Ralph Fox

Ralph Fox

Born	(1913-03-24)March 24, 1913 Morrisville, Pennsylvania, U.S.
Died	December 23, 1973(1973-12-23) (aged 60) Philadelphia, Pennsylvania, U.S.
Alma mater	Swarthmore College Johns Hopkins University Princeton University
Known for	Compact-open topology Fox derivative Fox n-coloring Fox–Artin arc
Scientific career
Fields	Mathematics
Thesis	On the Lusternick–Schnirelmann Category (1939)
Doctoral advisor	Solomon Lefschetz
Doctoral students	Alan J. Goldman Francisco Javier González-Acuña Harold W. Kuhn Samuel J. Lomonaco Jr. Barry Mazur John Milnor John R. Stallings

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. MR0140099
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

^ Ralph Fox at the Mathematics Genealogy Project
^ Fox, Ralph H. (1950). "Recent developments of knot theory at Princeton" (PDF). Proceedings of the International Congress of Mathematicians, Cambridge, Massachusetts, U.S.A., August 30–September 6, 1950. Vol. 2. pp. 453–458.
^ Neuwirth, Lee P. (1964). "Review: Introduction to knot theory by R. H. Crowell and R. H. Fox" (PDF). Bulletin of the American Mathematical Society. 70 (2): 235–238. doi:10.1090/s0002-9904-1964-11096-x.

External links

O'Connor, John J.; Robertson, Edmund F., "Ralph Fox", MacTutor History of Mathematics Archive, University of St Andrews
[1] Jozef H. Przytycki, Notes to the early history of the Knot Theory in Japan, 2001.

Authority control databases
International	ISNI VIAF FAST WorldCat
National	Germany United States Japan Czech Republic Netherlands Norway Israel
Academics	CiNii Mathematics Genealogy Project zbMATH MathSciNet
People	DDB
Other	IdRef SNAC

Read other articles:

Durian anggang

Durian anggang Durio acutifolius Status konservasiRentanIUCN34563 TaksonomiDivisiTracheophytaSubdivisiSpermatophytesKladAngiospermaeKladmesangiospermsKladeudicotsKladcore eudicotsKladSuperrosidaeKladrosidsKladmalvidsOrdoMalvalesFamiliMalvaceaeSubfamiliHelicteroideaeGenusDurioSpesiesDurio acutifolius lbs Durio acutifolius atau durian anggang adalah spesies pohon durian dalam keluarga Malvaceae . Ia endemik di Kalimantan . Julukan khusus, acutifolius</link> , yang berarti daun berduri, be...

Corregimiento de La Concepción

Corregimiento_de_La_Concepción

Corregimiento de La Concepción Corregimiento 1550-1786BanderaEscudo Otros nombres: Provincia de La Concepción Corregimientos coloniales de Chile en 1776Capital Ciudad de La Concepción del Nuevo ExtremoEntidad Corregimiento • País Imperio español • Virreinato Perú • Capitanía general ChileSubdivisiones La Concepción HualpénIdioma oficial EspañolReligión CatólicaHistoria • 1550 Establecido • 1786 DisueltoFronteras Río Biobio Sucedido por �...

Hidangan Kurdi

Artikel ini sebatang kara, artinya tidak ada artikel lain yang memiliki pranala balik ke halaman ini.Bantulah menambah pranala ke artikel ini dari artikel yang berhubungan atau coba peralatan pencari pranala.Tag ini diberikan pada Desember 2022. Artikel ini sebatang kara, artinya tidak ada artikel lain yang memiliki pranala balik ke halaman ini.Bantulah menambah pranala ke artikel ini dari artikel yang berhubungan atau coba peralatan pencari pranala.Tag ini diberikan pada Oktober 2022. Hidang...

羅生門 (電影)

本條目存在以下問題，請協助改善本條目或在討論頁針對議題發表看法。此條目需要补充更多来源。 (2018年3月17日)请协助補充多方面可靠来源以改善这篇条目，无法查证的内容可能會因為异议提出而被移除。致使用者：请搜索一下条目的标题（来源搜索：羅生門 (電影) — 网页、新闻、书籍、学术、图像），以检查网络上是否存在该主题的更多可靠来源（判定指引）。 �...

Rumah Sakit Katolik St. Vincentius a Paulo

Rumah_Sakit_Katolik_St._Vincentius_a_Paulo

Rumah Sakit KatolikSt. Vincentius a PauloYayasan ArnoldusGeografiLokasiJl. Diponegoro No.51, Kota Surabaya, Jawa TimurOrganisasiJenisBSejarahDibuka3 Mei 1925Pranala luarSitus webwww.rkzsurabaya.com Rumah Sakit Katolik St. Vincentius a Paulo (dikenal juga dengan nama RKZ Surabaya) adalah sebuah rumah sakit di Surabaya, Jawa Timur. Rumah Sakit ini merupakan salah satu bentuk keterlibatan dan pelayanan Gereja Katolik kepada semua orang, khususnya yang sakit dan miskin. Nama Patung Yesus di taman...

رموز هولندا الوطنية

رموز_هولندا_الوطنية

جزء من سلسلة حولثقافة هولندا التاريخ السكان اللغات المطبخ الاحتفالات الدين المسيحية الإسلام الأدب الموسيقى والفنون الموسيقى الفن الأساطير والفلكلور الأساطير الفلكلور الإعلام سينما التلفاز الرياضة المعالم قائمة مواقع التراث العالمي الرموز العلم الشعار عنت رموز هولندا ...

Elezioni generali nel Regno Unito del 2017

Elezioni_generali_nel_Regno_Unito_del_2017

Elezioni generali nel Regno Unito del 2017 Stato Regno Unito Data 8 giugno Assemblea Camera dei comuni Affluenza 68,8% (2,5%) Leader Theresa May Jeremy Corbyn Nicola Sturgeon Liste Conservatori Laburisti SNP Voti 13.667.21342,4% 12.874.98540,0% 977.5693,0% Seggi 317 / 650 262 / 650 35 / 650 Differenza % 5,5% 9,5% 1,7% Differenza seggi 14 30 21 Distribuzione del voto per collegio Primi ministri Theresa May (Governo May II, 2017-2019) Boris Johnson (Governo Johnson I, 2019) 2015 201...

Батиметрическая съёмка

Батиметрическая_съёмка

Рельеф морского дна возле Курильских островов Батиметрическая съёмка — процесс сбора данных о глубинах (батиметрии) в зоне исследования. Выполняется с помощью специальных технических средств, в особых случаях может выполняться вручную. После необходимой обработки �...

John Henry Barrows

John_Henry_Barrows

This article is written like a personal reflection, personal essay, or argumentative essay that states a Wikipedia editor's personal feelings or presents an original argument about a topic. Please help improve it by rewriting it in an encyclopedic style. (April 2021) (Learn how and when to remove this message) American clergyman John Henry Barrows5th President of Oberlin CollegeIn office1899 (1899) – June 3, 1902 (1902-06-03)Preceded byWilliam Gay Ballantine...

Renato Gojković

Renato_Gojković

Bosnian footballer (born 1995) Renato Gojković Gojković in 2022Personal informationDate of birth (1995-09-10) 10 September 1995 (age 28)Place of birth Tuzla, Bosnia and HerzegovinaHeight 1.95 m (6 ft 5 in)Position(s) Centre-backTeam informationCurrent team OrenburgNumber 15Youth career2002–2012 Sloboda TuzlaSenior career*Years Team Apps (Gls)2012–2013 Sloboda Tuzla 18 (3)2013–2015 Čelik Zenica 40 (0)2015–2018 Istra 1961 49 (0)2018–2019 Partizani 23 (2)2019–2...

國民陣線

国民阵线Barisan NasionalNational Frontباريسن ناسيونلபாரிசான் நேசனல்国民阵线标志简称国阵，BN主席阿末扎希总秘书赞比里署理主席莫哈末哈山总财政希山慕丁副主席魏家祥维纳斯瓦兰佐瑟古律创始人阿都拉萨成立1973年1月1日 (1973-01-01)[1]设立1974年7月1日 (1974-07-01)前身联盟总部马来西亚吉隆坡 50480 秋傑区敦依斯迈路太子世贸中心（英�...

تفشي الكوليرا في بابوا غينيا الجديدة 2009

تفشي_الكوليرا_في_بابوا_غينيا_الجديدة_2009

هذه المقالة يتيمة إذ تصل إليها مقالات أخرى قليلة جدًا. فضلًا، ساعد بإضافة وصلة إليها في مقالات متعلقة بها. (مايو 2021) تفشي الكوليرا في بابوا غينيا الجديدة 2009 كان انتشار الكوليرا على طول الساحل الشمالي لبابوا غينيا الجديدة. كان هذا أول تفشي للكوليرا في البلاد منذ 50 عامًا، وانت...

Філософія

Частина серії проФілософіяLeft to right: Plato, Kant, Nietzsche, Buddha, Confucius, AverroesПлатонКантНіцшеБуддаКонфуційАверроес Філософи Епістемологи Естетики Етики Логіки Метафізики Соціально-політичні філософи Традиції Аналітична Арістотелівська Африканська Близькосхідна іранська Буддій�...

Tarot series

1979–1980 fantasy novel/trilogy by Piers Anthony First omnibus edition (publ. Ace Books)Cover Artist: Kinuko Craft The Tarot series is a series of novels by Piers Anthony published in 1979 and 1980. It was originally written as a single novel, and has subsequently been published as such. Series The series consists of God of Tarot, Vision of Tarot, and Faith of Tarot.[1] Reception Greg Costikyan reviewed the series in Ares Magazine #2 and commented that Tarot is readable and pleasant...

Plessy v. Ferguson

Plessy_v._Ferguson

1896 U.S. Supreme Court case on racial segregation Plessy redirects here. Not to be confused with Plessey. 1896 United States Supreme Court casePlessy v. FergusonSupreme Court of the United StatesArgued April 13, 1896Decided May 18, 1896Full case nameHomer A. Plessy v. John H. FergusonCitations163 U.S. 537 (more)16 S. Ct. 1138; 41 L. Ed. 256; 1896 U.S. LEXIS 3390DecisionOpinionCase historyPriorEx parte Plessy, 11 So. 948 (La. 1892)SubsequentNoneHoldingThe separate but equal provision of priva...

Ride (lagu)

RideSingel oleh Twenty One Pilotsdari album BlurryfaceDirilis16 April 2016[1]Direkam2014–2015Genre Alternative hip hop reggae dub rock electropop Durasi3:34LabelFueled by RamenPenciptaTyler JosephProduserRicky ReedKronologi singel Twenty One Pilots Lane Boy (2015) Ride (2016) Heathens (2016) Video musikRide di YouTube Ride adalah lagu yang ditulis dan direkam oleh duo musik asal Amerika Serikat, Twenty One Pilots, dari album studio keempat mereka, Blurryface. Ride awalnya dirilis se...

Abdul Qadeer Khan

Abdul_Qadeer_Khan

Questa voce sull'argomento ingegneri è solo un abbozzo. Contribuisci a migliorarla secondo le convenzioni di Wikipedia. Segui i suggerimenti del progetto di riferimento. Abdul Qadeer Khan (2017) Abdul Qadeer Khan (in lingua urdu: عبدالقدیر خان; Bhopal, 1º aprile 1936 – Islamabad, 10 ottobre 2021[1]) è stato un ingegnere pakistano. Il secondo nome Qadeer è reso a volte come Quadeer, Qadir o Gadeer ed il suo nome è spesso abbreviato in A.Q.. Ingegnere metallurgi...

Pasquale Esposito

Pasquale_Esposito

Italian-born American tenor For the actor, see Pasquale Esposito (actor). Pasquale EspositoBackground informationOriginNaples, ItalyGenresOperatic popOccupation(s)TenorYears active1998-presentLabelsNotable RecordsWebsitewww.pasqualeesposito.comMusical artist Pasquale Esposito is an Italian-born American tenor. He has released ten albums and has toured internationally.[1] Early life and education Esposito is originally from Naples, Italy.[1] He has four sisters, and is the youn...

Incandescent light bulb

Incandescent_light_bulb

Electric light bulb with a resistively heated wire filament A 230-volt incandescent light bulb with a medium-sized E27 (Edison 27 mm) male screw base. The filament is visible as the mostly horizontal line between the vertical supply wires. A scanning electron microscope image of the tungsten filament of an incandescent light bulb Elaborate light in Denver, Colorado An incandescent light bulb, incandescent lamp or incandescent light globe is an electric light with a filament that is heate...

Netherlands at the 1924 Summer Olympics

Netherlands_at_the_1924_Summer_Olympics

Sporting event delegationNetherlands at the1924 Summer OlympicsFlag of the NetherlandsIOC codeNEDNOCDutch Olympic Committee*Dutch Sports FederationWebsitewww.nocnsf.nl (in Dutch)in ParisCompetitors177 (168 men and 9 women) in 17 sportsMedalsRanked 9th Gold 4 Silver 1 Bronze 5 Total 10 Summer Olympics appearances (overview)19001904190819121920192419281932193619481952195619601964196819721976198019841988199219962000200420082012201620202024Other related appearances1906 Intercalated Games Ath...