Effective de Rham Cohomology

Peter Scheiblechner (Hausdorff Center for Mathematics, Bonn)

Thursday 14 June 2012, 15.00, b-it 1.25 (cosec meeting room)

A long standing open problem in computational algebraic geometry is to find an algorithm which computes the topological Betti numbers of a semialgebraic set in single exponential time. There has been recent progress on the corresponding problem over the complex numbers. A fundamental theorem of Grothendieck states that the Betti numbers of a smooth complex variety can be computed via its algebraic de Rham cohomology, which is defined in terms of algebraic differential forms on the variety. In this talk we discuss degree bounds on these differential forms and their importance for the algorithmic computation of Betti numbers. We will start with a moderate introduction to algebraic topology and geometry, and finish with the latest of these results, which is a single exponential degree bound in the case of any smooth affine variety.