Equisingularity theory of complex analytic families

A fundamental goal of complex geometry is to describe the structure of singular sets. If the set is a member of an analytic family, then it is easier to predict when the set's structure is similar to that of most members. In general, the problem seems to be hard. The first step was made by Bernard Teissier (1972) who studied the case of analytic families of hypersurface germs with isolated singularities. Later on, Terry Gaffney and Steven Kleiman (1999) extended Teissier's work to analytic families of germs of isolated complete-intersection singularities or ICIS germs. They managed to develop algebraic methods to describe standard equisingularity conditions in terms of numerical invariants of individual members, rather than the family. These numerical invariants are certain Buchsbaum–Rim multiplicities that arise from the column space of the Jacobian module of the ICIS germ.
Gaffney and Kleiman studied various equisingularity conditions on families of ICIS germs. Specifically, they studied the Thom condition $A_{f}$ and the Whitney condition $W_{f}$ for a fixed function $f$ on the total space $X$. They proved that the constancy of two sequences of Milnor numbers, and the constancy of a single Buchsbaum–Rim multiplicity are necessary and sufficient conditions for the Thom and the Whitney equisingularity conditions to hold.
This talk will report on a current joint work of Steven Kleiman (MIT) and the speaker, which aims to generalize previous results to the case of arbitrary isolated singularities.