In the 1970's Zaslavsky initiated the modern study of hyperplane
arrangements. For a central hyperplane arrangement he showed
evaluating the characteristic polynomial at -1 gives the number of
regions in the complement of the arrangement, whereas for an affine
arrangement evaluating at +1 gives the number of bounded regions in
the complement. For central arrangements Bayer and Sturmfels proved
its flag f-vector can be determined by the intersection lattice.
Billera, Ehrenborg and Readdy make this map explicit using coalgebraic
techniques.

We extend the Billera-Ehrenborg-Readdy omega map to affine and toric
hyperplane arrangements. For toric arrangements, we also generalize
Zaslavsky's fundamental results on the number of bounded and unbounded
regions. We believe these results hint at a wealth of problems
involving regular subdivisions of manifolds. I will indicate a few of
these directions.

This is joint work with Richard Ehrenborg and Michael Slone.