Thom Polynomials and Nonassociative Hilbert Schemes

Thom polynomial is the characteristic cohomology class Poincaré dual to the cycle of fixed singularity type of generic differential mapping of smooth manifolds. We review the theory of Thom polynomials, their existence, methods of computations, and applications. We give a special consideration of stabilization of Thom polynomials with the growth of dimensions of manifolds participating in the mapping.
One of the methods of computation uses resolution of the singularity cycles. A particular construction of resolution is provided by the so called nonassociative Hilbert scheme. Using this approach we extend considerably the list of singularities with known Thom polynomials. For example, we are able to compute in a closed form the Thom polynomial of the third-order Thom–Boardman singularity type $\Sigma^{2,2,2}$.