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MIT Combinatorics Seminar

Affine and toric hyperplane arrangements

Margaret Readdy  (University of Kentucky, visiting MIT)

Wednesday, February 14, 2007    4:15 pm    Room 2-136


ABSTRACT

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.