**Ralph Fox** was an American mathematician who made important contributions to differential topology and knot theory.

- Swarthmore is less than 20 km from Philadelphia, so geographically this was not difficult, and Fox continued with these two studies for two years.
- After taking his first degree from Swarthmore, Fox went to Johns Hopkins University where he studied for a Master's Degree in mathematics before going to Princeton where he undertook research advised by the topologist Solomon Lefschetz.
- Fox spent the year after the award of his doctorate at the Institute for Advanced Studies at Princeton.
- The year after Fox was appointed as a lecturer at Princeton, Emil Artin left Indiana University at Bloomington and was appointed to Princeton.
- After arriving in Princeton he began to think again about braids and he began working with Fox on certain topological ideas.
- Fox became the leader of a very active group of young mathematicians at Princeton studying knots, links and three dimensional topology.
- In 1950 the International Congress of Mathematicians took place in Cambridge, Massachusetts, and Fox was an invited speaker giving the lecture Recent development of knot theory at Princeton on the work of his Princeton group.
- These ideas had not been published at the time Fox gave this talk but over the next few years he published them in a number of important articles.
- In the same 1950 lecture, Fox reported on work by his young student John Milnor on the total curvature of a knot.
- In the summer of 1951, Fox went to the Mathematics Institute of the Universidad Nacional Autónoma de México in Mexico City where he gave a series of lectures, supported by a Guggenheim Fellowship.
- A Fulbright grant allowed Fox to lecture at the Universities of Delft and Stockholm in 1952.
- In the spring of 1956 Fox was invited to give a series of lectures at Haverford College (an institution founded on Quaker values on a campus just outside Philadelphia) under the Philips Lecture Program.The Philips Grant consists of funds left by Haverford alumnus William Pyle Philips (who graduated in 1902) for (i) the purchase of rare books which the college would not otherwise buy and (ii) to invite distinguished scientists and statesmen to Haverford.
- Fox's lectures were written up as a book, with Richard Crowell, a doctoral student of Fox's who was awarded his doctorate in 1955, as co-author.
- Fox's definition suggested a natural generalization to the concept of a braid group on an arbitrary manifold, as the fundamental group of the space of nnn unordered, distinct points of that manifold.
- The present investigation begins with Fox's definition, and studies the algebraic and geometric properties of these braid groups on arbitrary manifolds.
- As a result of these visits, several Japanese knot theorists visited Princeton to collaborate with Fox's group there.
- Fox was very friendly and encouraging toward all young topologists and many of them received invitations to visit Princeton.
- Lee Neuwirth, from whom we quoted above, was one of Fox's doctoral students, being awarded a Ph.D. in 1959.
- Other doctoral students of Fox with biographies in this archive include John Stallings (Ph.D. 1959) and Barry Mazur (Ph.D. 1959).
- Fox's health deteriorated and in 1973 he entered the University of Pennsylvania Graduate Hospital to undergo open-heart surgery.
- Following his death Lee Neuwirth edited Introduction to Knots, Groups, and 3-manifolds, papers dedicated to the memory of R H Fox which was published by Princeton University Press in 1975.
- In this last regard Ralph Fox's life was particularly striking: the tradition of topology at Princeton owes much to his lively and highly imaginative presence.
- Ralph Fox had well defined tastes in mathematics.
- Let us end this biography by quoting the tongue-in-cheek advice that Fox gave Joan Birman when she complained that she was having to serve on too many committees.

Born 24 March 1913, Morrisville, Pennsylvania, USA. Died 23 December 1973, Philadelphia, USA.

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Origin Usa, Topology

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive