Georges Reeb

Reeb was born in Saverne, Bas-Rhin, Alsace, to Theobald Reeb and Caroline Engel. He started studying mathematics at University of Strasbourg, but in 1939 the entire university was evacuated to Clermont-Ferrand due to the German occupation of France.{{cite web |date= |title=Georges Reeb (1920 - 1993) |url=http://mathshistory.st-andrews.ac.uk/Biographies/Reeb.html |access-date=2020-02-10 |work=MacTutor History of Mathematics archive |publisher=University of St Andrews |via=st-andrews.ac.uk}}

After the war, he completed his studies and in 1948 he defended his PhD thesis, entitled Propriétés topologiques des variétés feuilletées [Topological properties of foliated manifolds] and supervised by Charles Ehresmann.{{Cite web |title=Georges Reeb - The Mathematics Genealogy Project |url=https://genealogy.math.ndsu.nodak.edu/id.php?id=96185 |access-date=2022-04-02 |website=genealogy.math.ndsu.nodak.edu}}

In 1952 Reeb was appointed professor at Université Joseph Fourier in Grenoble and in 1954 he visited the Institute for Advanced Study. From 1963 he worked at Université Louis Pasteur in Strasbourg.{{Cite journal |last=Diener |first=Francine |date=October 1993 |title=George Reeb (1920-1993) |url=https://smf.emath.fr/publications/la-gazette-des-mathematiciens-58-octobre-1993 |journal={{Interlanguage link|Gazette des mathématiciens|lt=Gazette des mathématiciens|fr}} |language=fr |volume=58 |pages=3}}

There, in 1965 he created with Jean Leray and Pierre Lelong the series of meeting Rencontres entre Mathématiciens et Physiciens Théoriciens. in 1966 Reeb and Jean Frenkel founded the Institute de Recherche mathématique Avancée, the first university laboratory associated to the Centre National de la Recherche Scientifique, which he directed between 1967 and 1972.{{cite web |last= |first= |date= |title=Some historical facts |url=http://www-irma.u-strasbg.fr/article217.html?lang=en |access-date=2020-02-10 |website=u-strasbg.fr |publisher=Institute for Advanced Mathematical Research, University of Strasbourg |archive-date=2013-10-02 |archive-url=https://web.archive.org/web/20131002234005/http://www-irma.u-strasbg.fr/article217.html?lang=en |url-status=dead }}

In 1967 he was President of the Société Mathématique de France{{Cite web |title=Liste anciens présidents {{!}} Société Mathématique de France |url=https://smf.emath.fr/la-smf/les-presidents-de-la-smf-depuis-1873 |access-date=2022-04-02 |website=smf.emath.fr}} and in 1971 he was awarded the {{Interlanguage link|Prize Petit d'Ormoy|lt=Prize Petit d'Ormoy|fr|Prix Petit d'Ormoy, Carrière, Thébault}}.

In 1991 Reeb received an honorary doctorate from Albert-Ludwigs-Universität Freiburg and from Université de Neuchâtel. He died in 1993 in Strasbourg when he was 72 years old.

File:Serre thom reeb 1949.jpg, Jean Arbault, Jean-Pierre Serre, {{Interlanguage link|Josiane Serre|lt=Josiane Serre|fr}}, {{Interlanguage link|Jean Braconnier (mathematician)|lt=Jean Braconnier|fr}} and Georges Reeb at the Mathematical Research Institute of Oberwolfach in 1949]]

Reeb was the founder of the topological theory of foliations, a geometric structure on smooth manifolds which partition them in smaller pieces. In particular, he described what is now called the Reeb foliation, a foliation of the 3-sphere, whose leaves are all diffeomorphic to $\backslash mathbb\{R\}^2$, except one, which is a 2-torus.{{cite web |last=Audin |first=Michèle |author-link=Michèle Audin |year=1953 |title=Differential Geometry, Strasbourg |url=https://www.ams.org/notices/200803/tx080300366p.pdf |access-date=2020-02-10 |work=Notices of the AMS |publisher=American Mathematical Society (published online 2008) |via=AMS.org}}

One of its first significant result, Reeb stability theorem, describes the local structure foliations around a compact leaf with finite holonomy group.

