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Heinz Hopf (19 November 1894 – 3 June 1971) was a German
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
who worked on the fields of
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...
and
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
.


Early life and education

Hopf was born in Gräbschen,
Germany Germany, officially the Federal Republic of Germany (FRG),, is a country in Central Europe. It is the most populous member state of the European Union. Germany lies between the Baltic and North Sea to the north and the Alps to the sou ...
(now , part of
Wrocław Wrocław (; , . german: Breslau, , also known by other names) is a city in southwestern Poland and the largest city in the historical region of Silesia. It lies on the banks of the Oder in the Silesian Lowlands of Central Europe, roughly ...
,
Poland Poland, officially the Republic of Poland, , is a country in Central Europe. Poland is divided into Voivodeships of Poland, sixteen voivodeships and is the fifth most populous member state of the European Union (EU), with over 38 mill ...
), 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. Hopf attended Karl Mittelhaus higher boys' school from 1901 to 1904, and then entered the König-Wilhelm- Gymnasium in Breslau. He showed mathematical talent from an early age. In 1913 he entered the
Silesian Friedrich Wilhelm University Silesian as an adjective can mean anything from or related to Silesia. As a noun, it refers to an article, item, or person of or from Silesia. Silesian may also refer to: People and languages * Silesians, inhabitants of Silesia, either a West S ...
where he attended lectures by Ernst Steinitz, Adolf Kneser,
Max Dehn Max Wilhelm Dehn (November 13, 1878 – June 27, 1952) was a German mathematician most famous for his work in geometry, topology and geometric group theory. Born to a Jewish family in Germany, Dehn's early life and career took place in Germany. H ...
,
Erhard Schmidt Erhard Schmidt (13 January 1876 – 6 December 1959) was a Baltic German mathematician whose work significantly influenced the direction of mathematics in the twentieth century. Schmidt was born in Tartu (german: link=no, Dorpat), in the Gover ...
, and
Rudolf Sturm Friedrich Otto Rudolf Sturm (6 January 1841 – 12 April 1919) was a German mathematician. His Ph.D. advisor was Heinrich Eduard Schroeter, and Otto Toeplitz Otto Toeplitz (1 August 1881 – 15 February 1940) was a German mathematician working ...
. When
World War I World War I (28 July 1914 11 November 1918), often abbreviated as WWI, was List of wars and anthropogenic disasters by death toll, one of the deadliest global conflicts in history. Belligerents included much of Europe, the Russian Empire, ...
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 Heidelberg (; Palatine German language, Palatine German: ''Heidlberg'') is a city in the States of Germany, German state of Baden-Württemberg, situated on the river Neckar in south-west Germany. As of the 2016 census, its population was 159,914 ...
(winter 1919/20 and summer 1920) and
Berlin Berlin is Capital of Germany, the capital and largest city of Germany, both by area and List of cities in Germany by population, by population. Its more than 3.85 million inhabitants make it the European Union's List of cities in the European U ...
(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 ''Über Zusammenhänge zwischen Topologie und Metrik von Mannigfaltigkeiten''), he proved that any simply connected complete Riemannian
3-manifold In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds ...
of constant sectional
curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the can ...
is globally isometric to Euclidean,
spherical A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ...
, or
hyperbolic space In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. ...
. He also studied the indices of zeros of vector fields on
hypersurface In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Eucl ...
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 In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
is independent of the choice of vector field and equal to the
Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological spac ...
of the manifold. This theorem is now called the
Poincaré–Hopf theorem In mathematics, the Poincaré–Hopf theorem (also known as the Poincaré–Hopf index formula, Poincaré–Hopf index theorem, or Hopf index theorem) is an important theorem that is used in differential topology. It is named after Henri Poincaré ...
. Hopf spent the year after his doctorate at the
