Eberhard Hopf

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Eberhard Hopf 1970

Eberhard Frederich Ferdinand Hopf (born April 4, 1902 in Salzburg , † July 24, 1983 in Bloomington (Indiana) ) was a German-American mathematician who made significant contributions to many areas of analysis and especially ergodic theory .

Life

Eberhard Hopf was born in Salzburg, Austria , the son of the German businessman and chocolate manufacturer Friedrich Hopf . He received his Abitur in 1920 at the Gymnasium zu Berlin-Friedenau , began studying mathematics and physics at the University of Berlin and in 1924 spent a semester in the same subjects at the University of Tübingen . He received his doctorate in 1926 (examination July 1925) under Erhard Schmidt and Issai Schur (2nd reviewer) and completed his habilitation in 1929 at the University of Berlin in mathematical astronomy , where he was a private lecturer. From 1926 to 1930 he was at the Astronomical Computing Institute in Berlin-Dahlem. In 1927 he had proven the maximum principle for elliptic differential equations of the second order. In 1929 he married Ilse Wolf, daughter of the musicologist Johannes Wolf . With her he had a daughter Barbara (* 1936).

In 1930 Hopf received a grant from the Rockefeller Foundation , which enabled him to stay at Harvard University with George David Birkhoff and at Cambridge University until 1932, where he dealt in particular with ergodic theory. In particular, the publication On time average theorem in dynamics is regarded by many as the first easily understandable work in modern ergodic theory. Another important contribution from this period is the Wiener-Hopf equation , which he developed in collaboration with Norbert Wiener ; this represents an integral equation that occurs, for example, in the theory of the radiation equilibrium of stellar atmospheres (a topic that Hopf dealt with) and has been used extensively in communications engineering and geophysics in a discrete variant ( Wiener filter ) since the 1960s .

With the support of Norbert Wiener, Hopf became assistant professor in 1931 at the mathematics faculty of the Massachusetts Institute of Technology . Here, too, he mainly devoted himself to the ergodic theory. At that time, he was already distinguished by his talent for explaining complex issues in an easily understandable way for colleagues and those outside the field.

He returned to Germany with his family in 1936 when he was offered to succeed Leon Lichtenstein as associate professor at the University of Leipzig , where he was professor from 1936 to 1944. During this time he also completed the book Ergodic Theory, which offers a concise description of the topic on 81 pages. Due to the Second World War, he was obliged to work at the German Research Institute for Gliding (Aviation Research Institute in Ainring ) in 1942 , which was entrusted with military developments for the Air Force. At the instigation of Oskar Perron , however, he received a professorship at the University of Munich in 1944 as the successor to Constantin Carathéodory , which he held until 1949. In 1942 he published his work on the Hopf bifurcation named after him but dating back to Henri Poincaré , the occurrence of periodic solutions in a steady state of an evolution equation at certain critical parameter values.

At the invitation of Richard Courant he went back to the United States in 1947 (in his own words as a paper clip scientist), where he was visiting professor at New York University ( Courant Institute ) in 1947/48 . He stayed in the USA because he had more time for research there, took on the US citizenship in 1949 and worked as a professor at Indiana University from 1948 , where he was interested in working with Clifford Truesdell , he was a research professor from 1962 and where he stayed until his retirement in 1972.

His results on the regularity and analyticity of the solutions of elliptic partial differential equations in the 1920s and 1930s found their way into textbooks as classic results. With the Hopf maximal ergodic lemma named after him, he succeeded in 1954 in extending the individual ergodic theorem to Markoff operators. In connection with the ergodic theory he also studied the behavior of geodetic curves on surfaces with negative curvature. Hopf made important contributions to the solution of the Navier-Stokes equations of hydrodynamics in two dimensions. In 1948 he first gave a model of a viscous liquid with turbulent solutions. In 1950 he investigated the mathematical basis of shock waves using the Burgers equation .

Awards and memberships

In 1971, Hopf was named a Gibbs Lecturer by the American Mathematical Society . In 1981 he received the Leroy P. Steele Prize from the American Mathematical Society.

In 1939 he became one of the editors of the Fundamental Mathematical Sciences and from 1952 to 1983 he was editor of the Journal of Rational Mechanics and Analysis .

In 1938 he was elected a member of the Saxon Academy of Sciences and in 1945 of the Bavarian Academy of Sciences , of which he became a full member in 1947 and a corresponding member after moving to the USA in 1949.

