Alexander Isaev (Australian National University, Canberra)
In our recent articles joint with M. Eastwood and J. Alper, it was conjectured that all rational GL(n)-invariant functions of homogeneous forms of degree d>2 on complex space C^n can be extracted, in a canonical way, from those of forms of degree n(d-2) by means of assigning to every form with nonvanishing discriminant the so-called associated form. While this surprising statement is interesting from the point of view of classical invariant theory, its original motivation was the reconstruction problem for isolated hypersurface singularities, which is the problem of finding a constructive proof of the well-known Mather-Yau theorem. Settling the conjecture is part of our program to solve the reconstruction problem for quasihomogeneous isolated hypersurface singularities. In my talk, I will give an overview of the recent progress on the conjecture. If time permits, I will further discuss the morphism that assigns to a nondegenerate form its associated form. This morphism is rather natural and deserves attention regardless of the conjecture. In particular, it leads to a natural contravariant.