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My main focus is on the applications of stratification in complex/abstract Algebraic geometry especially, from the scheme-theoretic viewpoint and (Added) Moduli spaces.

I have a vague feeling that how decomposing our (topological) space will be useful especially, for studying singularities, also looking at the given references in Wikipedia, it seems that its early applications are related to differential geometry and Morse theory.

I would appreciate any comments for clarifying this construction regarding its necessity and its usefulness.

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Your question seems a little odd to me. Algebraic varieties are by their nature stratified spaces. Think of $xy=0$ in the Euclidean plane, $\mathbb R^2$. –  Ryan Budney May 15 '12 at 4:29
    
@Ryan Budney: I have modified it a little. What I meant is like arxiv.org/pdf/alg-geom/9708014v1.pdf –  Ehsan M. Kermani May 15 '12 at 6:57

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