14 problems

Let $G_0$ be any connected unipotent group over ${\mathbb F}_q$, let $G=G_0\otimes_{{\mathbb F}_q}{\mathbb F}$ with Frobenius $\operatorname{Fr}$, and let…

Let $H$ be an algebraic group over the ground field, and call it easy if every $h$ lies in the neutral connected component of its centralizer. The easy-group characterization conje…

Let $G$ be a connected unipotent algebraic group, let $e$ be a minimal idempotent of ${\mathscr D}_G(G)$, and let its functional dimension be $d_e=(\dim G-n_e)/2$. The integrality…

The modularity conjecture. The subcategory ${\mathcal M}_e$ is closed under convolution and hence is a monoidal category with unit $e$ if the preceding perversity conjecture holds;…

Let $G$ be a connected unipotent algebraic group, let $e$ be a minimal idempotent of ${\mathscr D}_G(G)$, let ${\mathcal M}_e^{perv}$ be the full subcategory of perverse objects in…

For an integer $n$, let $\bbH$ be the affine group scheme of formal series $P(X)\in 1+XA[[X]]$ satisfying $P(X)P(Y)=P(X+Y+XY)$. The assignment $m\mapsto(1+X)^m$ defines a morphism…

Unique representatives conjecture. For every coadjoint orbit, there exists a unique linear form $\lambda\in S$ lying on that orbit.

Platonov–Potapchik's conjecture. If all primitive elements of $G$ are unipotent, then the group $G$ generated by $S$ is unipotent.

The dual canonical basis conjecture. For every $T\in {\rm SSYT}(k-1,[k])$,

Cluster compatibility conjecture. For any decorated reduced word for $w_0$, the set ${\bf X}_{\mathbb{I}}$ is a cluster for $U$ mutation-equivalent to an initial cluster attached t…

Verschiebung lower-bound conjecture. The essential dimension of $G$ over $k$ satisfies

Let $k$ be a field of positive characteristic, and let $G$ be a finite commutative unipotent group scheme over $k$. Let…

Let $\mathcal{O}$ be the ring of integers of a number field $K$, and let $\mathbf{G}$ be a unipotent group scheme arising from a finitely generated free torsion-free $\mathcal{O}$-…

Let $F$ be a field containing a primitive $p$-th root of unity, let $G=G_F(p)$ be the maximal pro-$p$ quotient of its absolute Galois group, and let $n\geq 3$ be an integer. A pro-…