MA 715, Hyperplane Arrangements.
A hyperplane arrangement
a collection of codimension 1 subspaces
in an n-dimensional vector space.
Already you can ask a number of questions
about a hyperplane 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.
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
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.
Peter Orlik and Hiroaki Terao,
Arrangements of Hyperplanes,
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
The topology of the complement of arrangements
Coxeter groups and reflection arrangements
Other topics, as time permits.
PREREQUISITES: A course in linear algebra. Knowledge of
algebraic topology (homology, cohomology, ...) is useful, but