6 problems

Let $q$ be a power of an odd prime, let $U_n(q)$ be the unitriangular group of degree $n$ over the finite field ${\mathbb F}_q$, and let $N_{n,e}(q)$ denote the number of irreducib…

Let $q$ be a power of an odd prime, let $U_n(q)$ be the unitriangular group of degree $n$ over the finite field ${\mathbb F}_q$, and let $e\in {\mathbb N}$. Lehrer's conjecture. Th…

Let $n\geqslant 2$ and let … Landesman's conjecture. There exists $m\geqslant 0$ such that … The conjecture predicts that, for every fixed degree $n\geqslant 2$, the quadratic poly…

Let $Q$ be an acyclic quiver and let $\mathbf{d}\in\mathbb{Z}_{\geq 0}^{Q_0}$. For each $\mathbf{e}\leq\mathbf{d}$, let $[Q,\mathbf{e}]$ denote the relevant commuting-pair count in…

Let $Q$ be an acyclic quiver, let $\mathbb{k}=\mathbb{F}_q$, and let $P\in\operatorname{rep}Q$ be a projective representation over $\mathbb{k}$. Consider the radical…

Van der Waerden's conjecture. For $n\geqslant 3$, we have