Fonts

Books:

  • Ergodic theory. Springer Verlag (Results of Mathematics and its Frontier Areas), 1937, Reprint 1970
  • Mathematical problems of radiation equilibrium , Cambridge University Press 1934, Archives

Articles (selection), except for the articles cited in the footnotes:

  • On the analytical character of the solutions to regular two-dimensional variation problems, Mathematische Zeitschrift, Volume 30, 1929, pp. 404-413
  • On the functional, in particular the analytical character of the solutions of elliptic partial differential equations of the second order, Mathematische Zeitschrift, Volume 34, 1931, pp. 193-233
  • Complete Transitivity and the Ergodic Principle, Proc. Nat. Acad. Sci., Vol. 18, 1932, pp. 204-209
  • Proof of Gibbs Hypothesis on Statistical Equilibrium, Proc. Nat. Acad. Sci., Vol. 18, 1932, pp. 333-340
  • On Causality, Statistics and Probability, J. Math. Phys., Volume 13, 1934, pp. 51-102
  • Statistics of the geodetic lines in manifolds of negative curvature, Dep. Sächs. Akad. Wiss. Leipzig, Volume 91, 1939, pp. 261-304
  • Statistics of the solutions to geodetic problems of the unstable type, Part II., Mathematische Annalen, Volume 117, 1940, pp. 590-608
  • Closed surfaces without conjugate points, Proc. Nat. Acad. Sci., Vol. 34, 1948, pp. 47-51
  • On S. Bernstein's theorem on surfaces z (x, y) of nonpositive curvature, Proc. At the. Math. Soc., Vol. 1, 1950, pp. 80-85
  • About the initial value problem for the basic hydrodynamic equations, Math. Nachrichten, Volume 4, 1951, pp. 213-231
  • Statistical Hydromechanics and Functional Calculus, J. Rational Mechanics and Analysis, Volume 1, 1952, pp. 87-123
  • Some topics of ergodic theory, CIME Rome 1960, pp. 1-64
  • An inequality for positive linear integral operators, J. Math. & Mech., Volume 12, 1963, pp. 683-692
  • Ergodic theory and the geodesic flow on surfaces of constant negative curvature, Bulletin Am. Math. Soc., Volume 17, 1971, pp. 863-877 (Gibbs Lecture 1971)

Collections of writings can be found in:

literature

  • In memoriam Eberhard Hopf, Indiana University Math. J., Volume 32, 1983, No. 6
  • Herbert Becker, obituary in the annual report Akad. Wiss. Leipzig, 1986
  • M. Denker: Eberhard Hopf, Annual Report DMV, Volume 92, 1990, No. 2
  • Andrzej Icha: Eberhard Hopf (1902-1983), Nieuw Archief voor Wiskunde, Volume 12, 1994, No. 1-2
  • Heinz Bauer : Eberhard Hopf April 17, 1902– July 24, 1983. In: Bavarian Academy of Sciences , Yearbook 1984, Munich 1984, pp. 254-256 ( online ).

Web links

Individual evidence

  1. ↑ Eberhard Hopf: About the connections between certain higher difference quotients of real functions of a real variable and their differentiability properties. North German Buchdr. u. Verlagsanst., Berlin 1926 ( Göttingen Digitization Center  ( page no longer available , search in web archives )  Info: The link was automatically marked as defective. Please check the link according to the instructions and then remove this notice. )@1@ 2Template: Toter Link / www-gdz.sub.uni-goettingen.de  
  2. ↑ Hopf, Elementary Remarks on the Solution of Partial Differential Equations of the Second Order of the Elliptical Type, Preuss Meeting Reports. Akad. Wiss., 1927, pp. 147-152
  3. ^ Hopf, remarks on a sentence by S. Bernstein on the theory from the elliptical differential equations, Math. Z., Volume 29, 1928, pp. 744-745
  4. ^ Eberhard Hopf: On the time average theorem in dynamics , Proceedings of the National Academy of Sciences USA, Volume 18, 1932, pp. 93-100
  5. ^ John J. O'Connor, Edmund F. Robertson :  Eberhard Hopf. In: MacTutor History of Mathematics archive .
  6. ^ Wiener, Hopf, On a class of singular integral equations, Preuss meeting reports. Akad. Wiss., Math.-Phys. Class, 1931, pp. 696-706
  7. ^ Hopf, branching off a periodic solution of a differential system, reports of the mathematical-physical class of the Saxons. Akad. Wiss. Leipzig, Volume 94, 1942, pp. 1-22
  8. ↑ Denker, Annual Report DMV, 1990, p. 48
  9. ↑ Denker, Annual Report DMV, Volume 92, 1990, No. 2, p. 48
  10. ↑ Short biography and picture
  11. ↑ E. Hopf, The General Temporally Discrete Markoff Process, Journal of Rational Mechanics and Analysis Vol. 3 (1954), pp. 13-45
  12. ^ Hopf, A mathematical example displaying features of turbulence, Comm. on Pure Appl. Math, Volume 1, 1948, pp. 303-320
  13. ^ Hopf, The partial differential equation , Comm. on Pure Appl. Math., Vol. 3, 1950, pp. 201-230
personal data
SURNAME Hopf, Eberhard
ALTERNATIVE NAMES Hopf, Eberhard Frederich Ferdinand (full name)
BRIEF DESCRIPTION German-American mathematician
DATE OF BIRTH April 4, 1902
PLACE OF BIRTH Salzburg
DATE OF DEATH July 24, 1983
Place of death Bloomington (Indiana)