Autumn/Winter 2001-2002, MWF 12:30-1:20
This course will be an introduction to oriented matroids.
Oriented matroids are combinatorial abstractions of convex polytopes,
directed graphs, real hyperplane arrangements and subspaces
of Euclidean space.
I will start with basic properties which oriented matroids
carry over from the geometric objects they are modeling: duality (Gale
or linear programming duality for polytopes, planar duality
for graphs and orthogonal complement for subspaces), contraction/deletion and
weak maps (small perturbations of a subspace).
I will consider axiomatics.
There turns out to be a surprisingly long list of different axiom
systems which all give rise to oriented
matroids. I will
discuss in particular the topological representation theorem of Folkman,
Lawrence, Edmonds and Mandel which gives the pseudosphere axiomatization.
I will discuss realization spaces. This is the difference between oriented matroids and the
geometric objects they model. This
includes the universality theorem of Mnëv which addresses
the question of what subspaces of the Grassmannian can be associated
to a single oriented matroid as well as theorems about the
computational problem of determining whether the realization
space is empty.
If there is time and interest there are various other
topics: the use of oriented matroids as models for vector bundles and characteristic
classes, the relationship between extension spaces of oriented matroids and
zonotopal tilings or questions about mutations, Baues' conjecture and the
connectivity of the category of oriented matroids.