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Abstract:
Equisingularity is the study of families of sets and mappings.
It seeks conditions ensuring that the members of the family are similar
in some way. Many of the most important equisingularity conditions can be
phrased in terms of limits of linear spaces. These conditions in turn can
be controlled by the theory of integral closure of modules. If the multiplicities
of the relevant modules are well defined, as is the case for families of complete
intersections with isolated singularities (ICIS), then the multiplicity can
be used to control the integral closure of these modules and hence the related
equisingularity condition. In this talk we describe how the multiplicity
of a pair of modules can be used to control equisingularity conditions for
families of spaces in which the multiplicity of the relevant modules is not
defined. For simplicity of exposition, we will describe these ideas in
the context of families of hypersurface singularities, where the singular
locus is an ICIS.
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