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Abstract:
Using invariants from commutative algebra to
count geometric objects is a basic idea in singularities. For example, the multiplicity of an ideal
is used to count the points of intersection of two analytic sets at a non-transverse intersection.
A problem with the multiplicity of an ideal or module is that it is only defined for modules and
ideals of finite colength. This is not the case for many situations of interest.
In this talk we will use pairs of modules and their multiplicities as a way around this difficulty.
We will describe the multiplicity-polar theorem which applies to families of pairs of modules,
linking the multiplicity at a general pair of the family to the multiplicity of the pair at
the special fiber. We will apply these ideas to the calculation of a certain intersection number
that
generalizes the Milnor number of an isolated singularity. Variations of this number appear in several different situations, such as:
The theory of differential forms on singular spaces
The theory of D-modules
The description of the Milnor fiber of a function with an isolated singularity
The theory of the Euler invariant, which in turn relates to the theory of characteristic
classes on singular spaces.
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