Editing Heinz Hopf

{{short description|German mathematician (1894–1971)}}
{{for|the Swedish actor|Heinz Hopf (actor)}}
{{Use dmy dates|date=May 2023}}
{{Infobox scientist
| name              = Heinz Hopf
| image             = ETH-BIB-Hopf, Heinz (1894-1971)-Portr 07480.jpg
| caption           = Hopf in 1954
| birth_date        = {{birth date|1894|11|19|df=y}}
| birth_place       = Gräbschen, [[Province of Silesia|Silesia]], [[Kingdom of Prussia|Prussia]], [[German Empire]]
| death_date        = {{death date and age|1971|06|03|1894|11|19|df=y}}
| death_place       = [[Zollikon]], [[Canton of Zürich|Zürich]], [[Switzerland]]
| fields            = Mathematics
| workplaces        = [[ETH Zürich]]
| alma_mater        = [[Humboldt University of Berlin|University of Berlin]]
| thesis_year       = 1925
| thesis_title      = Über Zusammmenhänge zwischen Topologie und Metrik von Mannigfaltigkeiten<ref>{{mathGenealogy|17409}}</ref>
| doctoral_advisor  = [[Erhard Schmidt]]<br>[[Ludwig Bieberbach]]
| doctoral_students = [[Beno Eckmann]]<br>[[Hans Freudenthal]]<br>[[Alfred Frölicher]]<br>[[Werner Gysin]]<br>[[Friedrich Hirzebruch]]<br>[[Michel Kervaire]]<br>[[Willi Rinow]]<br>[[Hans Samelson]]<br>[[Ernst Specker]]<br>[[Eduard Stiefel]]<br>[[James J. Stoker]]<br>[[Alice Roth]]
| known_for         = [[Almost complex manifold]]<br>[[H-space]]<br>[[Hopf algebra]]<br>[[Hopf conjecture]]<br>[[Hopf fibration]]<br>[[Hopf invariant]]<br>[[Hopf link]]<br>[[Hopf manifold]]<br>[[Hopf surface]]<br>[[Hopf theorem]]<br>[[Group cohomology#History and relation to other fields|Hopf's integral formula]]<br>[[Hopfion]]<br>[[Hopf–Rinow theorem]]<br>[[Stallings theorem about ends of groups|Freudenthal-Hopf theorems]]<br>[[Killing–Hopf theorem]]<br>[[Poincaré–Hopf theorem]]<br>[[Grid cell topology]]<br>[[Spherical space form conjecture]]<br>[[Sphere theorem]]
| awards            = 
}}

'''Heinz Hopf''' (19 November 1894 – 3 June 1971) was a German [[mathematician]] who worked on the fields of [[dynamical systems]], [[topology]] and [[geometry]].<ref>{{cite book|editor=I.M. James|title=History of Topology|url=https://books.google.com/books?id=7iRijkz0rrUC&pg=PA991|date=24 August 1999|publisher=Elsevier|isbn=978-0-08-053407-7|page=991}}</ref>

==Early life and education==
Hopf was born in Gräbschen, [[German Empire]] (now {{ill|Grabiszyn|pl|Grabiszyn-Grabiszynek}}, part of [[Wrocław]], [[Poland]]), the son of Elizabeth (née Kirchner) and Wilhelm Hopf. His father was born Jewish and converted to Protestantism a year after Heinz was born; his mother was from a Protestant family.<ref>{{cite web| url= http://www-history.mcs.st-andrews.ac.uk/Printonly/Hopf.html |title= Heinz Hopf| publisher= University of St Andrews}}</ref><ref>{{cite web | url= http://www.robertnowlan.com/pdfs/Hopf,%20Heinz.pdf | archive-url= https://web.archive.org/web/20110106023144/http://www.robertnowlan.com/pdfs/Hopf,%20Heinz.pdf | url-status= dead | archive-date= 6 January 2011 | title= Hopf, Heinz | website= RobertNowlan.com }}</ref>

Hopf attended [[Karl Mittelhaus]] higher boys' school from 1901 to 1904, and then entered the König-Wilhelm-[[Gymnasium (school)|Gymnasium]] in Breslau. He showed mathematical talent from an early age. In 1913 he entered the [[Silesian Friedrich Wilhelm University]] where he attended lectures by [[Ernst Steinitz]], [[Adolf Kneser]], [[Max Dehn]], [[Erhard Schmidt]], and [[Rudolf Sturm]]. When [[World War I]] broke out in 1914, Hopf eagerly enlisted. He was wounded twice and received the iron cross (first class) in 1918.

