ISBN 978-0-387-943 -(hardcover) ISBN 978-1-4613-8431-1 (eBook) ISBN 978----(softcover)

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A hyperplane arrangement is a finite set of hyperplanes through the origin in a finite-dimensional real vector space. Such an arrangement divides the vector space into a finite set of regions. Every such region determines a partial order on the set of all regions in which these are ordered according to their combinatorial distance from the fixed base region. We show that the base region is simplicial whenever the poset of regions is a lattice and that conversely this condition is sufficient for the lattice property for three-dimensional arrangements, but not in higher dimensions. For simplicial arrangements, the poser of regions is always a lattice. In the case of supersolvable arrangements (arrangements for which the lattice of intersections of hyperplanes is supersolvable), the poset of regions is a lattice if the base region is suitably chosen. We describe the geometric structure of such arrangements and derive an expression for the rank-generating function similar to a known one for Coxeter arrangements. For arrangements with a lattice of regions we give a geometric interpretation of the lattice property in terms of a closure operator defined on the set of hyperplanes. The results generalize to oriented matroids. We show that the adjacency graph (and poset of regions) of an arrangement determines the associated oriented matroid and hence in particular the lattice of intersections.

These lectures on the combinatorics and geometry of 0/1-polytopes are meant as an introduction and invitation. Rather than heading for an extensive survey on 0/1polytopes I present some interesting aspects of these objects; all of them are related to some quite recent work and progress. 0/1-polytopes have a very simple definition and explicit descriptions; we can enumerate and analyze small examples explicitly in the computer (e. g. using polymake). However, any intuition that is derived from the analysis of examples in "low dimensions" will miss the true complexity of 0/1-polytopes. Thus, in the following we will study several aspects of the complexity of higher-dimensional 0/1-polytopes: the doubly-exponential number of combinatorial types, the number of facets which can be huge, and the coefficients of defining inequalities which sometimes turn out to be extremely large. Some of the effects and results will be backed by proofs in the course of these lectures; we will also be able to verify some of them on explicit examples, which are accessible as a polymake database.