*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.

*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..

Quantization theory, using techniques from Lie groupoids and Lie algebroids and from -algebras. Superselection sectors and symmetry in low-dimensional quantum field theory, using von Neumann algebras and braided monoidal categories. Theory of quantum groupoids. Quantization of anomalous gauge theories, particularly involving actions of loop groups.

Reduction from non-equivariant symplectic actions of loop groups. q-Hamiltonian reduction formalism. Geometric quantization of moduli spaces of flat connections on Riemann surfaces with boundary.

*Symplectic Geometry*

The central themes of this programme are symplectic and contact geometry, spaces of non-positice curvature, differential geometry in complex vector bundles, and the arithmetic and geometric study of algebraic varieties. One of our aims is to strengthen the interaction with the theoretical physics group around prof. P. van Baal, principally in the areas of symplectic geometry and gauge theory.