# Forschung/Research

I consider myself as a geometer. The central objects of modern geometry are manifolds. These are spaces which locally look like Euclidean spaces. They are models for most physical spaces. For example, the surface of the earth is a 2-dimensional manifold called the 2-sphere. I am mostly interested in the interplay between local and global aspects of manifolds. In geometry, one adds more information to a manifold, namely a metric which allows to speak about the distance between two points. In most of my research the metric does not play a role. But it is almost always in the background.

I am particularly interested in the classification of manifolds. This tries to single out certain characteristics of a manifold, ideally in such a way that they determine the manifold completely. The classification of all manifolds is impossible (in the strict sense it is an unsolvable problem). I always was attracted by classes of manifolds which occur in "daily life", e.g. in algebraic geometry as solutions of certain polynomial equations, in differential geometry as manifolds with special curvature properties or in physics. The most important tools for classification are homology theories. These are very delicate algebraic measures for the complexity of a space. Apart from using for this the established homology theories, I have constructed various new homology theories.

Recently, I was particularly attracted by a problem which is around for several decades, namely the question whether manifolds admitting a symmetry are less typical than those admitting no symmetry. On the one hand I am interested in constructing examples of asymmetric manifolds and on the other hand I am thinking about a precise formulation of the above vague statement.