University of Göttingen The University of Göttingen, officially the Georg August University of Göttingen, (german: Georg-August-Universität Göttingen, known informally as Georgia Augusta) is a public research university in the city of Göttingen, Germany. Founded i ...
, where David Hilbert,
Richard Courant Richard Courant (January 8, 1888 – January 27, 1972) was a German American mathematician. He is best known by the general public for the book '' What is Mathematics?'', co-written with Herbert Robbins. His research focused on the areas of real ...
,
Carl Runge Carl David Tolmé Runge (; 30 August 1856 – 3 January 1927) was a German mathematician, physicist, and spectroscopist. He was co-developer and co-eponym of the Runge–Kutta method (German pronunciation: ), in the field of what is today known ...
, and
Emmy Noether Amalie Emmy Noether Emmy is the ''Rufname'', the second of two official given names, intended for daily use. Cf. for example the résumé submitted by Noether to Erlangen University in 1907 (Erlangen University archive, ''Promotionsakt Emmy Noeth ...
were working. While there he met
Pavel Alexandrov Pavel Sergeyevich Alexandrov (russian: Па́вел Серге́евич Алекса́ндров), sometimes romanized ''Paul Alexandroff'' (7 May 1896 – 16 November 1982), was a Soviet mathematician. He wrote about three hundred papers, ma ...
and began a lifelong friendship. In 1926 Hopf moved back to
Berlin Berlin is Capital of Germany, the capital and largest city of Germany, both by area and List of cities in Germany by population, by population. Its more than 3.85 million inhabitants make it the European Union's List of cities in the European U ...
, where he gave a course in combinatorial topology. He spent the academic year 1927/28 at
Princeton University Princeton University is a private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the n ...
on a Rockefeller fellowship with Alexandrov.
Solomon Lefschetz Solomon Lefschetz (russian: Соломо́н Ле́фшец; 3 September 1884 – 5 October 1972) was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear ...
, Oswald Veblen and J. W. Alexander were all at Princeton at the time. At this time Hopf discovered the
Hopf invariant In mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between n-spheres. __TOC__ Motivation In 1931 Heinz Hopf used Clifford parallels to construct the '' Hopf map'' :\eta\colon S^3 \t ...
of maps S^3 \to S^2 and proved that the
Hopf fibration In the mathematical field of differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in terms of circles and an ordinary sphere. Discovered by Heinz Ho ...
has invariant 1. In the summer of 1928 Hopf returned to Berlin and began working with
Pavel Alexandrov Pavel Sergeyevich Alexandrov (russian: Па́вел Серге́евич Алекса́ндров), sometimes romanized ''Paul Alexandroff'' (7 May 1896 – 16 November 1982), was a Soviet mathematician. He wrote about three hundred papers, ma ...
, at the suggestion of Courant, on a book on
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...
. Three volumes were planned, but only one was finished. It was published in 1935. In 1929, he declined a job offer from
Princeton University Princeton University is a private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the n ...
. In 1931 Hopf took
Hermann Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is ass ...
's position at
ETH (colloquially) , former_name = eidgenössische polytechnische Schule , image = ETHZ.JPG , image_size = , established = , type = Public , budget = CHF 1.896 billion (2021) , rector = Günther Dissertori , president = Joël Mesot , ac ...
, in
Zürich , neighboring_municipalities = Adliswil, Dübendorf, Fällanden, Kilchberg, Maur, Oberengstringen, Opfikon, Regensdorf, Rümlang, Schlieren, Stallikon, Uitikon, Urdorf, Wallisellen, Zollikon , twintowns = Kunming, San Francisco Zürich () i ...
. 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 Nazism ( ; german: Nazismus), the common name in English for National Socialism (german: Nationalsozialismus, ), is the far-right politics, far-right Totalitarianism, totalitarian political ideology and practices associated with Adolf Hit ...
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 New York University (NYU) is a private research university in New York City. Chartered in 1831 by the New York State Legislature, NYU was founded by a group of New Yorkers led by then- Secretary of the Treasury Albert Gallatin. In 1832, ...
and Stanford University. He served as president of the
International Mathematical Union The International Mathematical Union (IMU) is an international non-governmental organization devoted to international cooperation in the field of mathematics across the world. It is a member of the International Science Council (ISC) and supports ...
from 1955 to 1958.


