Equisingularity and the theory of Integral Closure
This short course will study equisingularity, and the related ideas from commutative algebra - closure operations and multiplicities, which are bound up with it. A family of sets or mappings is called equisingular if all of the members of the family “are alike” in some way. Different notions of equisingularity arise from different meanings of “all alike”.
Some of the notions of equisingularity we will study are: constant analytic type (all family members are equivalent by biholomorphic coordinate changes), constant topological type (equivalence by homeomorphisms) constant bi-Lipschitz type (bi-Lipschitz homeomorphisms) and Whitney equisingularity (described below).
As a point moves along a curve, by calculus we associate an infinitesimal object to it at each moment - its velocity vector. If we think of the members of the family as parametrized by time, their evolution gives rise to infinitesimal structures on the members of the family. Whitney equisingularity is a strong form of constant topological type which preserves some of the infinitesimal structure associated to the singular points of the family.
These infinitesimal structures are best studied through modules, and the modules associated to the members of the family are studied using tools from commutative algebra. Often they are defined using a closure operation such as integral closure or Lipschitz saturation, ideas which we will study. If we associate an ideal or module to a member of a family, then algebraic notions such as the multiplicity of an ideal or module can be used to describe the infinitesimal geometry of the members of the family. We will show how multiplicities describe and control the geometry of their associated families, and how geometry provided motivation for the development of these algebraic ideas.
We will study families of functions, hypersurfaces and complete intersections with isolated singularities, and as time allows, hypersurfaces with non-isolated singularities and determinantal singularities which illustrate possible ways of developing the theory where traditional algebraic tools fail.