18 problems

Dyson's rank equidistribution conjecture. For all nonnegative integers $n$, one has

Beck's conjecture. The quantity $e(n)$ equals the total number of distinct parts of all partitions of $2n+2$ with rank $n+1$.

Mao's conjectured inequalities. Computation evidence suggests that

Let $\mathbb{K}$ be a finite field, let $d\geq 2$ be an integer, and let … Here $\operatorname{PR}_{\mathbb{K}}(f)$ is the partition rank and $\operatorname{AR}_{\mathbb{K}}(f)$ is…

Let $\mathbb{K}$ be a field, let $d\geq 2$ be an integer, and let … Here $\operatorname{PR}_{\mathbb{K}}(f)$ denotes the partition rank of $f$, and $\overline{\mathbb{K}}$ is an al…

Analytic rank–partition rank conjecture. For every $d\geq 2$, there is a constant $C_d$ such that

For a multilinear map, let the partition rank measure its decomposition complexity and let the analytic rank measure its randomness over a finite field. Gowers–Wolf conjecture. The…

Adiprasito–Kazhdan–Ziegler conjecture. The bound in Theorem $$ should hold with linear bounds; equivalently, the relevant strength and partition-rank quantities over $K$ should be…

Lovett–Adiprasito–Kazhdan–Ziegler conjecture. There exists a constant $C_d$ such that for any finite field and multilinear form $f$ we have

Partition-rank conjecture. There exists a constant $C_{p,d,S}^{\operatorname{pr}}>0$ such that, for every such $m$, if

Let $d\ge2$ be an integer and let $R$ be a non-empty family of partitions on $\lbrack d\rbrack$. For an order-$d$ tensor $T$, let $\operatorname{d\mathit{R}rk}T$ denote its disjoin…

Let $d\ge2$ be a positive integer and let $R$ be a non-empty family of partitions on $\lbrack d\rbrack$. For every positive integer $l$, let $\operatorname{\mathit{R}rk}$ denote th…

Let $d\ge2$ be an integer and let $R$ be a non-empty family of partitions on $\lbrack d\rbrack$. Let $\operatorname{\mathit{R}rk}$ denote the rank associated with $R$. Linear bound…

Let $d \ge 2$, $s \ge 1$ be positive integers and let $R$ be a non-empty family of partitions on $\lbrack d \rbrack$. For order-$d$ tensors $T_1,\dots,T_s$ and $a\in\mathbb{F}^s$,…

Let $G$ be a vector space over $\mathbb{F}_p$, and let $alpha\colon G^k\tomathbb{F}_p$ be an $r$-approximately symmetric multilinear form, meaning that the partition rank of…

Symmetrization conjecture. If

Constant-factor equivalence conjecture. For every $k ge 2$ there is a constant $C=C(k)$ such that, for every finite field $F$ and every $k$-tensor $T$ over $F$,

Let $\ell>3$ be prime and let $d$ satisfy … For residues $r_1$ and $r_2$ modulo $\ell$, consider the rank-difference generating function … where $N(r,\ell,n)$ counts partitions of…