Eberhard Hopf
Encyclopedia
Eberhard Frederich Ferdinand Hopf (April 4, 1902, Salzburg
, Austria-Hungary
– July 24, 1983, Bloomington, Indiana
) was a mathematician
and astronomer
, one of the founding fathers of ergodic theory
and a pioneer of bifurcation theory
who also made significant contributions to the subjects of partial differential equation
s and integral equation
s, fluid dynamics
, and differential geometry. The Hopf maximum principle
is an early result of his (1927) which is one of the most important techniques in the theory of elliptic partial differential equations.
, Austria-Hungary
, but his scientific career was divided between Germany
and the United States
. He received his Ph.D. in Mathematics in 1926 and his Habilitation
in Mathematical Astronomy from the University of Berlin
in 1929.
In 1930 he received a fellowship from the Rockefeller Foundation
to study classical mechanics
with George Birkhoff
at Harvard, but his appointment was at the Harvard College Observatory. In late 1931, with the help of Norbert Wiener
, Hopf joined the Department of Mathematics of the Massachusetts Institute of Technology
, accepting the position of Assistant Professor. While at MIT, Hopf did much of his work on ergodic theory
.
In Cambridge Hopf worked on many mathematical and astronomical subjects. His paper On time average theorem in dynamics, which appeared in the Proceedings of the National Academy of Sciences
, is considered by many to be the first readable paper in modern ergodic theory. His book Mathematical problems of radiative equilibrium first appeared in 1934 and was reprinted in 1964. Another important contribution from this period is the theory of Wiener-Hopf equations, which he developed in collaboration with Norbert Wiener. By 1960, a discrete version of these equations was being extensively used in electrical engineering and geophysics, their use continuing until the present day. During this time, Hopf gained a reputation for his ability of illuminating the most complex subjects for his colleagues and even for non-specialists. Because of this talent, many discoveries and proofs of other mathematicians became easier to understand after they had been described by Hopf.
In 1936 Hopf received and accepted an offer of a full professorship from the University of Leipzig
. Hopf, with his wife Ilse and their infant daughter Barbara, returned to Germany, which by this time was under the control of the Nazi Party.
The book Ergodentheorie, most of which was written when Hopf was still at the Massachusetts Institute of Technology, was published in 1937. In that book, containing only 81 pages, Hopf presented a precise and elegant summary of ergodic theory. In 1939 Hopf established ergodicity of the geodesic flow on compact
manifolds of constant negative curvature
. In 1940 Hopf was on the list of the invited lecturers to the International Congress of Mathematicians
to be held in Cambridge, Massachusetts
. Because of the start of World War II
, however, the Congress was cancelled.
In 1942 Hopf was drafted to work in the German Aeronautical Institute. In 1944, one year before the end of World War II, Hopf was appointed to a professorship at the University of Munich. In 1947, at the behest of Richard Courant
he returned to the United States, where he presented the definitive solution of Hurewicz's problem.
On 22 February 1949 Hopf became a US citizen and joined Indiana University at Bloomington as a Professor of Mathematics. In 1962 he was made Research Professor of Mathematics, staying in that position until his death.
Hopf was never forgiven by many people for his moving to Germany in 1936, where the Nazi party was in power. As a result, most of his work in ergodic theory and topology was neglected or even attributed to others in the years following the end of World War II. An example of this was the expulsion of Hopf's name from the discrete version of the Wiener–Hopf equations, which were frequently referred to as "Wiener filter
".
Gibbs
Lecturer.
Salzburg
-Population development:In 1935, the population significantly increased when Salzburg absorbed adjacent municipalities. After World War II, numerous refugees found a new home in the city. New residential space was created for American soldiers of the postwar Occupation, and could be used for...
, Austria-Hungary
Austria-Hungary
Austria-Hungary , more formally known as the Kingdoms and Lands Represented in the Imperial Council and the Lands of the Holy Hungarian Crown of Saint Stephen, was a constitutional monarchic union between the crowns of the Austrian Empire and the Kingdom of Hungary in...
– July 24, 1983, Bloomington, Indiana
Bloomington, Indiana
Bloomington is a city in and the county seat of Monroe County in the southern region of the U.S. state of Indiana. The population was 80,405 at the 2010 census....