After the war Hopf continued his mathematical education in [[Heidelberg]] (winter 1919/20 and summer 1920)<ref>{{cite web| url=http://histmath-heidelberg.de/homo-heid/hopf.htm |title=Heinz Hopf|publisher=Historia Mathematica Heidelbergensis}}</ref> and [[Berlin]] (starting in winter 1920/21). He studied under [[Ludwig Bieberbach]] and received his doctorate in 1925.

==Career==
In his dissertation, ''Connections between topology and metric of manifolds'' ([[German language|German]]: ''Über Zusammenhänge zwischen Topologie und Metrik von Mannigfaltigkeiten''), he proved that any simply connected complete [[Riemannian manifold|Riemannian]] [[3-manifold]] of constant sectional [[curvature]] is globally [[Isometry|isometric]] to [[Euclidean space|Euclidean]], [[Elliptic geometry|spherical]], or [[hyperbolic space]]. He also studied the indices of zeros of [[vector field]]s on [[hypersurface]]s, and connected their sum to curvature. Some six months later he gave a new proof that the sum of the indices of the zeros of a vector field on a [[manifold]] is independent of the choice of vector field and equal to the [[Euler characteristic]] of the manifold. This theorem is now called the [[Poincaré–Hopf theorem]].

Hopf spent the year after his doctorate at the [[University of Göttingen]], where [[David Hilbert]], [[Richard Courant]], [[Carl Runge]], and [[Emmy Noether]] were working. While there he met [[Pavel Alexandrov]] and began a lifelong friendship.

In 1926 Hopf moved back to [[Berlin]], where he gave a course in [[combinatorial topology]]. He spent the academic year 1927/28 at [[Princeton University]] on a Rockefeller fellowship with Alexandrov. [[Solomon Lefschetz]], [[Oswald Veblen]] and [[James Waddell Alexander II|J. W. Alexander]] were all at Princeton at the time. At this time Hopf discovered the [[Hopf invariant]] of maps <math>S^3 \to S^2</math> and proved that the [[Hopf fibration]] has invariant 1. In the summer of 1928 Hopf returned to Berlin and began working with [[Pavel Alexandrov]], at the suggestion of Courant, on a book on [[topology]]. Three volumes were planned, but only one was finished. It was published in 1935.

In 1929, he declined a job offer from [[Princeton University]]. In 1931 Hopf took [[Hermann Weyl]]'s position at [[ETH Zurich|ETH]], in [[Zürich]]. Hopf received another invitation to Princeton in 1940, but he declined it. Two years later, however, he was forced to file for Swiss citizenship after his property was confiscated by [[Nazi Germany|Nazi]]s, his father's conversion to Christianity having failed to convince German authorities that he was an "[[Aryan]]".

In 1946/47 and 1955/56 Hopf visited the United States, staying at Princeton and giving lectures at [[New York University]] and [[Stanford University]]. He served as president of the [[International Mathematical Union]] from 1955 to 1958.<ref>{{Cite web|url=http://www.mathunion.org/organization/ec/ec-1952-2014/|title=International Mathematical Union (IMU): IMU Executive Committees 1952–2014|website=www.mathunion.org|language=en|archive-url=https://web.archive.org/web/20150108154651/http://www.mathunion.org/organization/ec/ec-1952-2014|archive-date=8 January 2015|url-status=dead|access-date=20 March 2017}}</ref>

==Personal life==
In October 1928 Hopf married Anja von Mickwitz (1891–1967).

