PREFACE

In 1983,1 began work on my dissertation, "Families of Hypersurfaces with One-dimension-

al Singular Sets", at Duke University. In that paper, I attempted to describe two numbers

which one could effectively calculate from the defining equation of a hypersurface with a

one-dimensional singular set - two numbers which control the topology and geometry in a

similar fashion to how the Milnor number controls the topology and geometry for isolated

hypersurface singularities.

Since that time, my work has centered around finding numerical data which "control"

various topological and geometric properties of complex analytic singularities.

In 1987, while at The University of Notre Dame, I defined the Le cycles and the Le

numbers for non-isolated hypersurface singularities. The Le numbers are a generalization of

the Milnor number, and they control the singularities; the constancy of the Le numbers in

a family implies the constancy of the Milnor fibres in the family, and also implies Thorn's

df condition holds. My work on Le numbers from 1987 to the present is contained in my

recent monograph, Le Cycles and Hypersurface Singularities.

In 1988, I came to Northeastern University, and was immediately asked by Terry Gaffney

how to generalize the Le numbers of a hypersurface to the case of complete intersections. My

answer to this was that I thought the generalization would have two distinct pieces: the first

piece should be a method for associating numbers to an arbitrary constructible complex of

sheaves on a complex analytic space - one should recover the Le numbers of a hypersurface

by applying this new method to the complex of vanishing cycles of the defining equation

of the hypersurface. The second piece should be to decide what complex of sheaves should

play the role of the sheaf of vanishing cycles in the case of a complete intersection.

This first piece - finding a method for associating numbers to a constructible complex of

sheaves - is described in my 1994 paper, "Numerical Invariants of Perverse Sheaves". As the

title indicates, a number of results for Le numbers only generalize nicely in the case where

the underlying complex is actually a perverse sheaf.

After this first piece was completed, it became apparent that the second piece of the

generalization was not to replace the sheaf of vanishing cycles by some other complex.

Rather, one should continue to use the vanishing cycles a function, but the function could

now have an arbitrarily singular domain. This changed the problem to one of finding a

sufficiently algebraic characterization of the vanishing cycles - one that actually allows one

to effectively produce the numbers that should control the singularities.

The first part of such an algebraic description of the vanishing cycles appears in my paper

"Hypercohomology of Milnor Fibres", and the final piece appears in the paper "Critical

Points of Functions on Singular Spaces".

Having completed all of the above pieces, I thought it would be a relatively simple matter

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