Dieter Kotschick (born 1963) is a German mathematician, specializing in differential geometry and topology.

Biography

At age fifteen, Kotschick moved from

Transylvania Transylvania ( or ; ; or ; Transylvanian Saxon dialect, Transylvanian Saxon: ''Siweberjen'') is a List of historical regions of Central Europe, historical and cultural region in Central Europe, encompassing central Romania. To the east and ...

to Germany. He first studied at

Heidelberg University Heidelberg University, officially the Ruprecht Karl University of Heidelberg (; ), is a public research university in Heidelberg, Baden-Württemberg, Germany. Founded in 1386 on instruction of Pope Urban VI, Heidelberg is Germany's oldest unive ...

and then at the

University of Bonn The University of Bonn, officially the Rhenish Friedrich Wilhelm University of Bonn (), is a public research university in Bonn, North Rhine-Westphalia, Germany. It was founded in its present form as the () on 18 October 1818 by Frederick Willi ...

. He received his doctorate from the

University of Oxford The University of Oxford is a collegiate university, collegiate research university in Oxford, England. There is evidence of teaching as early as 1096, making it the oldest university in the English-speaking world and the List of oldest un ...

in 1989 under the supervision of

Simon Donaldson Sir Simon Kirwan Donaldson (born 20 August 1957) is an English mathematician known for his work on the topology of smooth function, smooth (differentiable) four-dimensional manifolds, Donaldson–Thomas theory, and his contributions to Kähl ...

with thesis ''On the geometry of certain 4-manifolds'' and held postdoctoral positions at

Princeton University Princeton University is a private university, private Ivy League research university in Princeton, New Jersey, United States. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial ...

and the

University of Cambridge The University of Cambridge is a Public university, public collegiate university, collegiate research university in Cambridge, England. Founded in 1209, the University of Cambridge is the List of oldest universities in continuous operation, wo ...

. He became a professor at the

University of Basel The University of Basel (Latin: ''Universitas Basiliensis''; German: ''Universität Basel'') is a public research university in Basel, Switzerland. Founded on 4 April 1460, it is Switzerland's oldest university and among the world's oldest univ ...

in 1991 and a professor at the

Ludwig Maximilian University of Munich The Ludwig Maximilian University of Munich (simply University of Munich, LMU or LMU Munich; ) is a public university, public research university in Munich, Bavaria, Germany. Originally established as the University of Ingolstadt in 1472 by Duke ...

in 1998. Kotschick has been a member of the

Institute for Advanced Study The Institute for Advanced Study (IAS) is an independent center for theoretical research and intellectual inquiry located in Princeton, New Jersey. It has served as the academic home of internationally preeminent scholars, including Albert Ein ...

three times (1989/90, 2008/09 and 2012/13). In 2012 he was elected a Fellow of the

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

. In 2009, he solved a 55-year-old open problem posed in 1954 by

Friedrich Hirzebruch Friedrich Ernst Peter Hirzebruch ForMemRS (17 October 1927 – 27 May 2012) was a German mathematician, working in the fields of topology, complex manifolds and algebraic geometry, and a leading figure in his generation. He has been described as ...

, which asks "which linear combinations of

Chern number In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since become fundamental concepts in many branches o ...

s of smooth complex

projective varieties In algebraic geometry, a projective variety is an algebraic variety that is a closed subvariety of a projective space. That is, it is the zero-locus in \mathbb^n of some finite family of homogeneous polynomials that generate a prime ideal, the ...

are topologically invariant". He found that only linear combinations of the

Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's ...

and the Pontryagin numbers are invariants of orientation-preserving

diffeomorphism In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable. Definit ...

s (and thus according to Sergei Novikov also of oriented

homeomorphism In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ...

s) 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 Telstar is a football made by Adidas. The iconic 32-panel alternating black-and-white design of the ball, based on the work of Eigil Nielsen, has since become a global standard design used to portray a football in different media. History The ba ...

soccer ball A football or soccer ball is the ball used in the sport of association football. The ball's spherical shape, as well as its size, mass, and material composition, are specified by Law 2 of the Laws of the Game maintained by the International Fo ...

, ''i.e.'' special tilings with pentagons and hexagons on the sphere.Kotschic
''The topology and combinatorics of soccer balls''
American Scientist, July/August 2006 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 In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrical tha ...

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

fullerene A fullerene is an allotropes of carbon, allotrope of carbon whose molecules consist of carbon atoms connected by single and double bonds so as to form a closed or partially closed mesh, with fused rings of five to six atoms. The molecules may ...

s and polyhedra that Kotschick calls ''generalized soccer balls''.

Selected publications

* * * '
Gauge theory is dead! Long live gauge theory!
'' (

PDF Portable document format (PDF), standardized as ISO 32000, is a file format developed by Adobe Inc., Adobe in 1992 to present documents, including text formatting and images, in a manner independent of application software, computer hardware, ...

- File, 95 kB), Notices of the AMS 42, March 1995, pp. 335–338 (on the Seiberg-Witten Theory) *
Topologie und Kombinatorik des Fußballs
', Spektrum der Wissenschaft, 24 June 2006 *

References

External links

{{DEFAULTSORT:Kotschick, Dieter 20th-century German mathematicians 21st-century German mathematicians Alumni of the University of Oxford Academic staff of the University of Basel Academic staff of the Ludwig Maximilian University of Munich Fellows of the American Mathematical Society Differential geometers German topologists 1963 births Living people Heidelberg University alumni University of Bonn alumni