  Let $d\ge 3$, $n\ge 2$. The object of our study is the morphism $\Phi$,
introduced in earlier articles by J. Alper, M. Eastwood and the author, that
assigns to every homogeneous form of degree $d$ on ${\mathbb C}^n$ for which
the discriminant $\Delta$ does not vanish the so-called associated form, which
is a form of degree $n(d-2)$ on the dual space. This morphism is
${\mathrm{SL}}_n$-equivariant and is of interest in connection with the
well-known Mather-Yau theorem, specifically, with the problem of explicit
reconstruction of an isolated hypersurface singularity from its Tjurina
algebra. Letting $p$ be the smallest integer such that the product
$\Delta^p\Phi$ extends to the entire affine space of degree $d$ forms, one
observes that the extended map defines a contravariant. In the present paper we
survey known results on the morphism $\Phi$ as well as the contravariant
$\Delta^p\Phi$, and state several open problems. Our goal is to draw the
attention of complex analysts and geometers to the concept of the associated
form and the intriguing connection between complex singularity theory and
invariant theory revealed through it.
