Introduction to the papers of R. Thom and J. Mather

Abstract

Immediately following the commentary below, the previously published article by R. Thom is printed in its entirety: R. Thom, Ensembles et morphismes stratifiés, Bulletin of the American Mathematical Society 75 (1969), no. 2, 240-284 (French). This is followed by the first publication of the 1970 lecture notes of J. Mather, Notes on topological stability.. This includes the Cantor set, the Sierpiński sponge, the Snowflake, and other sets of fractional Hausdorff dimension. How does one prove that this sort of behavior cannot happen when f is an analytic function or an algebraic function? These questions were approached about 80 years ago when it was shown ([38, 15, 13, 16]) that algebraic sets could be triangulated. 1 For many years the 1932 paper [13] was cited as the only known proof that complex algebraic sets were locally contractible. But these papers are difficult to follow, and for decades the triangulability of algebraic and analytic sets was treated with some suspicion. Later articles, such as [30,3,18] and especially [10,11,8], finally put these questions to rest. However, the interesting structure of the singularities of an algebraic set is not easily2 described using simplices, and people began to look for a more intelligent way to decompose an algebraic set into (fewer, larger) pieces, starting with the nonsingular part.Hassler Whitney struggled with these questions for decades. In 1946 he wrote Complexes of manifolds [40], in which he considered spaces that were glued together out of smooth manifolds much in the way that a cell complex is glued together from cells. In 1957 Whitney showed in [41] that it is possible, in any algebraic set, to choose an open dense nonsingular part such that its complement is an algebraic set of smaller dimension. Therefore this procedure can be repeated so as to give a finite filtration by closed subsets X 0 ⊂ X 1 ⊂ · · · ⊂ X n such that X r+1 is obtained from X r by attaching a (possibly empty) smooth manifold S r+1 := X r+1 − X r of dimension r + 1.One might hope that the resulting decomposition into pieces X = r S r is locally trivial-that any two sufficiently nearby points x, y ∈ S r should have neighborhoods that are isomorphic (in a sense to be made precise below) by an isomorphism (a homeomorphism, or perhaps, a diffeomorphism) that preserves the induced filtrations. In 1962 René Thom made a first attempt in [34] to make precise such a notion of a locally trivial stratification. He proposed that each stratum S should have a 1 In other words, every compact algebraic set is homeomorphic to a finite simplicial complex.

Arc-wise analytic stratification, Whitney fibering conjecture and Zariski equisingularity

“…Stratifications are often considered with extra regularity conditions such as Whitney's conditions (a) and (b) or the (w) condition of Verdier. For more details and insight we refer the reader to [69], [70], [67], [15], [64], [17], [16] and the references therein. Recall that for a real analytic stratification the (w) condition implies the conditions (a) and (b), see [64].…”

Simulating sticky particles: A Monte Carlo method to sample a stratification

“…We make the assumption that for each I ∈ I, Q I has full rank everywhere in M I (Assumption 1.) The set S is a stratification, given some additional mild but technical conditions on the relationship between manifolds 1 [19], [18]. We assume these conditions hold, and refer to S as a stratification hereafter.…”

Support Theorems for Algebraic Maps

“…For a beautiful discussion of this theory, including historical remarks see [25], Part I, Chapter 1, and [22]. Roughly speaking one can decompose every singular algebraic variety Z in a finite disjoint union of nonsingular ones, the strata, in such a way that the variety Z is "equisingular" along every stratum.…”