Programme leaders: G.B.M. van der Geer, A. van de Ven
The research in this program deals with the study of algebraic and analytic varieties and with mathematical physics. Central themes in the research activity are: varieties of small dimension (curves, surfaces, threefolds), moduli spaces, complex and real algebraic geometry, quantum field theory and string theory. In this field the Netherlands possesses an important tradition and enjoys a strong international position and there are renowned research groups at UvA, VU and RUL.
Although algebraic and analytic geometry were
central fields in pure mathematics, recently important new
and surprinsing applications of these fields have been found in
various areas in and outside mathematics. New developments in the
mathematical aspects of quantum field theory have led to an
ongoing series of new revolutionary connections between algebraic
geometry and theoretical physics and these turn out to be very
fruitful for both fields. New applications of curves over finite
fields have been found in coding theory, cryptography and financial
mathematics.
Central research themes are:
Real Algebraic Geometry
Here one studies
algebraic varieties with the real numbers as base field. Special
interest was given to mappings between real algebraic varieties,
cycle and homology classes in real algebraic geometry, the
underlying real structure of complex varieties and the study of
complete intersections in differential topology and analytic
geometry.
Curves and Abelian Varieties and their Moduli
Special attention was devoted to the
determination of natural cycle classes on moduli spaces in
positive characteristic and to complete subvarieties. Also moduli
spaces of vector bundles on abelian surfaces were studied. Another
focus was the topic of curves over finite fields. Here research was
directed to finding curves with many rational points. These are
relevant for coding theory, cryptography and low-descrepancy
sequencies.
Mathematical Physics
The main focus is the
mathematical aspects of quantum field theory. There is special
emphasis on the relations with algebraic geometry via such topics
as quantum cohomology, mirror symmetry, moduli of curves and
Riemann surfaces, topological field theory and manifold invariants.
Research focuses on (non)-perturbative string theory, conformal
field theory and gauge theories.
Complex Algebraic Varieties
Special attention was paid to moduli of vector bundles on surfaces
in projective three-space, and to Higgs bundles on non-Kaehler
varieties. Other foci were Noether-Lefshetz properties of general
hypersurfaces and holomorphic maps between complex varieties, in
particular those of dimension three.