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In the 1990ies Vassiliev applied his spectral sequence to the study of knot and link invariants in 3-space, obtaining what he called finite type invariants.

The role of these invariants (now known as Vassiliev knot invariants) in the algebra of all invariants is that of polynomials in the algebra of smooth functions.

It puts the so-called polynomial invariants, including most up-to-date quantum ones, into a framework of very general nature, extending our knowledge of the very essence of these invariants, and changing the settings of the problems dramatically.

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Among other results of significant interest by Vassiliev, one should mention the most general formulas of Picard–Lefschetz type for the ramification of cycles on singular complex varieties, his study of the algebraicity of the potential of an algebraic hypersurface in Euclidean space, the generalization of Newton's nonquadrability theorem to the multidimensional case.

A great achievement is the explicit computation of Petrovskii characteristic class for simple singularities, which solves the Atiyah–Bott–Grding problem on lacunas of hyperbolic partial differential equations.

Vassiliev constructed, under very general assumptions, a spectral sequence computing the corresponding homology and yielding powerful invariants.

He developed the stability theory for the cohomology of complements to discriminants.




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