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1.2. Algebraic and Analytic Geometry

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.

next up previous contents
Next: 1.3. Topology and Dynamical Up: Algebra and Geometry Previous: 1.1 Number Theory

Fri Mar 20 16:01:06 MET 1998