180 problems

For every connected complex reductive algebraic group $G$ and every involutive automorphism $\theta:G\to G$, let $H=G^\theta$ and let $\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{s}$…

Let $X$ be an affine variety of dimension at least $2$, and let … be generated by finitely many ${\mathbb G}_a$-subgroups. Suppose that $G$ acts $2$-transitively on one of its orbi…

Let $X$ be an irreducible affine variety. For locally nilpotent derivations $\partial_1,\ldots,\partial_k$ of ${\mathbb K}[X]$, let … A Lie-algebra closure conjecture asserts that…

Let $R$ be a finitely generated commutative ring with unity that is infinitely generated as a $\b7\mathbb{Z}$-module. For $n\b7\geq 2$, let $\b7\mathbf{A}_n(R)$ be the corresponding…

Finite generation conjecture. The group $\Gamma$ is never finitely generated.

Let $G$ be a split semisimple algebraic group with Weyl group $W$, and let $L^{u,v}=N\overline{u}N\cap B_-vB_-$ be the reduced double Bruhat cell for $u,v\in W$. For a reduced word…

Katsylo's conjecture. The following properties are equivalent: $\dim V=\dim W$; and there exists a $G$-equivariant birational map

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 $V$ be a symplectic vector space, let $X'\subset \P(V)$ be an irreducible Legendrian subvariety, let $g\in\mathfrak{sp}(V)$ be a nilpotent endomorphism, and let $m$ be an integ…

Let $G$ be a reductive group over a field $k$ of good characteristic, with Lie algebra ${\mathfrak g}$. Let ${\cal N}_1$ be the restricted nilpotent cone, let $e\in{\cal N}_1$, and…

Let $C$ be a smooth projective geometrically connected curve over $k$, of genus $g$, and let $G$ be a split semisimple connected algebraic group over $k$. Denote by…

Let $G$ be a connected reductive group over $F$, let $b\in G(L)$, and let $\mu$ be a dominant coweight such that the affine Deligne–Lusztig variety $X_\mu(b)$ is non-empty. Write…

Weil's conjecture. If $G$ is simply connected, then

Harder's conjecture. If ${\mathcal G}$ is split, the Tamagawa number of $G$ equals the number of connected components of the moduli space of ${\mathcal G}$-torsors on $X$.

Geometric divisibility sequence conjecture. The geometric divisibility sequence $(D_{n\mathcal{P}})_{n\geq 1}$ corresponding to $\mathcal{P}$ satisfies

Let $G$ be a ${\mathord{\mathbb{Q}}}$-simple algebraic ${\mathord{\mathbb{Q}}}$-group, and let $G_{\mathord{\mathbb{Z}}}$ denote its integral points. A subgroup is left orderable i…

Let $\operatorname{SO}_{2m-1}$ and $\operatorname{SO}_{2m}$ denote the special orthogonal groups of the indicated dimensions, and let $\operatorname{cd}(G)$ denote the canonical di…

Let $G$ be an algebraic group over $\mathbb R$, let … and let $p$ be a prime number. Regard $G(\mathbb C)$ as a $\Gamma$-space and let $G(\mathbb C)^{\operatorname{top}}$ denote it…

Let $G$ be an algebraic group over an algebraically closed field $k$, and let $n$ be a positive integer such that the characteristic of $k$ does not divide $n$. The simplicial sche…

Let $G$ be an algebraic group and $H\subseteq G$ an observable subgroup. Write $W(H)=N_G(H)/H$ for the corresponding quotient of the normalizer. The subgroup $H$ is reductive if it…

Let $K={\bf F}_q(T)$, let $G/K$ be an isotropic, semisimple, simply-connected algebraic group, and let $Z$ be its center. Let ${\bf A}^{\mathrm{fin}}_K$ denote the finite adeles of…

Serre's Conjecture I. Does

Let $K$ be a finitely generated field, let $X$ be a smooth projective variety over $K$, and consider a $\mathds{Q}$-compatible system of Galois representations … Let…

Let $k$ be the base field, and let $G$ be a simply connected, almost $k$-simple, $k$-isotropic algebraic group over $k$. Let $G(k)^+$ denote the abstract subgroup of $G(k)$ generat…

Let $S$ be a Noetherian scheme and let $G$ be an isotrivial reductive group over $S$, meaning that $G$ becomes split after a finite étale cover of $S$. A $G$-equivariant $S$-fiberw…