Lefschetz properties for ideals of powers of linear forms

If $R/I$ is an artinian algebra over an infinite field $K$, and if $L$ is a general linear form, then the Lefschetz question asks for which $k$ does the multiplication map $\times L^k : [R/I]_{j-k} \rightarrow [R/I]_j$ have maximal rank for all $j$? If this is true for $k=1$ then we say $R/I$ has the Weak Lefschetz Property (WLP). If it is true for all $k$ then we say that $R/I$ has the Strong Lefschetz Property (SLP). But it can happen that this holds for some $k$ but not others. At the end of a paper from 2010, Mir\’o-Roig, Nagel and I gave some computer evidence suggesting that the question might be an interesting one for ideals generated by powers of general linear forms. This generated a number of papers by ourselves and others. Now the situation is largely understood for WLP and few variables, and largely wide open for more variables or higher $k$. I’ll give an overview of the current state of affairs and try to indicate some interesting open directions. I’ll also describe some of the methods that have been used in the study of this problem, especially a duality result of Emsalem and Iarrobino to translate the problem to a study of fat points in projective space.