Personal life

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


Honors and awards

He received honorary doctorates from
Princeton University Princeton University is a private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the n ...
, the
University of Freiburg The University of Freiburg (colloquially german: Uni Freiburg), officially the Albert Ludwig University of Freiburg (german: Albert-Ludwigs-Universität Freiburg), is a public research university located in Freiburg im Breisgau, Baden-Württe ...
, the
University of Manchester The University of Manchester is a public university, public research university in Manchester, England. The main campus is south of Manchester city centre, Manchester City Centre on Wilmslow Road, Oxford Road. The university owns and operates majo ...
, the
University of Paris The University of Paris (french: link=no, Université de Paris), Metonymy, metonymically known as the Sorbonne (), was the leading university in Paris, France, active from 1150 to 1970, with the exception between 1793 and 1806 under the French Revo ...
, the Free University of Brussels, and the
University of Lausanne The University of Lausanne (UNIL; french: links=no, Université de Lausanne) in Lausanne, Switzerland was founded in 1537 as a school of Protestant theology, before being made a university in 1890. The university is the second oldest in Switze ...
. In 1949 he was elected a corresponding member of the
Heidelberg Academy of Sciences The Heidelberg Academy of Sciences and Humanities (German: ''Heidelberger Akademie der Wissenschaften''), established in 1909 in Heidelberg, Germany, is an assembly of scholars and scientists in the German state of Baden-Wuerttemberg. The Academ ...
. He was elected to the United States National Academy of Sciences in 1957, the
American Academy of Arts and Sciences The American Academy of Arts and Sciences (abbreviation: AAA&S) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, ...
in 1961, and the
American Philosophical Society The American Philosophical Society (APS), founded in 1743 in Philadelphia, is a scholarly organization that promotes knowledge in the sciences and humanities through research, professional meetings, publications, library resources, and communi ...
in 1963. He was an Invited Speaker at the
International Congress of Mathematicians The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the Nevanlinna Prize (to be rename ...
(ICM) in
Zürich , neighboring_municipalities = Adliswil, Dübendorf, Fällanden, Kilchberg, Maur, Oberengstringen, Opfikon, Regensdorf, Rümlang, Schlieren, Stallikon, Uitikon, Urdorf, Wallisellen, Zollikon , twintowns = Kunming, San Francisco Zürich () i ...
in 1932 and a Plenary Speaker at the ICM in
Cambridge, Massachusetts Cambridge ( ) is a city in Middlesex County, Massachusetts, United States. As part of the Greater Boston, Boston metropolitan area, the cities population of the 2020 United States Census, 2020 U.S. census was 118,403, making it the fourth most ...
in 1950. In memory of Hopf,
ETH Zürich (colloquially) , former_name = eidgenössische polytechnische Schule , image = ETHZ.JPG , image_size = , established = , type = Public , budget = CHF 1.896 billion (2021) , rector = Günther Dissertori , president = Joël Mesot , ac ...
awards the Heinz Hopf Prize for outstanding scientific work in the field of
pure mathematics Pure mathematics is the study of mathematical concepts independently of any application outside mathematics. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications ...
.


See also

*
Co-Hopfian group In the mathematical subject of group theory, a co-Hopfian group is a group that is not isomorphic to any of its proper subgroups. The notion is dual to that of a Hopfian group, named after Heinz Hopf. Formal definition A group ''G'' is called ...
* Cohomotopy group *
EHP spectral sequence In mathematics, the EHP spectral sequence is a spectral sequence used for inductively calculating the homotopy groups of spheres localized at some prime ''p''. It is described in more detail in and . It is related to the EHP long exact sequence o ...
* Hopfian group * Hopfian object *
Quantum group In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which are quasitriangular Hopf algebra ...


Selected publications

* Alexandroff P., Hopf H. Topologie Bd.1 — B: 1935 * *


References


Further reading

* *


External links

*
"On the ''curvature integra'' of closed hypersurfaces," transl. by D. H. Delphenich

"Vector fields in ''n''-dimensional manifolds," transl. by D. H. Delphenich
{{DEFAULTSORT:Hopf, Heinz 1894 births 1971 deaths Differential geometers ETH Zurich faculty Geometers 20th-century German mathematicians Jewish emigrants from Nazi Germany to Switzerland 20th-century Swiss mathematicians Topologists University of Göttingen faculty Foreign associates of the National Academy of Sciences Scientists from Wrocław Recipients of the Iron Cross (1914), 1st class Humboldt University of Berlin alumni Presidents of the International Mathematical Union Members of the American Philosophical Society