13 problems

Let $\mathcal{P}_1\in U_1\otimes\cdots\otimes U_n$ and $\mathcal{P}_2\in V_1\otimes\cdots\otimes V_n$ be persistent tensors, where $U_1,\ldots,U_n$ and $V_1,\ldots,V_n$ are vector…

Boundary support-functional conjecture. For every $\theta\in\Theta_{(\varkappa)}$, one has

Tame-wild orbit-closure conjecture. As long as an orbit closure intersection problem is not efficiently solvable using quiver representation techniques, it is…

TOCI-completeness conjecture. These three orbit closure intersection problems are also $\textnormal{\textbf{TOCI}}$-complete.

Let $k\geq 3$ and let $f$ be a $k$-tensor. Denote its asymptotic symmetric subrank by $\operatorname{\widetilde{Q}_{s}}(f)$ and its asymptotic subrank by $\widetilde{Q}(f)$. Asympt…

Exponent-two conjecture. It is generally conjectured that

For $\alpha,\beta\in\mathbb{N}$ with $\alpha\leq\beta$ and $\beta$ even, fix a set $\mathcal U$ of $\beta$ vectors in $\mathbb{C}^{\alpha}$ generically. A Latin Rectangle for…

Let $U$, $V$, and $W$ be finite-dimensional complex vector spaces, and let the polynomials described by the multilinear-algebra construction for equations of the slice-rank variety…

Let ${\cal S}\in\mathrm{S}^d\mathord{\mathbb C}^n$ be a symmetric tensor. Nonsingularity-decision conjecture. With probability $1$, one can decide in time polynomial in $d$ whether…

Fix $n\in\mathord{\mathbb N}$ with $2<n$, and let ${\cal S}\in\mathrm{S}^d\mathord{\mathbb C}^n$ be nonsingular. Let $\mathbf H$ denote the polynomial map whose fixed points are us…

Let $n\geq 3$ be fixed, and let ${\cal S}\in\mathrm{S}^d\mathord{\mathbb C}^n$ be a symmetric $d$-qudit. For most such tensors, consider the spectral norm $\|{\cal S}\|_{\sigma}$.…

The exponent-two conjecture. The exponent satisfies

Let $M_{\langle n,n,n\rangle}$ denote the tensor encoding multiplication of two $n\times n$ matrices, and let $\underline{R}(M_{\langle n,n,n\rangle})$ be its border rank. For ever…