*Programme leaders: R.H. Dijkgraaf, G.B.M. van der Geer*

Central research themes are:
*Real Algebraic Geometry.*
The study of algebraic varieties with the real numbers as base field.
Special interest is given to mappings between real algebraic varieties,
cycle classes and homology classes in real algebraic geometry,
approximations of smooth mappings between non-singular real algebraic
varieties by algebraic morphisms. Semi-algebraic sets, complexification
of real algebraic varieties.
*Arithmetic Geometry.*
One studies polynomial equations over the rational numbers or over
the integers. The goal of arithmetic geometry is to understand the relations
between algebraic geometry and number theory. Modular varieties and modular
forms play a key role.
*Algebraic Geometry of Curves and Abelian Varieties.*
The study of moduli spaces of abelian varieties and curves both
in characteristic zero and characteristic p. Also the moduli spaces
of vector bundles and K3 surfaces are studied. Curves and varieties over
finite fields are studied in relation with coding theory.
*Mathematical Physics* The
study of mathematical aspects of quantum field theory and string
theory. Special emphasis on the relations with algebraic geometry,
such as quantum cohomology, mirror symmetry, moduli space of Riemann
surfaces. Topological field theory and manifold invariants.
Nonperturbative string theory, string duality and extended objects
such as D-branes.