Macaulay inverse systems and computation of cohomology rings

It was observed by Pukhlikov and Khovanskii that the BKK theorem implies that the volume polynomial on the space of polytopes is the Macaulay generator of the cohomology ring of a smooth projective toric variety. This provides a way to express the cohomology ring of toric variety as a quotient of the ring of differential operators with constant coefficients by the annihilator of an explicit polynomial. The crucial ingredient of this observation is an explicit expression for the Macaulay generator of graded Gorenstein algebras generated in degree 1.

In my talk I will explain this construction in detail, then I will tell about recent results on explicit expression for the Macauley generator of an arbitrary algebra with Gorenstein duality. Finally, if time permits, I will show how these results yield to the computation of the cohomology rings of more general classes of algebraic varieties.