5 problems

QF-1 implication conjecture. If

QF-1 characterisation conjecture. The following conditions are equivalent:

Let $n\geq 2$ and let $k$ satisfy $2\leq k\leq 2n-2$. A Nakayama algebra is a finite-dimensional algebra whose indecomposable modules are uniserial; it is connected when its underl…

Let $n\geq 2$ and let $k$ satisfy $2\leq k\leq 2n-2$. A connected Nakayama algebra is a connected finite-dimensional Nakayama algebra, and $\operatorname{gldim}(A)$ denotes its glo…

Let $n$ and $r$ be integers with $2\leq r\leq 2n-2$. Consider Nakayama algebras with $n$ simple modules and global dimension $r$. Existence conjecture. For every such $n$ and $r$,…