Docent lecture, Nancy Abdallah

Abstract: Inspired by the Lefschetz hyperplane theorem and the Hard Lefschetz theorem in algebraic geometry, Lefschetz properties reveal remarkable hidden structures in graded Artinian algebras. In the past few decades, the study of the Lefschetz properties in the algebraic setting has blossomed, resulting in new connections across several fields of mathematics.

In this talk, I explore the weak and strong Lefschetz properties of graded Artinian Gorenstein algebras and explain how they impose strong constraints on their Hilbert functions, including unimodality. A central theme is to understand when these properties hold and why they fail.

I will present an overview of recent developments in this area including my own recent results and will conclude with a selection of open problems.