David Alkass: The Strong Lefschetz Property and Geometric Aspects of Graded Artinian Gorenstein Algebras

This thesis studies graded Artinian Gorenstein algebras arising from Macaulay inverse systems, with emphasis on Hilbert functions, Lefschetz properties, higher Hessians, and zero-dimensional projective schemes. Given a reduced set of points $X = \{P_1, \dots, P_s\} \subset \mathbb{P}^2$, we consider inverse system generators $F = \sum_{i=1}^s \alpha_i L_i^d$, where the $L_i$ correspond to the points of $X$, and study the associated algebra $A = S/\operatorname{Ann}_S(F)$. Using the Maeno--Watanabe criterion and an explicit Cauchy--Binet expansion of higher Hessians, we prove that if $X$ has maximal Hilbert function and $s\leq 9$, then $A$ satisfies the Strong Lefschetz Property. The thesis also reformulates Hessian vanishing as a toric-linear incidence problem between a hypersimplex-type monomial image and a linear kernel determined by the point configuration.