# Steven Kleiman

MIT

April 20, 2007 from 16:00 to 18:00 (Montreal/Miami time) On location

Colloquium presented by **Steven Kleiman (MIT)**

When should we say that a singularity on one complex algebraic variety is the "same" as that on another? If the two varieties are plane curves, then we have known the answer for eighty years, but in other cases, the notion of equisingularity has been "elusive," as Zariski put it. Yet, if one variety is a special member of an algebraic family and the other a general member, then it is easier to say when; namely, the total space should satisfy the Thom--Whitney conditions. Thus we seek numerical invariants whose constancy across the members of a family is necessary and sufficient for the the Thom--Whitney conditions to hold. For isolated complete-intersection singularities (ICIS), such numerical invariants arise from the Buchsbaum--Rim multiplicity of the column space of the Jacobian matrix; the proof involves the theory of integral dependence. This talk will review equisingularity theory, and describe some recent efforts to extend it from ICIS to more general singularities.

**Address**