15 problems

Let $\mathsf k$ be an infinite field, let $B$ be a nilpotent matrix, and let $\mathfrak D(P)$ denote the maximum Jordan partition arising from a nilpotent matrix commuting with one…

Cyclicity conjecture. If $G_{\mathcal L}$ is finite, then $G_{\mathcal L}$ is cyclic.

Invariant D-module abelianization conjecture. The pull-back induces an isomorphism

Let $n\in\{8,16\}$, and let $L\subset\mathbb{R}_{sym}^{n\times n}$ be a subspace satisfying … and such that $\boldsymbol{\mathrm{P}}(L^\vee)\cap\boldsymbol{\mathrm{Z}}$ exists and…

Let $k$ be a base field, let $R$ be a coefficient ring, and let $\operatorname{GL}_n$ be the general linear group over $k$. Suppose the algebraic analogs $X_m(m,n)$ of the filtrati…

Let $\mathbb G$ be a linear algebraic group over a field of characteristic $p$, let $\mathbb G_{(1)}$ denote its first Frobenius kernel, and let $\mathbb G(\mathbb F_p)$ denote its…

Let $\mathbb{F}$ be a field of characteristic different from $2$, let $V$ be a finite-dimensional vector space over $\mathbb{F}$, and let $b$ be a non-degenerate skewsymmetric bili…

Global-to-local finiteness conjecture. The map

Finiteness conjecture for forms with good reduction. For every reductive algebraic $K$-group $G$, there are only finitely many $K$-isomorphism classes of $K$-forms of $G$ having go…

Degree-zero word reformulation. The image of $w$ on $G$ must be

Let $K$ be an algebraically closed field of characteristic $0$, let $w$ be a group word in $x_1,\dots,x_m$, and let $\operatorname{Im} w$ denote the image of its word map on…

Let $K$ be an algebraically closed field of characteristic $0$, let $w(x_1,\dots,x_m)$ be a nontrivial group word, and let $\operatorname{PSL}_2(K)$ denote the projective linear gr…

Let $G$ be an infinite reductive group over a field of positive characteristic, and let $G^{0}$ denote its connected component. Assume that $G^{0}$ is not a torus. Infinite delta-i…

Truncated affine Grassmannian affine-paving conjecture. The truncated affine Grassmannians for $\mathrm{GL}_{r+1}$ admit affine pavings.

Let $F={\mathbb{C}}(z)$ and let $\phi:F\to F$ be given by $\phi(z)=z+1$. For a linear algebraic group ${\mathcal{G}}$ over ${\mathbb{C}}$, a difference Galois group means the Galoi…