Most of my research deals with the interplay between geometry, combinatorics and group theory. In this context I study regular and chiral polytopes. In recent research, dealing with the symmetry of geometric figures, regular polytopes are defined as combinatorial objects, which locally have structure of a classical regular polytope or a regular tessellation. Globally regular polytopes may be of topological type other then spherical. The concept, introduced by Grünbaum and Danzer, was influenced by Coxeter's extensive work on regularity of geometric figures. Chiral polytopes are combinatorial objects that have maximal rotational symmetry but are not symmetrical by reflections. This abstract concept generalizes the well-known phenomenon from crystallography, physics and biology. I also study realizations of polytopes. By a realization of a polytope we mean an image of its vertices under a mapping into some Euclidean space with the property that each automorphism of the polytope corresponds to some isometry of the realization.
