Title: The Milnor algebra of a smooth hypersurface
Abstract: I will recall the role that the Jacobian ideal of a smooth homogeneous form plays in the deformation theory of the corresponding hypersurface (or the hypersurface singularity). I will explain how Macaulay's theorem and Koszul complex enter the picture on the algebraic side. On the geometric side, if time permits, I will sketch Donagi's generic Torelli theorem for hypersurfaces, and Mather-Yau theorem for isolated hypersurface singularities.
Title: Stability of associated forms
Abstract: In a recent joint work with Alexander Isaev, we proved that the Macaulay inverse system of the Milnor algebra of a smooth hypersurface is GIT polystable. I will explain the proof of this (and in fact of a slightly more general) result. I will then present some applications, including a purely invariant-theoretic Mather-Yau theorem for isolated homogeneous hypersurface singularities, and a criterion for direct sum decomposability of smooth forms