First, let me recall briefly Vassiliev's theory (the way I understand it). I will skip many important details on the way.

Vassiliev's invariants first appeared as certain cohomology classes of the space of all knots in the 3-space. The strategy is as follows: instead of the space of all knots, which is infinite-dimensional, consider the space $V_d$ of maps from $\mathbf{R}$ to $\mathbf{R}^3$ given by polynomials of some fixed degree $d$ behaving in a prescribed way at infinity. This is a finite dimensional space and non-knots form a hypersurface $\Sigma_d$ in it. Using the Alexander duality one can reduce the computation of the cohomology of the complement of $\Sigma_d$ in $V_d$ to computing the Borel-Moore homology of $\Sigma_d$. [Note: there is a price to pay for replacing an open set with its complement -- we loose the information of how $\Sigma_d$ lies in $V_d$.] The Borel-Moore homology of $\Sigma_d$ can be computed by constructing a suitable semi-simplical resolution of $\Sigma_d$ (i.e. a semi-simplicial space with the geometric realization properly homotopy equivalent to $\Sigma_d$).

The geometric realization of a semi-simplicial space admits a natural filtration, which gives a spectral sequence converging to the Borel-Moore homology of $\Sigma_d$. There is a way to embed $V_d$ into some $V_{d'},d'>d$ so that knots are mapped to knots and unknots to unknots. Moreover, spectral sequences for $d'$ and $d$ are mapped to one another. This corresponds to the restriction map in cohomology, but constructing the maps on the level of spectral sequences is non-trivial. As $d$ goes to $\infty$ a part of the $E^1$ term of the spectral sequence stabilizes. It is a difficult theorem (Kontsevich-Vassiliev) that (at least over the rationals) the second and higher differentials vanish on the stable part and that if an element in the stable part is killed, it is killed in the first sheet by some stable element for all sufficiently large $d$.

So letting $d$ go to $\infty$ we get a vector subspace (in fact, a subring) in the cohomology of the space of all knots (equipped with the Whitney topology), together with a filtration on it. Vassiliev's conjecture says that this subspace is in fact the whole of the cohomology. By taking the $H^0$ part we get Vassiliev invariants and the Vassiliev filtration on them.

The most non-trivial part of the above is the actual construction of the semi-simplicial resolution of $\Sigma_d$ (since this gives a spectral sequence that collapses very fast).

The same strategy can be applied in many other cases, such as spaces of smooth functions without complicated singularities, spaces of mappings from $m$-dimensional CW-complexes to $m-1$ connected ones, spaces of smooth projective hypersurfaces, classical Lie groups etc. It can also be applied to spaces of knots in 3-manifolds other than $\mathbf{R}^3$ and in higher-dimensional manifolds. For $\mathbf{R}^n,n>3$ this gives a complete description of the cohomology: it is isomorphic to the cohomology of the Hochschild complex of the Poisson operad for $n$ odd and of the Gerstenhaber operad for $n$ even, as shown by V. Turchin arXiv:math/0010017. The case $n=3$ is much trickier. Some of these applications can be considered higher dimensional analogs of Vassiliev knot invariants.

Most of the above is described in detail in "Complements of the Discriminants of Smooth Maps" by Vassiliev. A brief synopsis can be found in Vassiliev's ICM 1994 talk.

useof phrases such as "politically incorrect" is often interpreted as an indication that the speaker does not appreciate the power of judicious word choice. $\endgroup$ – S. Carnahan♦ May 6 '10 at 4:18