*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.