At age fifteen, Kotschick moved from
Transylvania to Germany. He first studied at
Heidelberg University and then at the
University of Bonn. He received his doctorate from the
University of Oxford in 1989 under the supervision of
Simon Donaldson with thesis
On the geometry of certain 4-manifolds and held postdoctoral positions at
Princeton University and the
University of Cambridge. He became a professor at the
University of Basel in 1991 and a professor at
LMU Munich in 1998. Kotschick has been a member of the
Institute for Advanced Study three times (1989/90, 2008/09 and 2012/13). In 2012 he was elected a Fellow of the
American Mathematical Society. In 2009, he solved a 55-year-old open problem posed in 1954 by
Friedrich Hirzebruch, which asks "which linear combinations of
Chern numbers of smooth complex
projective varieties are topologically invariant". He found that only linear combinations of the
Euler characteristic and the
Pontryagin numbers are invariants of orientation-preserving
diffeomorphisms (and thus according to
Sergei Novikov also of oriented
homeomorphisms) of these varieties. Kotschick proved that if the condition of orientability is removed, only multiples of the Euler characteristic can be considered among the Chern numbers and their linear combinations as invariants of diffeomorphisms in three and more complex dimensions. For homeomorphisms, he showed that the restriction on the dimension can be omitted. In addition, Kotschick proved further theorems about the structure of the set of Chern numbers of smooth complex-projective manifolds. He classified the possible patterns on the surface of an
Adidas Telstar soccer ball,
i.e. special
tilings with pentagons and hexagons on the sphere. In the case of the sphere, there is only the
standard football (12 black pentagons, 20 white hexagons, with a pattern corresponding to an
icosahedral root) provided that "precisely three edges meet at every vertex". If more than three faces meet at some vertex, then there is a method to generate infinite sequences of different soccer balls by a topological construction called a
branched covering. Kotschick's analysis also applies to
fullerenes and polyhedra that Kotschick calls
generalized soccer balls. ==Selected publications==