American mathematician (1913–1973)
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 Americanmathematician . As a professor atPrinceton University , he taught and advised many of the contributors to theGolden Age ofdifferential topology , and he played an important role in the modernization ofknot theory and of bringing it into the mainstream.
Ralph Fox attendedSwarthmore College for two years, while studying piano at the Leefson Conservatory of Music inPhiladelphia . He earned a master's degree fromJohns Hopkins University , and aPhD degree fromPrinceton University in 1939. His doctoral dissertation,On the Lusternick–Schnirelmann Category , was directed bySolomon Lefschetz .[ 1] Lusternik–Schnirelmann category , and certainly never published on the subject again.) He directed 21 doctoral dissertations, including those ofJohn Milnor ,John Stallings ,Francisco González-Acuña ,Guillermo Torres-Diaz andBarry Mazur , and supervisedKen Perko 's undergraduate thesis.
He was an Invited Speaker at theInternational Congress of Mathematicians held in 1950 inCambridge, Massachusetts .[ 2] Foxn -coloring of knots, theFox–Artin arc , and thefree differential calculus . He also identified thecompact-open topology onfunction spaces as being particularly appropriate forhomotopy theory .
Aside from his strictly mathematical contributions, he was responsible for introducing several basic phrases toknot theory : the phrasesslice knot ribbon knot Seifert circle Seifert surface
He popularized the playing of the game ofGo at both Princeton and theInstitute for Advanced Study .
Selected publications [ edit ] 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 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 .
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