Ott-Heinrich Keller

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Keller (left) with Hellmuth Kneser

Eduard Ott-Heinrich Keller (born June 22, 1906 in Frankfurt am Main ; † December 5, 1990 in Halle an der Saale ) was a German mathematician who dealt with geometry, topology and algebraic geometry.

life and work

Keller studied at the universities of Vienna , Berlin , Göttingen and Frankfurt. In Frankfurt he became a member of the Corps Austria . In 1931 he did his doctorate in Frankfurt under Max Dehn ( on the complete filling of space with cubes ). He then worked as an assistant in Frankfurt and from 1931 at the TH Berlin with Georg Hamel , where he completed his habilitation in 1933 and became an adjunct professor in 1941. During the Second World War he taught mathematics and mechanics at the Mürwik Naval School in Flensburg - Mürwik . In 1946 he was a professor in Münster and from 1947 a full professor at the TH Dresden . From 1952 he was the successor of Heinrich Wilhelm Ewald Jung Professor at the University of Halle , where he retired in 1971.

Keller dealt with geometry, among other things. In his dissertation, he set up Keller's conjecture about filling the d-dimensional space with d-dimensional cubes of the same size, which he proved in 1937 for dimensions d = 5.6 (and Oskar Perron in 1940 for dimensions d less than or equal to 6). The assumption says that with such a filling at least two cubes have a whole (d − 1) -dimensional side in common. It is related to a conjecture by Hermann Minkowski about Diophantine approximation (which geometrically expresses itself in an analogous conjecture only for lattice arrangements of cubes). In 1992 Jeffrey Lagarias and Peter Shor showed by a counterexample that it is wrong in more than 9 dimensions. In 2000 it was shown by John Mackey by an 8-dimensional counterexample that it is wrong for more than 7 dimensions. In 2020 Joshua Brakensiek, Marijn Heule, John Mackey and David Narváez showed that the conjecture is also correct for the case d = 7. Keller had already suspected the falsity of the conjecture in higher dimensions in his original work.

In algebraic geometry, he dealt among other things with Cremona transformations (for example in his habilitation in 1933), which are important for the classification and resolution of singularities of algebraic curves. Here he set up what was later called the Jacobi conjecture by Shreeram Abhyankar and others. The name comes from the fact that the Jacobide terminant J of a system F of n polynomials in n variables (over an algebraically closed field) is included in the conjecture. The conjecture was proven in a few special cases , such as in the 1970s by Tzuong-Tsieng Moh and others for two polynomials in two variables with degrees less than or equal to 100. Keller also wrote papers on the ideal theoretical structure of algebraic geometry and on topological investigations of algebraic surfaces.

In 1960 he received the National Prize of the GDR III. Science and technology class . In 1961 he was President of the German Mathematicians Association . He was a member of the Saxon Academy of Sciences and the Leopoldina .

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literature

Web links

Individual evidence

  1. ↑ Jürgen Herrlein : Corpslist - Directory of the members of the Corps Austria 1861-2001 , p. 130, serial no. 366
  2. ↑ Ott-Heinrich Keller: A sentence about the complete fulfillment of the 5- and 6-dimensional space with cubes , Crelles Journal 177, 1937, pp. 61-64
  3. ↑ Jeffrey C. Lagarias, Peter W. Shor: Keller's cube-tiling conjecture is false in high dimensions , Bulletin AMS 27, 1992, pp. 279-283
    Chuanming Zong: What is known about unit cubes , Bulletin AMS 42, 2005, p 181-211
  4. ^ John Mackey: A cube tiling of dimension eight with no facesharing , Discrete & Computational Geometry 28, 2002, pp. 275-279
  5. ^ A Fleet of Computers Helps Settle a 90-Year-Old Math Problem
  6. ↑ Ott-Heinrich Keller: Whole Cremona Transformations , monthly books for mathematics 47, 1939, pp. 299-306. van Essen zur Jacobivermutung, Dutch, Nieuw Archief voor Wiskunde 204, PDF file ( Memento from January 26, 2005 in the Internet Archive )
  7. ↑ The Jacobite terminant J is itself a polynomial here, and a necessary condition for the existence of an inverse function to F defined everywhere is that J is a constant not equal to zero, otherwise it would have zeros. The presumption says that this condition is also sufficient.
  8. ↑ Keller uses the spelling homoiomorph for homeomorph
personal data
SURNAME Keller, Ott-Heinrich
ALTERNATIVE NAMES Keller, Eduard Ott-Heinrich (full name)
BRIEF DESCRIPTION German mathematician
DATE OF BIRTH June 22, 1906
PLACE OF BIRTH Frankfurt am Main
DATE OF DEATH 5th December 1990
Place of death Halle on the Saale