) was a mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
and astronomer
Astronomer
An astronomer is a scientist who studies celestial bodies such as planets, stars and galaxies.Historically, astronomy was more concerned with the classification and description of phenomena in the sky, while astrophysics attempted to explain these phenomena and the differences between them using...
, one of the founding fathers of ergodic theory
Ergodic theory
Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....
and a pioneer of bifurcation theory
Bifurcation theory
Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations...
who also made significant contributions to the subjects of partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
s and integral equation
Integral equation
In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. There is a close connection between differential and integral equations, and some problems may be formulated either way...
s, fluid dynamics
Fluid dynamics
In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics and hydrodynamics...
, and differential geometry. The Hopf maximum principle
Hopf maximum principle
The Hopf maximum principle is a maximum principle in the theory of second order elliptic partial differential equations and has been described as the "classic and bedrock result" of that theory...
is an early result of his (1927) which is one of the most important techniques in the theory of elliptic partial differential equations.
Biography
Eberhard Hopf was born in SalzburgSalzburg
-Population development:In 1935, the population significantly increased when Salzburg absorbed adjacent municipalities. After World War II, numerous refugees found a new home in the city. New residential space was created for American soldiers of the postwar Occupation, and could be used for...
, Austria-Hungary
Austria-Hungary
Austria-Hungary , more formally known as the Kingdoms and Lands Represented in the Imperial Council and the Lands of the Holy Hungarian Crown of Saint Stephen, was a constitutional monarchic union between the crowns of the Austrian Empire and the Kingdom of Hungary in...
, but his scientific career was divided between Germany
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
and the United States
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
. He received his Ph.D. in Mathematics in 1926 and his Habilitation
Habilitation
Habilitation is the highest academic qualification a scholar can achieve by his or her own pursuit in several European and Asian countries. Earned after obtaining a research doctorate, such as a PhD, habilitation requires the candidate to write a professorial thesis based on independent...
in Mathematical Astronomy from the University of Berlin
Technical University of Berlin
The Technische Universität Berlin is a research university located in Berlin, Germany. Translating the name into English is discouraged by the university, however paraphrasing as Berlin Institute of Technology is recommended by the university if necessary .The TU Berlin was founded...
in 1929.
In 1930 he received a fellowship from the Rockefeller Foundation
Rockefeller Foundation
The Rockefeller Foundation is a prominent philanthropic organization and private foundation based at 420 Fifth Avenue, New York City. The preeminent institution established by the six-generation Rockefeller family, it was founded by John D. Rockefeller , along with his son John D. Rockefeller, Jr...
to study classical mechanics
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
with George Birkhoff
George David Birkhoff
-External links:* − from National Academies Press, by Oswald Veblen....
at Harvard, but his appointment was at the Harvard College Observatory. In late 1931, with the help of Norbert Wiener
Norbert Wiener
Norbert Wiener was an American mathematician.A famous child prodigy, Wiener later became an early researcher in stochastic and noise processes, contributing work relevant to electronic engineering, electronic communication, and control systems.Wiener is regarded as the originator of cybernetics, a...
, Hopf joined the Department of Mathematics of the Massachusetts Institute of Technology
Massachusetts Institute of Technology
The Massachusetts Institute of Technology is a private research university located in Cambridge, Massachusetts. MIT has five schools and one college, containing a total of 32 academic departments, with a strong emphasis on scientific and technological education and research.Founded in 1861 in...
, accepting the position of Assistant Professor. While at MIT, Hopf did much of his work on ergodic theory
Ergodic theory
Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and related problems. Its initial development was motivated by problems of statistical physics....
.
In Cambridge Hopf worked on many mathematical and astronomical subjects. His paper On time average theorem in dynamics, which appeared in the Proceedings of the National Academy of Sciences
Proceedings of the National Academy of Sciences
The Proceedings of the National Academy of Sciences of the United States of America, usually referred to as PNAS, is the official journal of the United States National Academy of Sciences...
, is considered by many to be the first readable paper in modern ergodic theory. His book Mathematical problems of radiative equilibrium first appeared in 1934 and was reprinted in 1964. Another important contribution from this period is the theory of Wiener-Hopf equations, which he developed in collaboration with Norbert Wiener. By 1960, a discrete version of these equations was being extensively used in electrical engineering and geophysics, their use continuing until the present day. During this time, Hopf gained a reputation for his ability of illuminating the most complex subjects for his colleagues and even for non-specialists. Because of this talent, many discoveries and proofs of other mathematicians became easier to understand after they had been described by Hopf.
