10 problems

Minor-equation conjecture. The equations given by the vanishing of every size-$n$ minor of this matrix define the other component or components of the diagonal commutator scheme.

Independence and completeness conjecture. The explicit set of invariants constructed from $M^{\alpha \beta}$ and $N_{\alpha \beta}$ is independent and complete.

Two-sided Hadwin–Larson conjecture. For ${\underline{A}},{\underline{B}}\in M_n^m$, their ${\rm GL}_n$-orbits coincide if and only if

Two-sided Curto–Herrero conjecture. For every ${\underline{A}},{\underline{B}}\in M_n^m$, the ${\rm GL}_n$-orbits of ${\underline{A}}$ and ${\underline{B}}$ coincide if and only if

Let $M_n(F)^ell$ be the variety of $ell$-tuples of $n\times n$ matrices, and let $\mathcal G_n(\ell)$ be the group of automorphisms induced by automorphisms of the free algebra wit…

Determinant conjecture. The determinant $|X(t)|$ can be obtained as a component of the solution of a linear system of differential equations with constant coefficients, whose coeff…

Let $K$ be a field, let $n,m\in\mathbb{N}$, and let $C_{n,m,K}$ be the smallest integer $d$ such that all monomials of degree $d$ in the free non-unital associative algebra…

Bounded-degree generation theorem. The algebra $R^{GL(n)}$ is generated by elements of degree at most $C_{n,d}$.

Bounded-degree generation conjecture. The algebra $R^{GL(n)}$ is generated by its elements of degree at most $C_{n,d}$.

Let $p$ be the characteristic of the ground field, let $R^{S\!p(n)}$ be the invariant ring for $n\times n$ generic matrices under the symplectic group, and let $K_{\infty}$ denote…