Methods of combinatorial geometry in action (Q1818423)

This is an elaboration of the author's inaugural lecture at the Technical University of Berlin. By means of the following three examples he shows how methods of combinatorial or discrete geometry may be applied in other fields of mathematics. (1) Algebraic geometry: triangulations of lattice polytopes lead to the resolution of toric singularities. (2) Algebraic topology: the combinatorics and cohomology of arrangements are found in the analysis of the configuration spaces of spheres. (3) Complex function theory: circle packings may help to understand and construct conformal mappings. There are several very instructive figures. In particular, the last example is illustrated by a beautiful ``proof by pictures''.