held in 1950 in

Cambridge, Massachusetts Cambridge ( ) is a city in Middlesex County, Massachusetts, United States. As part of the Boston metropolitan area, the cities population of the 2020 U.S. census was 118,403, making it the fourth most populous city in the state, behind Boston ...

. 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 and ...

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 vector ...

as being particularly appropriate for homotopy theory. Aside from his strictly mathematical contributions, he was responsible for introducing several basic phrases to knot theory: the phrases ''

slice knot A slice knot is a 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 or smoothly slice knot, if it is the boundary of an embedded disk in ...

'', ''

ribbon knot In the mathematical area of knot theory, a ribbon knot is a knot that bounds a self-intersecting disk with only ''ribbon singularities''. Intuitively, this kind of singularity can be formed by cutting a slit in the disk and passing another part o ...

'', 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 example ...

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

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

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) (ISSN 0072-5285) 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 ...

, 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 Topologists Princeton University faculty Swarthmore College alumni People from Morrisville, Pennsylvania Mathematicians from Pennsylvania