13 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…

Parity-controlled rank conjecture. Outside a subset of $\mathbb{Q}$ of density zero, $r_t$ is either $r$ or $r+1$, with the choice determined by the parity given by…

Modified Nagao conjecture. The source suggests

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 $p$ be a fixed prime. For each of the families of rational elliptic curves $\mathcal E_{\mathrm{pr}}$, $\mathcal E_{\mathrm{all}}$, and $\mathcal E_{\mathrm{ht}}$, let…

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…

Rank conjecture. For sufficiently large $n$, all such submatrices have, with high probability, rank at least

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 $k$-…

SU-rank one generalization conjecture. One should have

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…

Let $E$ be an elliptic curve over $\mathbb{Q}$, let $r$ be its rank, and let $w\in\{\pm1\}$ be its root number, defined as the sign of the functional equation of $L(E,s)$. The pari…

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