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American mathematician
This article is about the American mathematician. For the English communist writer, see
Ralph Winston Fox .
Ralph Hartzler Fox (March 24, 1913 – December 23, 1973) 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 of
knot theory and of bringing it into the mainstream.
Biography
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
PhD degree from
Princeton University in 1939. His doctoral dissertation, On the Lusternick–Schnirelmann Category , was directed by
Solomon Lefschetz .
[1] (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 Stallings ,
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 .
[2] 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 phrases 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 game of
Go at both Princeton and the
Institute for Advanced Study .
Selected publications
Introduction to Knot Theory ,
Richard H. Crowell and Ralph H. Fox, Reprint of the 1963 original,
Graduate Texts in Mathematics , No. 57, Springer-Verlag, New York-Heidelberg, 1977.
ISBN
0-387-90272-4
[3]
"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.
MR
0140099
Fox, Ralph H. (1970).
"Metacyclic invariants of knots and links" .
Canadian Journal of Mathematics . 22 (2): 193–201.
doi :
10.4153/CJM-1970-025-9 .
MR
0261584 .
S2CID
121427629 .
Fox, Ralph H. (1966).
"Rolling" .
Bulletin of the American Mathematical Society . 72, Part 1: 162–164.
doi :
10.1090/s0002-9904-1966-11467-2 .
MR
0184221 .
Fox, Ralph H.; Smythe, Neville F. (1964).
"An ideal class invariant of knots" .
Bulletin of the American Mathematical Society . 15 (5): 707–709.
doi :
10.1090/s0002-9939-1964-0165516-2 .
MR
0165516 .
Chen, Kuo Tsai; Fox, Ralph H.;
Lyndon, Roger C. (1958). "Free differential calculus. IV. The quotient groups of the lower central series".
Annals of Mathematics . 68 (1): 81–95.
doi :
10.2307/1970044 .
JSTOR
1970044 .
MR
0102539 .
Fox, Ralph H. (1945).
"On topologies for function spaces" .
Bulletin of the American Mathematical Society . 51 (6): 429–432.
doi :
10.1090/s0002-9904-1945-08370-0 .
MR
0012224 .
Blankinship, William A.; Fox, Ralph H. (1950).
"Remarks on certain pathological open subsets of 3-space and their fundamental groups" .
Proceedings of the American Mathematical Society . 1 (5): 618–624.
doi :
10.1090/s0002-9939-1950-0042120-8 .
MR
0042120 .
Fox, Ralph H. (February 1945).
"Torus Homotopy Groups" .
Proceedings of the National Academy of Sciences of the United States of America . 31 (2): 71–74.
Bibcode :
1945PNAS...31...71F .
doi :
10.1073/pnas.31.2.71 .
PMC
1078755 .
PMID
16588687 .
Fox, Ralph H. (1943).
"On fibre spaces. I" .
Bulletin of the American Mathematical Society . 49 (8): 555–557.
doi :
10.1090/s0002-9904-1943-07969-4 .
MR
0008702 .
Fox, Ralph H. (1943).
"On fibre spaces. II" .
Bulletin of the American Mathematical Society . 49 (10): 733–735.
doi :
10.1090/s0002-9904-1943-08015-9 .
MR
0009109 .
Fox, Ralph H. (1942).
"A characterization of absolute neighborhood retracts" . Bulletin of the American Mathematical Society . 48 (4): 271–275.
doi :
10.1090/s0002-9904-1942-07652-x .
MR
0006508 .
Fox, Ralph H. (January 15, 1940).
"On Homotopy and Extension of Mappings" . Proceedings of the National Academy of Sciences of the United States of America . 26 (1): 26–28.
Bibcode :
1940PNAS...26...26F .
doi :
10.1073/pnas.26.1.26 .
PMC
1078000 .
PMID
16577957 .
Fox, Ralph; Neuwirth, Lee (1962).
"The braid groups" . Mathematica Scandinavica . 10 : 119–126.
doi :
10.7146/math.scand.a-10518 .
MR
0150755 .
References
External links
International National Academics Other