The
Research of Gunther Cornelissen
Picture: a dyadic
icosahedron (picture by Bill Casselman from the cover of the
A.M.S. Notices, vol. 52)
Research
Topics
Gunther Cornelissen was born in Gent (Belgium) in 1971. He received his
PhD in 1997 under the supervision of Jan Van Geel (Gent) and
Ernst-Ulrich Gekeler (Saarbrucken). After a 4 year post-doc position at
the Max-Planck-Institute for Mathematics in Bonn and a visiting
position in Leuven, he joined the faculty of Utrecht University, first
as lecturer and since 2007 as full professor.
He works
mainly in algebraic and arithmetic geometry, automorphic forms, logic,
number theory and noncommutative geometry.
In his 1997 PhD (funded by the Belgian NSF), he studied the
distribution of zeros of Eisenstein series for function fields, with
applications to supersingularity of Drinfeld modules. In connection
with this gave criteria for the existence of rational 2-power torsion
points on Jacobians of hyperelliptic curves over finite fields. He also
applied the theory of Castelnuovo-Mumford regularity to rings of
Drinfeld modular forms.
With Fumiharu Kato and Aristides
Kontogeorgis, he has worked on orbifold curve uniformization over
fields of positive characteristic (funded by the Belgian and Dutch NSF
and the Max-Planck-Society). This includes a sharp upper bound on the
number of automorphisms of a Mumford curve in any characteristic, the
solution of the "Hurwitz Group" problem in this situation, a study of
the analytic equivariant deformation theory for Mumford curves, and a
comparison of this to the algebraic theory.
With Fumiharu
Kato, Ariane Mezard and Jakub Byszewski, he has been working on a long
term project (first funded by the Belgian NSF, then the Dutch NSF and a
Franco-Dutch collaboration grant) to completely understand the
deformation theory of weakly ramified group actions on curves, and its
local counterpart. Noteworthy results are the computation of the
(local) versal equicharacteristic deformation functor with Kato, the
mixed-characteristic functor with Mezard, and the proof of universality
for most of those with Byszewski.
With Karim Zahidi,
Thanases Pheidas and Shasha Shlapentokh, he has worked on undecidable
diophantine problems over the rational numbers. With Zahidi, he proved
that the existence of a diophantine model of the integers in the
rational numbers defies a conjecture of Mazur, and he found a
one-universal-quantifier definition of the integers in the rationals,
based on a conjecture about elliptic divisibility sequences. He also
studied diophantine storing and other relations between undecidability
and elliptic curves.
With Oliver Lorscheid, he studied the
theory of toroidal automorphic forms (funded by the dutch NSF). This is
part of a previously dormant approach to the Riemann Hypothesis
initiated by Don Zagier in the 1970's. Results include a study of such
automorphic forms for function fields of class number one, and some
structural results for the space of such forms over number fields,
using multiple Dirichlet series.
With Matilde Marcolli,
Kamran Reihani and Alina Vdovina, he has worked on the relation between
spectral triples (a.k.a. noncommutative Riemannian geometries) and
rigidity phenomena for classical spaces such a Riemann surfaces and
graphs or buildings.
He is currently (funded by Dutch NSF) studying zeta functions in Riemannian and noncommutative geometry, diff-invariant distances between Riemannian manifolds, Anabelian geometry of quantum statistical mechanical systems for number fields (with Marcolli), and Chern-Simons vortices in higher genus (with Akerblom, Stavenga and van Holten).
Latest three publications on arxix being uploaded...
