MA 715 - Hyperplane Arrangements

MA 715, Hyperplane Arrangements.

A hyperplane arrangement is a collection of codimension 1 subspaces in an n-dimensional vector space. Already you can ask a number of questions about a hyperplane arrangement, such as:

How many chambers (maximal open regions) does this arrangement cut up the plane?
Is there a polytope corresponding to this arrangement?
What is the topology of the complement of this arrangement?

The first question was considered in Zaslavksy's dissertation, which has been hailed as the cornerstone of the study of hyperplane arrangements. Since then the theory of hyperplanes has developed to include many areas of mathematics, including geometry, algebra and topology. Surprisingly many of these results can be reduced to understanding the combinatorial structure of an arrangement.

For the majority of the course we will follow Orlik and Terao's book on hyperplane arrangements, augmented with more recent results discovered within the past decade.

COURSE OUTLINE

Introduction to hyperplane arrangements
The intersection lattice, the lattice of regions and oriented matroids
The characteristic polynomial
Supersolvable and graphic arrangements
The module of derivations
Free arrangements
The topology of the complement of arrangements
Coxeter groups and reflection arrangements
Other topics, as time permits.

TEXTBOOK: Peter Orlik and Hiroaki Terao, Arrangements of Hyperplanes, Springer-Verlag, 1992.

PREREQUISITES: A course in linear algebra. Knowledge of algebraic topology (homology, cohomology, ...) is useful, but not necessary.

http://www.ms.uky.edu/~readdy/715/Hyperplane/