George Mouselli: Apolarity Theory and Macaulay's Theorem

In this talk, we mainly consider two polynomial rings in the same number of variables with coefficients over the same field k, where we want to think of the polynomials in one ring as partial differential operators acting on the polynomials in the other. We first introduce Apolarity Theory and Inverse Systems, and how this can be used as an efficient method to compute the Hilbert function of quotient rings of polynomial rings. The final aim of the talk is to explain the idea of Artinian Gorenstein rings along with explicit examples, and how Macaulay's theorem (1916) provides a complete characterization of these rings.