Isolated hypersurface singularities and associated forms (Alexander Isaev - Australian National University, Australia)

Let <nobr>d≥3</nobr>, <nobr>n≥2</nobr>. The object of our study is the morphism <nobr>Φ</nobr>, introduced by J. Alper, M. Eastwood and the speaker, that assigns to
every homogeneous form of degree <nobr>d</nobr> on a complex <nobr>n</nobr>-dimensional space for
which the discriminant <nobr>Δ</nobr> does not vanish a form of degree <nobr>n(d−2)</nobr> on
the dual space, called the associated form. This morphism is <nobr>\SLn</nobr>-
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. In this talk I
will survey known results on the morphism <nobr>Φ</nobr> 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.