7 problems

Let $S$ be a polynomial ring in $n$ variables, let $I\sim\mathcal{I}(n,D,p)$, and consider the random Artinian algebra … where $n\geq 3$. Set $p=1/D^2$. The vanishing WLP probabili…

Let $a\geq b\geq c$, set $d=a+b+c$, and define … The weak Lefschetz failure conjecture. The algebra $R_{a,b,c}$ fails the weak Lefschetz property if and only if both of the followi…

SLP classification conjecture. The equigenerated monomial almost complete intersections of codimension three with the SLP are precisely those given by

Let $\mathbb{k}[x_1,\ldots,x_n]$ be a polynomial ring. For a positive integer $u$ with $\binom{n}{2}<u\leq\binom{n+1}{2}$, let $\operatorname{Lex}(u)$ and…

Characteristic bound conjecture. Then $R/I$ has the weak Lefschetz property in every characteristic $p>\frac{1}{2}(a+b+c)$.

The odd axial-puncture criterion.

Smoktunowicz–Bell conjecture. Every prime monomial algebra $A$ is either PI, primitive, or has nonzero Jacobson radical.