| The famous Mather-Yau theorem says that two isolated hypersurface singularities are biholomorphically equivalent if and only if their moduli algebras are isomorphic. However, the reconstruction problem of explicitly recovering the singularity from its moduli algebra is open. In the case of quasi-homogeneous hypersurface singularities, Eastwood and Isaev proposed an invariant-theoretic approach to the reconstruction problem. For homogeneous hypersurface singularities, Alper and Isaev gave a geometric invariant theory reformulation of this approach, in which GIT stability of the Hilbert points of the Milnor algebra of the singularity plays a key role. In particular, Alper and Isaev pose several problems concerning GIT stability of the associated form and the gradient point of a homogeneous form, which they recently solved in the binary case and in the case of generic forms in any number of variables. In my talk, I will explain these recent developments. I will then proceed to prove semistability of the gradient point of a semistable form, and semistability of the associated form of a non-degenerate form. This will answer several (but not all) questions of Alper and Isaev. |