13 problems

Let $\omega$ denote the exponent of matrix multiplication, equivalently the asymptotic exponent governing the border rank of the matrix multiplication tensors. Exponent-two conject…

Valiant's conjecture. Valiant's conjecture for $R$ is

Simultaneous double product property conjecture. For arbitrarily large $n$, there exists an abelian group $H$ with $n$ pairs of subsets $A_i,B_i$ satisfying the simultaneous double…

Strong USP capacity conjecture. The strong USP capacity equals

Let $\Pi_d$ be the polynomial set associated with degree-$d$ approximation conditions, and let $\overline{\Pi^*_{2^7}}$ denote the closure of the polynomial set obtainable with sev…

Let $\Pi_d$ be the polynomial set associated with degree-$d$ approximation conditions, and let $\overline{\Pi^*_{2^6}}$ denote the closure of the polynomial set obtainable with six…

Let $m>2$ and let $\Pi^*_{2^m}$ denote the polynomial set associated with evaluation schemes using $2^m$ matrix-matrix multiplications. Numerical computations suggest that generic…

Let $\mathrm{val}(\mathbb Z_n)$ denote the value parameter for the cyclic group $\mathbb Z_n$, as defined in the paper. The near-linear value conjecture. For every $\varepsilon>0$,…

Let $G$ be a finite abelian group, and let $(A_i,B_i)_{i=1}^n$ be pairs of subsets satisfying the simultaneous double product property (SDPP): for each $i$, the equation…

Let $[n]=\{1,\ldots,n\}$, and let $S\subseteq[n]^2$ be skew corner-free, meaning that $S$ contains no three points of the form … with $\delta\ne0$. The skew corner-free set conject…

Let $\mathcal{A}$ be an algorithm of length $r$ computing the matrix multiplication tensor $\langle m,n,p\rangle$. Its automorphism group is denoted by $Aut(\mathcal{A})$. Let…

Let $sM_{\langle 3\rangle}$ denote the skew matrix multiplication polynomial of size $3$, and let ${\bold R}_s$ denote its symmetric rank, namely the least number of cubes of linea…

Commutative coherent configuration conjecture. There exist commutative coherent configurations ${\cal C}_n$ realizing $\langle n,n,n\rangle$ and of rank $n^{2+o(1)}$.