His works on foliations had also applications in Morse theory. In particular, the Reeb sphere theorem says that a compact manifold with a function with exactly two critical points is homeomorphic to the sphere. In turn, in 1956 this was used to prove that the Milnor spheres, although not diffeomorphic, are homeomorphic to the sphere $S^7$.{{Cite journal |last=Milnor |first=John |author-link=John Milnor |date=1956 |title=On Manifolds Homeomorphic to the 7-Sphere |url=https://www.jstor.org/stable/1969983 |journal=Annals of Mathematics |volume=64 |issue=2 |pages=399–405 |doi=10.2307/1969983 |jstor=1969983 |issn=0003-486X}}

Other important geometric concepts named after him include the Reeb graph{{Cite journal |last1=Shinagawa |first1=Y. |last2=Kunii |first2=T.L. |last3=Kergosien |first3=Y.L. |date=1991 |title=Surface coding based on Morse theory |url=https://ieeexplore.ieee.org/document/90568 |journal=IEEE Computer Graphics and Applications |volume=11 |issue=5 |pages=66–78 |doi=10.1109/38.90568 |s2cid=29897524 |issn=0272-1716}} and the Reeb vector field associated to a contact form.

Towards the end of his career, Reeb become a supporter of the theory of non-standard analysis by Abraham Robinson, coining the slogan "The naïve integers don't fill up $\backslash mathbb\{N\}$"{{Google books|id=a5spdc-gKqUC|page=4|title=Nonstandard Analysis in Practice}}. Edited by Francine Diener, Marc Diener.{{Cite journal |last=Nelson |first=Edward |date=1995 |title=Ramified recursion and intuitionism |url=https://web.math.princeton.edu/~nelson/papers/ramrec.pdf |journal=Presented to Colloque Trajectorien: à la mémoire de Georges Reeb et Jean-Louis Callot |location=Strasbourg/Obernai}} and working on its applications to dynamical systems.{{Cite book |last1=Diener |first1=Francine |url=https://www.worldcat.org/oclc/300057457 |title=Analyse non standard |last2=Reeb |first2=Georges |date=1989 |publisher=Herman |isbn=2-7056-6109-3 |location=Paris |language=fr |trans-title=Non standard analysis |oclc=300057457}}

• with Wu Wen-Tsün: Sur les espaces fibrés et les variétés feuilletées, 1952{{cite journal| last= Chern| first= Shiing-Shen| authorlink=Shiing-Shen Chern |title= Review: Sur les espaces fibrés et les variétés feuilletées by W. T. Wu and G. Reeb| journal= Bulletin of the American Mathematical Society |year= 1953| volume= 59 |pages= 258–263| doi= 10.1090/S0002-9904-1953-09700-2| doi-access= free}}
• with A. Fuchs: Statistiques commentées, 1967
• with J. Klein: Formules commentées de mathématiques: Programme P.C., 1971
• Feuilletages: résultats anciens et nouveaux (Painlevé, Hector et Martinet), 1974

• {{cite journal| title= Sur les points singuliers d'une forme de Pfaff complètement intégrable ou d'une fonction numérique| journal= C. R. Acad. Sci. Paris| volume= 222| year= 1946| pages= 847–849}}
• {{cite journal| title= Variétés feuilletées, feuilles voisines| journal= C. R. Acad. Sci. |place= Paris| volume= 224| year= 1947| pages= 1613–1614}}
• {{cite journal| title= Sur certaines propriétés topologiques des variétés feuilletées| journal= Actualités Sci. Ind., Publ. Inst. Math. Univ. Strasbourg| volume= 11 | pages= 5–89, 155–156| number= 1183 | publisher= Hermann & Cie.|place= Paris| year= 1952}}
• with André Haefliger: {{cite journal| title= Variétés (non séparées) à une dimension et structures feuilletées du plan| journal= Enseignement Math.| volume= 2| number= 3 |year= 1957| pages= 107–125}}

• Séminaire Nicolas Bourbaki
• Séminaire Nicolas Bourbaki (1950–1959)

{{reflist}}{{commons category|Georges Henri Reeb (mathematician)}}{{Authority control}}

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Category:1920 births

Category:1993 deaths

Category:People from Saverne

Category:Academic staff of the University of Strasbourg

Category:20th-century French mathematicians

Category:Topologists

Category:Geometers