An artinian graded algebra, $A$, is said to have the weak Lefschetz property (WLP) if multiplication by a general linear form has maximal rank in every degree. A vast quantity of work has been done studying and applying this property, touching on numerous and diverse areas of algebraic geometry, commutative algebra and combinatorics. Amazingly, though, much of this work has a ``common ancestor" in a theorem originally due to Stanley, although subsequently reproved by others. In this paper we describe the different directions in which research has moved starting with this theorem, and we discuss some of the open questions that continue to motivate current research.

There is a natural epimorphism from the symmetric algebra to the Rees algebra of an ideal. When this epimorphism is an isomorphism, we say that the ideal is of linear type. Given two determinantal rings over a field, we consider the diagonal ideal, kernel of the multiplication map. We prove in many cases that the diagonal ideal is of linear type and recover the defining ideal of the Rees algebra. In our cases, the special fiber rings of the diagonal ideals are the homogeneous coordinate rings of the join varieties.

We consider the non-positivity of the Hilbert coefficients for a parameter ideal of a commutative Noetherian local ring. In particular, we show that the second Hilbert coefficient of a parameter ideal of depth at least $d-1$ is always non-positive and give a condition for the coefficient to be zero. With the added condition that the depth of the associated graded ring also be at least $d-1$ we show $e_{i}(q) \leq 0$ for $i=1, \ldots, d$.

Understanding the complement of the co\-amoeba of a (reduced) $A$-discriminant is one approach to studying the monodromy of solutions to the corresponding system of $A$-hypergeometric differential equations. Nilsson and Passare described the structure of the coamoeba and its complement (a zonotope) when the reduced $A$-discriminant is a function of two variables. Their main result was that the coamoeba and zonotope form a cycle which is equal to the fundamental cycle of the torus, multiplied by the normalized volume of the set $A$ of integer vectors. That proof only worked in dimension two. Here, we use simple ideas from topology to give a new proof of this result in dimension two, one which can be generalized to all dimensions.

We construct explicitly a minimal free resolution of the defining ideal of an orbit closure arising from a representation of the non-equioriented $A_3$ quiver. The resolution is a generalization of Lascoux's resolution for determinantal ideals.

The case of non-equioriented $A_3$ quiver is made special by the fact that, in this case, every orbit closure admits a so-called $1$-step desingularization. Using the resolution we give a description of the minimal set of generators of the defining ideal. The resolution also allows us to read off some geometric properties of the orbit closure, like normality and Cohen-Macaulay. In addition, we give a characterization for the orbit closure to be Gorenstein.