8 problems

Let a partial matrix have a bipartite graph whose vertices represent its rows and columns and whose edges represent specified entries. A fully specified submatrix is one in which e…

Average-rank conjecture. One has

Order of rank bias. If $r\geq0$ and $\mathcal E_r$ is infinite, then there exists $\delta>0$ such that, for every $\epsilon>0$, if $\phi(N)\ll(\log N)^{\delta-\epsilon}$ then…

Let $B_n$ be an $n\times n$ random sign matrix with independent entries taking the values $1$ and $-1$ with probability $1/2$ each, and let $k$ be any positive integer. Rank conjec…

Let a binary form of degree $kd$ be expressed as a sum of $k$-th powers of binary forms of degree $d$, and let $\mathrm{rk}^{\max}_k(2,kd)$ denote the maximum of the resulting $k$-…

Wilton's inclusion/exclusion Hanna Neumann conjecture. For all such $H$ and $K$,

Finite-rank conjecture. Suppose that $p_1$ is minimal in the sense of $L_1$ and ACFA. If $p_1$ is grouplike in the sense of $L_1$ and ACFA, then $p$ has finite rank in the sense of…

Andrews' smallest-parts moment inequality. For $k$ even, $k\geq2$, and $n\geq1$,