Herbert Busemann

Busemann (second from right) with Werner Fenchel , Alexander Alexandrow , Børge Jessen (1954)

Herbert Busemann (born May 12, 1905 in Berlin ; † February 3, 1994 ) was a German-American mathematician .

Live and act

Busemann was born in Berlin as the son of Finance Director Alfred Busemann . He went to school in Frankfurt and Essen , was originally supposed to become a businessman like his father (which led to two and a half - according to Busemann - lost years in the business world after graduating from high school) and studied mathematics in Munich , Göttingen , Paris and Rome . He received his doctorate in 1931 at the University of Göttingen under Richard Courant ( On the geometries in which the "circles with infinite radius" are the shortest lines ), the dissertation of Pawel Sergejewitsch Alexandrow was suggested and partly written in protest against Courant . According to Busemann, Courant in Göttingen took a very conservative view of mathematics (for example he tried to suppress the Lebesgue integral) and the Russian guests like Alexandrow filled a gap by representing modern concepts such as algebraic topology. As the economic depression prevailed at the time, Courant had difficulties financing his institute and urged Busemann to become an unpaid assistant to him and Gustav Herglotz (which enabled other young mathematicians to get a job), knowing that Busemann's father was in good financial shape. Courant also contacted Busemann's father directly for financial support for his institute. In 1933 Busemann, who had a Jewish grandfather, emigrated from Germany to Copenhagen , where he was a lecturer at the university. From 1936 to 1939 he was at the Institute for Advanced Study at Princeton and in 1939 an instructor at Swarthmore College and Johns Hopkins University . From 1940 he was an instructor and then an assistant professor at the Illinois Institute of Technology . He later described the time as miserable because the head of the mathematics department basically did not want any foreigners or mathematical research, but on the other hand he was urged to do so by the university management. He kept in touch with Courant in the USA, but more on practical matters. In 1945 he was an assistant professor at Smith College in Northampton and in 1947 a professor at the University of Southern California , where he stayed until his retirement in 1970 and was Distinguished Professor in 1964.

Busemann mainly dealt with differential geometry , the geometry of convex surfaces, geodetic curves , Finsler geometry and the basics of geometry in the context of the Riemann-Helmholtz spatial problem and isoperimetric problems . The Busemann function is named after him. He dealt extensively with geometries in which the straight line is the shortest connection (subject of the fourth of Hilbert's problems ). After Papadopoulos, he worked in the USA in relative isolation (except for his doctoral students, with whom he had a close relationship) and his work was appreciated in the Soviet Union, where Alexander Danilowitsch Alexandrow and his school carried out a similar return to fundamental principles of geometry Euclid's sense followed (synthetic global geometry), but in the West only with the successes of William Thurston and Michail Leonidowitsch Gromow from the 1980s. In addition, he did not follow the respective mathematical trends, but his own ideas. In an article in the Los Angeles Times of June 14, 1985 on the occasion of the Lobachevsky Prize for Busemann, he is quoted: "If I have one merit, it is that I am not influenced by what other people do," and: "Anyone seemingly difficult problem can be conquered with very simple methods. This is a characteristic of a lot of my work. I see a simple geometric consideration that others have overlooked. "

The Busemann-Petty problem is named after him and his doctoral student Clinton Myers Petty (1956). Let K, L be symmetrical convex bodies in n-dimensional Euclidean space. If the (n-1) -dimensional volume of all hypersurface cuts through the origin of K is greater than or equal to that of L, is the n-dimensional volume of K then also greater than or equal to that of L? According to Busemann and Petty, this is the case when L is a sphere. In general it is correct up to n = 4, but wrong in higher dimensions. First, Claude Ambrose Rogers showed with DG Larman that it is wrong in twelve and more dimensions (1975). In 1988, Erwin Lutwak proved that the problem has a positive solution precisely when the bodies are the intersection bodies he introduced . Richard J. Gardner showed the validity for n = 3 in 1994 and Gao Yong Zhang for n = 4 (1994). Alexander Koldobsky , Richard Gardner, and T. Schlumprecht in 1999 finally gave uniform proof in all dimensions .

He was a member of the Royal Danish Academy of Sciences . In 1984 he received the Lobachevsky Medal (for Geometry of Geodesics ). He was President of the California Area of ​​the Mathematical Association of America and on the Council of the American Mathematical Society . In 1971 he received an honorary doctorate from the University of Southern California.

He was linguistically gifted and spoke French, Spanish, Italian, Russian and Danish in addition to German and English. He could also read Latin, ancient Greek, Arabic and Swedish. For Mathematical Reviews he wrote and translated many Russian mathematical papers. He read Homer's Odyssey and Plato regularly. He started painting when he retired. He had been married since 1939 but had no children.

Fonts

• Introduction to algebraic manifolds. Princeton University Press 1939.
• with Paul J. Kelly : Projective geometry and projective metrics. Academic Press 1953, Dover 2006.
• Convex surfaces. Interscience 1958, Dover 2008.
• The Geometry of Geodesics. Academic Press 1955, Dover 2005.
• Metric methods in Finsler spaces and in the foundations of geometry. Princeton University Press, Oxford University Press 1942.
• with Bhalchandra Phadke: Spaces with distinguished geodesics. Dekker 1987.
• Recent synthetic differential geometry. Springer 1970.
• Selected Works, Ed. Athanase Papadopoulos. 2 volumes, Springer Verlag, 2018.

literature

• Anikó Szabó: eviction, return, reparation. Göttingen university professor in the shadow of National Socialism, with biographical documentation of the dismissed and persecuted university professors: University of Göttingen - TH Braunschweig - TH Hannover - University of Veterinary Medicine Hannover. Wallstein, Göttingen 2000, p. 482, ISBN 978-3-89244-381-0 (= publications of the working group history of Lower Saxony (after 1945), volume 15, also dissertation at the University of Hanover 1998).
• Benjamin H. Yandell: The honors class. Hilbert's problems and their solvers. AK Peters, Natick MA 2001.
• Athanase Papadopoulos: Herbert Busemann, Notices of the AMS. Volume 65, No. 3, March 2018, pp. 341–343 ( ams.org PDF).

Web links

• Literature by and about Herbert Busemann in the catalog of the German National Library
• Biography (spanish)
• Herbert Busemann's estate in the German Exile Archive of the German National Library

Individual evidence

1. ↑ Busemann, after Papadopoulos, Busemann, Notices AMS, March 2018, p. 341.
2. ↑ Busemann after Papadopoulos, Notices AMS, March 2018, p. 342.
3. ↑ Papadopoulos, Busemann, Notices AMS, March 2018, p. 342.
4. ↑ Papadopoulos, Busemann, Notices AMS, March 2018, p. 343.
5. ↑ Busemann, Petty, Problems on convex bodies, Mathematica Scandinavica, Volume 4, 1956, pp. 88-94.
6. ^ Busemann-Petty problem , Mathworld
7. ^ Larman, Rogers: The existence of a centrally symmetric convex body with central sections that are unexpectedly small. In: Mathematika. Volume 22, 1976, pp. 164-175.
8. ↑ Gardner, Koldobsky, Schlumprecht: An analytic solution to the Busemann-Petty problem on sections of convex bodies. In: Annals of Mathematics. Volume 149, 1999, pp. 691-703.
9. ↑ Of this book, Busemann said that the emphasis there was more on the radical new methods than on the results. Quoted in Papadopoulos, Notices AMS, March 2018, p. 343.