Examples of reconstruction of homogeneous isolated hypersurface singularities from their milnor algebras

By the Mather-Yau theorem, a complex hypersurface germ V with isolated singularity is completely determined by its moduli algebra A(V). The proof of the theorem does not provide an explicit procedure for recovering V from A(V), and finding such a procedure is a long-standing open problem. In a recent joint paper with N. Kruzhilin, we proposed an explicit way for reconstructing V from A(V) up to biholomorphic equivalence under the assumption that the singularity of V is homogeneous, in which case A(V) coincides with the Milnor algebra of V. In the present note, we review this result and give examples of application of our method to the Milnor algebras of simple elliptic singularities.