==Honors and awards==
He received honorary doctorates from [[Princeton University]], the [[University of Freiburg]], the [[University of Manchester]], the [[University of Paris]], the [[Free University of Brussels (1834–1969)|Free University of Brussels]], and the [[University of Lausanne]]. In 1949 he was elected a corresponding member of the [[Heidelberg Academy of Sciences]]. He was elected to the United States National Academy of Sciences in 1957, the [[American Academy of Arts and Sciences]] in 1961,<ref>{{Cite web |title=Heinz Hopf |url=https://www.amacad.org/person/heinz-hopf |access-date=14 November 2022 |website=American Academy of Arts & Sciences |language=en}}</ref> and the [[American Philosophical Society]] in 1963.<ref>{{Cite web |title=APS Member History |url=https://search.amphilsoc.org/memhist/search?creator=Heinz+Hopf&title=&subject=&subdiv=&mem=&year=&year-max=&dead=&keyword=&smode=advanced |access-date=14 November 2022 |website=search.amphilsoc.org}}</ref> He was an Invited Speaker at the [[International Congress of Mathematicians]] (ICM) in [[Zürich]] in 1932 and a Plenary Speaker at the ICM in [[Cambridge, Massachusetts]] in 1950.<ref>{{cite book|author=Hopf, H.|chapter=Die ''n''-dimensionalen Sphären und projektiven Räume in der Topologie|title=''In:'' Proceedings of the International Congress of Mathematicians, Cambridge, Massachusetts, U.S.A., August 30 – September 6, 1950|volume=1|pages=193–202|year=1950|chapter-url=http://www.mathunion.org/ICM/ICM1950.1/Main/icm1950.1.0193.0202.ocr.pdf|access-date=5 December 2017|archive-date=28 December 2013|archive-url=https://web.archive.org/web/20131228051341/http://www.mathunion.org/ICM/ICM1950.1/Main/icm1950.1.0193.0202.ocr.pdf|url-status=dead}}</ref>

In memory of Hopf, [[ETH Zürich]] awards the [[Heinz Hopf Prize]] for outstanding scientific work in the field of [[pure mathematics]].

==See also==
* [[Co-Hopfian group]]
* [[Cohomotopy group]]
* [[EHP spectral sequence]]
* [[Hopfian group]]
* [[Hopfian object]]
* [[Quantum group]]

==Selected publications==
* Alexandroff P., Hopf H. Topologie Bd.1 — B: 1935
* {{Citation | last1=Hopf | first1=Heinz | author1-link=Heinz Hopf | title=Selecta Heinz Hopf | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Herausgegeben zu seinem 70. Geburtstag von der Eidgenössischen Technischen Hochschule Zürich | mr=0170777 | year=1964}}
* {{Citation | last1=Hopf | first1=Heinz | author1-link=Heinz Hopf | title=Collected papers/Gesammelte Abhandlungen | url=https://books.google.com/books?id=DI4iWNdLDwAC | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-57138-4 | mr=1851430 | year=2001}}

==References==
{{reflist}}

==Further reading==
* {{cite web|last=Bagni|first= Giorgio T. |url=http://www.icmihistory.unito.it/portrait/hopf.php|title=Heinz Hopf| publisher=History of ICMI web-site}}
* {{citation|last=Hilton|first= Peter J.|authorlink=Peter Hilton| title=Heinz Hopf|journal=[[Bulletin of the London Mathematical Society]]|year=1972|volume= 4 |issue=2|pages= 202–217| doi=10.1112/blms/4.2.202}}

==External links==
{{commons category|Heinz Hopf (mathematician)}}
* {{MacTutor Biography|id=Hopf}}
* [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/hopf_-_curvatura_integra.pdf "On the ''curvature integra'' of closed hypersurfaces," transl. by D. H. Delphenich]
* [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/hopf_-_vector_fields.pdf "Vector fields in ''n''-dimensional manifolds," transl. by D. H. Delphenich]

{{Authority control}}

{{DEFAULTSORT:Hopf, Heinz}}
[[Category:1894 births]]
[[Category:1971 deaths]]
[[Category:Differential geometers]]
[[Category:Academic staff of ETH Zurich]]
[[Category:German geometers]]
[[Category:20th-century German mathematicians]]
[[Category:Jewish emigrants from Nazi Germany to Switzerland]]
[[Category:20th-century Swiss mathematicians]]
[[Category:German topologists]]
[[Category:Academic staff of the University of Göttingen]]
[[Category:Foreign associates of the National Academy of Sciences]]
[[Category:Scientists from Wrocław]]
[[Category:Recipients of the Iron Cross (1914), 1st class]]
[[Category:Humboldt University of Berlin alumni]]
[[Category:Presidents of the International Mathematical Union]]
[[Category:Members of the American Philosophical Society]]