This is a non-technical decription of my principal field of research: Complex Algebraic Geometry, i.e the study of algebraic varieties. These are the solutions of a finite set of polynomials. In complex geometry one only looks at complex solutions: the circle x²+y²=1 should be the same as x²+y²=-1 which has no real solutions. Also, in order not to lose solutions at infinity, one looks at homogeneous equations. In our example, replacing x²+y²=1 by x²+y²=z² one adds the two ``circle points'' (1,i,0) and (1,-i,0) at infinity. Finally, self-crossings such as for two intersectiong lines should be avoided. The objects thus described are called ``projective manifolds''. They can be viewed as algebraic objects (solution sets of equations), geometric or topological objects (``manifolds''), and as such one can study them by means of algebraic, topological, differential geometric and complex-analytic tools. The first basic invariant is the number of free parameters, the dimension: our example, the circle is 1-dimensional, essentially described by the angle parameter (in complex geometry this must be replaced by a complex parameter). A circle is an example of an algebraic curve or Riemann surface (its dimension is 1). A variety of dimension 2 is called an algebraic surface. More generally, a complex surface is a complex manifold of dimension 2, i.e it locally looks like an open subset of C²

Within the large field of complex algebraic geometry my specialization comprizes several subfields, the two most important of which are the fields of complex surfaces and Hodge theory.

The classification of algebraic surfaces is largely due to Castelnuovo (1865-1952) and Enriques(1871-1946). Their classification has been extended to include the non-algebraic surfaces by Kodaira (1915-1995). There is much recent activity, including by physicists who are using detailed results about K3-surfaces for string theory (for instance this prepublication).

One of the goals of Hodge theory is to measure and describe what happens if you introduce extra ``parameters'' in the equations. In our example, one might want to vary the radius of the circle. A related question is to describe the subsets of a given variety which themselves are given by supplementary polynomial equations; these are called ``subvarieties, or more generally ``algebraic cycles''. The theory has been named after Sir William Hodge (1903-1975) who boosted research in this area after he formulated a conjecture. In fact it is one of the millenium prize questions posed by the Clay Foundation.

Hodge theory has links to other fields. It has for instance been discovered that Hodge theory is a natural tool to attack the so called ``mirror conjecture'', a conjecture posed by mathematical physicists. See for instance this book.