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Abstract: Buchsbaum and Rim developed the notion of the multiplicity of a
module which bears their name in the mid-60's. This multiplicity plays an
important role in the theory of equisingularity. Although much studied in
recent years, it is still easy to ask questions whose answers are unknown.
In this talk, we will
discuss the multiplicity of an ideal, then of a module, and connect these
notions to the geometry of polyhedra for monomial ideals and modules. We
give a formula for the Buchsbaum-Rim multiplicity of a module which is a
direct sums of ideals in terms of the "mixed multiplicities" of the
summands. For monomial modules we show this formula relates the volume of
the polyhedron associated with the module to the mixed covolumes of the
polyhedra associated with the summands.
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schenck@neu.edu