In 1936 Hopf received and accepted an offer of a full professorship from the University of Leipzig
University of Leipzig
The University of Leipzig , located in Leipzig in the Free State of Saxony, Germany, is one of the oldest universities in the world and the second-oldest university in Germany...
. Hopf, with his wife Ilse and their infant daughter Barbara, returned to Germany, which by this time was under the control of the Nazi Party.
The book Ergodentheorie, most of which was written when Hopf was still at the Massachusetts Institute of Technology, was published in 1937. In that book, containing only 81 pages, Hopf presented a precise and elegant summary of ergodic theory. In 1939 Hopf established ergodicity of the geodesic flow on compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
manifolds of constant negative curvature
Hyperbolic manifold
In mathematics, a hyperbolic n-manifold is a complete Riemannian n-manifold of constant sectional curvature -1.Every complete, connected, simply-connected manifold of constant negative curvature −1 is isometric to the real hyperbolic space Hn. As a result, the universal cover of any closed manifold...
. In 1940 Hopf was on the list of the invited lecturers to the International Congress of Mathematicians
International Congress of Mathematicians
The International Congress of Mathematicians is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union ....
to be held in Cambridge, Massachusetts
Cambridge, Massachusetts
Cambridge is a city in Middlesex County, Massachusetts, United States, in the Greater Boston area. It was named in honor of the University of Cambridge in England, an important center of the Puritan theology embraced by the town's founders. Cambridge is home to two of the world's most prominent...
. Because of the start of World War II
World War II
World War II, or the Second World War , was a global conflict lasting from 1939 to 1945, involving most of the world's nations—including all of the great powers—eventually forming two opposing military alliances: the Allies and the Axis...
, however, the Congress was cancelled.
In 1942 Hopf was drafted to work in the German Aeronautical Institute. In 1944, one year before the end of World War II, Hopf was appointed to a professorship at the University of Munich. In 1947, at the behest of Richard Courant
Richard Courant
Richard Courant was a German American mathematician.- Life :Courant was born in Lublinitz in the German Empire's Prussian Province of Silesia. During his youth, his parents had to move quite often, to Glatz, Breslau, and in 1905 to Berlin. He stayed in Breslau and entered the university there...
he returned to the United States, where he presented the definitive solution of Hurewicz's problem.
On 22 February 1949 Hopf became a US citizen and joined Indiana University at Bloomington as a Professor of Mathematics. In 1962 he was made Research Professor of Mathematics, staying in that position until his death.
Hopf was never forgiven by many people for his moving to Germany in 1936, where the Nazi party was in power. As a result, most of his work in ergodic theory and topology was neglected or even attributed to others in the years following the end of World War II. An example of this was the expulsion of Hopf's name from the discrete version of the Wiener–Hopf equations, which were frequently referred to as "Wiener filter
Wiener filter
In signal processing, the Wiener filter is a filter proposed by Norbert Wiener during the 1940s and published in 1949. Its purpose is to reduce the amount of noise present in a signal by comparison with an estimation of the desired noiseless signal. The discrete-time equivalent of Wiener's work was...
".
Awards and honors
In 1971 Hopf was the American Mathematical SocietyAmerican Mathematical Society
The American Mathematical Society is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, which it does with various publications and conferences as well as annual monetary awards and prizes to mathematicians.The society is one of the...
Gibbs
Josiah Willard Gibbs
Josiah Willard Gibbs was an American theoretical physicist, chemist, and mathematician. He devised much of the theoretical foundation for chemical thermodynamics as well as physical chemistry. As a mathematician, he invented vector analysis . Yale University awarded Gibbs the first American Ph.D...
Lecturer.
See also
- Hopf alternative
- Hopf bifurcationHopf bifurcationIn the mathematical theory of bifurcations, a Hopf or Poincaré–Andronov–Hopf bifurcation, named after Henri Poincaré, Eberhard Hopf, and Aleksandr Andronov, is a local bifurcation in which a fixed point of a dynamical system loses stability as a pair of complex conjugate eigenvalues of...
- Hopf–Cole transformation (see Burgers' equationBurgers' equationBurgers' equation is a fundamental partial differential equation from fluid mechanics. It occurs in various areas of applied mathematics, such as modeling of gas dynamics and traffic flow...
) - Wiener–Hopf method