Stefan Kebekus

Fellowship 2013

ARCHIVE

Stefan Kebekus, born in 1970, is Professor of Mathematics at the University of Freiburg. His research interests lie in Complex Geometry, a branch of Pure Mathematics with connections to Number Theory, Cryptography, Theoretical Physis, and many other fields.

Kebekus studied Mathematics at the Ruhr-Universität Bochum (Germany) and also obtained his PhD there in 1996. Subsequent to his studies, he was Scientific Assistant at the University of Bayreuth (Germany), Guest Researcher at Kyoto University (Japan), Visiting Professor at the University of Washington in Seattle (U.S.A.), Professeur Invite at Strasbourg, Grenoble and Rennes, and Heisenberg-fellow of the Deutsche Forschungsgemeinschaft (German National Science Foundation). He obtained his Habilitation in 2001 from the University of Bayreuth. From 2003-2008, Kebekus was Professor of Mathematics the University of Cologne (Germany). He moved to Freiburg in 2008. Kebekus is an editor of the journal „Algebraic Geomety“. He is currently vice director of Freiburg's Research Training Group „Cohomological Methods in Geometry“. From 2006 to 2008, Kebekus was director of the Research Training Group „Global Structures in Geometry and Analysis“. He was a member of the Mathematical Sciences Research Institute in Berkeley (U.S.A.) in 2009.

Rational Points, Rational Curves and Automorphisms of Special Varieties

Fellows Fribourg-Strasbourg: Carlo Gasbarri, Stefan Kebekus and Gianluca Pacienza
Post-docs: Lionel Darondeau, Sergei Kovalenko and Tommaso Matteini

The attraction of mathematicians towards Diophantine equations dates as long as the Greeks but renewed in the xxth century with the geometrization of this subject. In the case of curves (which, roughly speaking corresponds to diophantine equations in two variables), a geometric invariant, the Euler characteristic, describes qualitatively the solutions in the ring of integers of a number fields (i. e. the set of integral numbers):

• when it is negative, solutions are ultimately dense and come in abundance
• solutions are still ultimately dense when the Euler characteristic is zero, but they are much sparser (logarithmic growth);
• on the opposite, there are only finitely many solutions when the Euler characteristic is positive.

The Euler characteristic also discriminates the analysis of these curves: it explains for the existence of holomorphic functions from the whole complex plane, or of meromorphic function from the Riemann sphere, valued in the given Riemann surface.
All of these questions concerning curves reveal profound interactions between arithmetics, geometry and analysis of curves and have been fully understood in the xixth and xxth centuries (theorems of Mordell–Weil, Siegel, Faltings, theorems of Picard, Nevanlinna theory).
Problems in higher dimensions are the subject of intense activity. The results are much more sparse, but there is a reasonably well-established philosophy : the conjectures of Lang–Vojta, Campana, and Green–Griffiths. However, the picture still remains largely conjectural; for example, despite important partial results, none of the cases of the above trichotomy is fully understood. In the actual state of knowledge, most geometers have split their forces for a long time: some study arithmetic, other the geometry of rational curves, other the geometry and the classification of varieties.
This project aspires to be the meeting point of some mathematicians all working in algebraic geometry, but whose interests and expertise cover all of these three aspects, united in the same philosophy and ultimately similar technical needs.

The main themes of the project are:

• The arithmetic of higher dimensional projective varieties over function fields.
• The geometry of rational curves in higher dimensional varieties and its relationship with classification. 
• The geometry of the affine varieties and its relationship with classification problems.