This paper is dedicated to those administrators of research who realize how much damage is done by the evaluation of mathematical research solely by the rankings of the journals in which it is published, or more generally by bibliometric indices. We hope that their lucidity will become widespread in all countries. AbstractThis text presents several aspects of the theory of equisingularity of complex analytic spaces from the standpoint of Whitney conditions. The goal is to describe from the geometrical, topological, and algebraic viewpoints a canonical locally finite partition of a reduced complex analytic space X into nonsingular strata with the property that the local geometry of X is constant on each stratum. Local polar varieties appear in the title because they play a central role in the unification of viewpoints. The geometrical viewpoint leads to the study of spaces of limit directions at a given point of X ⊂ C n of hyperplanes of C n tangent to X at nonsingular points, which in turn leads to the realization that the Whitney conditions, which are used to define the stratification, are in fact of a Lagrangian nature. The local polar varieties are used to analyze the structure of the set of limit directions of tangent hyperplanes. This structure helps in particular to understand how a singularity differs from its tangent cone, assumed to be reduced. The multiplicities of local polar varieties are related to local topological invariants, local vanishing Euler-Poincaré characteristics, by a formula which turns out to contain, in the special case where the singularity is the vertex of the cone over a reduced projective variety, a Plücker-type formula for the degree of the dual of a projective variety. arXiv:1607.07979v3 [math.AG]

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We study the behavior of limits of tangents in topologically equivalent spaces. In the context of families of generically reduced curves, we introduce the $s$-invariant of a curve and we show that in a Whitney equisingular family with the property that the $s$-invariant is constant along the parameter space, the number of tangents of each curve of the family is constant. In the context of families of isolated surface singularities, we show through examples that Whitney equisingularity is not sufficient to ensure that the tangent cones of the family are homeomorphic. We explain how the existence of exceptional tangents is preserved by Whitney equisingularity but their number can change.

We introduce a higher-order version of the tangent map of a morphism and find a matrix representation. We then apply this matrix to solve a conjecture by T. Yasuda regarding the semigroup of the higher Nash blowup of formal curves. We first show that the conjecture is true for toric curves. We conclude by exhibiting a family of non-monomial curves where the conjecture fails.

Let (X, 0) be a reduced, equidimensional germ of analytic singularity with reduced tangent cone (CX,0, 0). We prove that the absence of exceptional cones is a necessary and sufficient condition for the smooth part X 0 of the specialization to the tangent cone ϕ : X → C to satisfy Whitney's conditions along the parameter axis Y . This result is a first step in generalizing to higher dimensions Lê and Teissier's result for hypersurfaces of C 3 which establishes the Whitney equisingularity of X and its tangent cone under this conditions.