Dieter Kotschick

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Dieter Kotschick (* 1963 ) is a German mathematician who deals with differential geometry and topology .

Kotschick moved from Transylvania to Germany when he was fifteen . He studied first in Heidelberg and then in Bonn, did his doctorate in 1989 at the University of Oxford with Simon Donaldson ( On the geometry of certain 4-manifolds ) and was a post-doc at the University of Cambridge . He became professor at the University of Basel in 1991 and professor at the Ludwig Maximilians University in Munich in 1998 . Kotschick was a member of the Institute for Advanced Study three times (1989/90, 2008/09 and 2012/13). He is a fellow of the American Mathematical Society .

In 2009 he solved an open problem by Friedrich Hirzebruch (1954), more than 50 years old , which asks which Chern numbers are topological invariants of smooth complex-algebraic varieties. He found that only linear combinations of Euler's invariant and the Pontryagin numbers are invariants of orientation- preserving diffeomorphisms (and thus, according to Sergei Novikow , of oriented homeomorphisms as well ) of these varieties. Kotschick proved that, if the condition of orientability is abandoned, only multiples of the Euler characteristic come into question among the Chern numbers and their linear combinations as invariants of diffeomorphisms in three and more complex dimensions. For homeomorphisms he showed that there is no restriction on the dimension. In addition, Kotschick proved further theorems about the structure of the space of Chern numbers of smooth complex-projective manifolds .

He classified the possible patterns on the surface of a soccer ball, that is, special tiling with pentagons and hexagons on the sphere. In the case of the sphere, there is only the standard soccer ball (12 black pentagons, 20 white hexagons; it corresponds to a truncated icosahedron ) and its branched overlays as a solution, with a higher gender of the surface there are more solutions. The analysis also has application to fullerenes .

Fonts

  • On manifolds homeomorphic to . Invent. Math. 95 (1989) no. 3, 591-600.
  • with H. Endo: Bounded cohomology and non-uniform perfection of mapping class groups. Invent. Math. 144 (2001) no. 1, 169-175.
  • Gauge theory is dead! Long live gauge theory! ( PDF file, 95 kB), Notices of the AMS 42, March 1995, pp. 335–338 (English; on the Seiberg-Witten theory)
  • Topology and Combinatorics of Football , Spectrum of Science, June 24, 2006
  • with J. Amorós, M. Burger, K. Corlette, D. Toledo: Fundamental groups of compact Kähler manifolds. Mathematical Surveys and Monographs, 44. American Mathematical Society, Providence, RI, 1996. xii + 140 pp. ISBN 0-8218-0498-7

Web links

References

  1. ^ Kotschick, Dieter in the directory A community of scholars of the IAS
  2. ↑ Friedrich Hirzebruch: Some problems on differentiable and complex manifolds , Annals of Mathematics, Vol. 60, 1954, pp. 213-236
  3. ↑ defined by zeros of polynomials in complex
  4. ↑ Kotschick Characteristic numbers of algebraic varieties , Proceedings National Academy of Sciences, Vol. 106, 2009, 10014, Online . Also university protocols
  5. ↑ the sides of the pentagons may only meet hexagons, those of the hexagons alternate with pentagons and hexagons
  6. ^ Column Mathematical Entertainments , Spectrum of Science, July 2006, Braungardt, Kotschick The Classification of Soccer Patterns , Math. Semesterberichte, Vol. 54, 2007, pp. 53-68, Kotschick The topology and combinatorics of soccer balls , American Scientist, July / August 2006
personal data
SURNAME Kotschick, Dieter
BRIEF DESCRIPTION German mathematician
DATE OF BIRTH 1963