We extend results proved by the second author (Amer. J. Math., 2009) for nonnegatively curved Alexandrov spaces to general compact Alexandrov spaces $X$ with curvature bounded below. The gradient flow of a geodesically convex functional on the quadratic Wasserstein space $\left( \mathcal{P}\left( X \right),\,{{W}_{2}} \right)$ satisfies the evolution variational inequality. Moreover, the gradient flow enjoys uniqueness and contractivity. These results are obtained by proving a first variation formula for the Wasserstein distance.

For ${\cal X}$ a stratification with suitable regularity, in particular for any Whitney stratification and, via regular embedding, for any abstract stratified set, time-dependent vector fields are used to prove an extension theorem for diffeomorphisms near the identity defined on strata of a given dimension. Then it is shown that after isotopy a stratified map $h\,{:}\,{\cal Z} \,{\longrightarrow}\, {\cal X}$ can be made transverse to a fixed stratified map $g\,{:}\, {\cal Y} \,{\longrightarrow}\, {\cal X}$.

Strata in manifold stratified spaces are shown to have neighborhoods that are teardrops of manifold stratified approximate fibrations (under dimension and compactness assumptions). This is the best possible version of the tubular neighborhood theorem for strata in the topological setting. Applications are given to replacement of singularities, to the structure of neighborhoods of points in manifold stratified spaces, and to spaces of manifold stratified approximate fibrations.

The paper shows that, for subanalytic stratifications, Lipschitz equisingularity as defined by Mostowski is preserved after intersection with generic wings, that is, $L$-regularity implies $L^*$-regularity. This was one of the conditions required of a good equisingularity notion by Teissier in his foundational 1974 Arcata paper.

Previous authors have shown that Lipschitz equisingularity is generic, implies bilipschitz triviality, and hence topological triviality, and implies equimultiplicity.

In this paper, we investigate stratification theory in terms of the defining equations of strata and maps (without tube systems), offering a concrete approach to show that some given family is topologically trivial. In this approach, we consider a weighted version of $\left( w \right)$-regularity condition and Kuo’s ratio test condition.

When a pair of adjacent C1 strata (X, Y) is Whitney (a)- regular at a point , there is, up to homeomorphism, just one germ at 0 of intersections S ∩ X where S is a Lipschitz transversal to Y at 0 (S is the graph of a Lipschitz map from The proof uses a blowing-up construction and a strong form of Verdier's (w)-regularity.