I will give a non-technical survey over the Farrell-Jones Conjecture and its application. It aims at the computation of the algebraic K- and L-groups of group rings of discrete groups, which are computed in terms of equivariant homology groups of certain classifying spaces. It has many applications to prominent problems in topology, geometry, and algebra. Examples are the Borel Conjecture, which predicts the topological rigidity of closed aspherical manifolds, and the Idempotent Conjecture, which predicts that the complex group ring of a torsionfree group has no non-trivial idempotents. The Conjecture is known for instance for hyperbolic and CAT(0)-groups. Its proof uses methods from equivariant homotopy theory, controlled topology, dynamical systems, and geometric group theory. At the end I will report on a very recent formulation and proof of Arthur Bartels and myself of a version of the Farrell-Jones Conjecture for the Hecke algebra of reductive p-adic groups which is of great interest to the theory of smooth representations of such groups.
We discuss a system of 16 quadratic equations in 24 variables that arises in the study of Lie algebras. The solutions are the Lie algebra structures on a 4-dimensional vector space. There are four irreducible components of dimension 11. We compute their degrees and Hilbert polynomials, and thereby answer a 1999 question by Kirillov and Neretin. This is joint work with Laurent Manivel and Svala Sverrisdottir.
