393 problems

Grothendieck–Serre conjecture. Every étale $G$-torsor over $A$ that becomes trivial after base change to $K$ is already trivial over $A$.

Donkin-type tilting conjecture. For every $\lambda\in X_{n-1}(T)$, the tensor product

Let $F$ be a global field, let $G$ be an algebraic group defined over $F$, and let $\tau(G)\in\mathbb{R}$ denote its Tamagawa number, the volume of…

Let $X$ be an affine algebraic variety defined over an algebraically closed field, and let $G$ be a subgroup of $\operatorname{Aut}(X)$ generated by a finite collection of…

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…

Genus finiteness conjecture. The set ${\bf gen}(G)$ is finite.

Let $G$ be a connected, simply connected, complex simple algebraic group, and let a Poisson bracket on $G$ belong to a Belavin–Drinfeld class. Gekhtman–Shapiro–Vainshtein conjectur…

Panyushev's conjecture. Given orthogonal subsets $S,S'\subset\Psi(\mathfrak p^{\mathrm u})$, one has

Let $K$ be a finitely generated field, let $V$ be a divisorial set of discrete valuations of $K$, and let $G$ be a connected reductive $K$-group. For each $v otin V$, let $Kv$ deno…

Mumford's conjecture. Every smooth affine geometrically reductive group scheme over $k$ is reductive, and every smooth affine reductive group scheme over $k$ is geometrically reduc…

Let $G$ be a semisimple algebraic group over $k$, let $r\geq 1$, let $M$ be a rational $G$-module, and let $\operatorname{St}_r$ denote the $r$-th Steinberg representation. A good…

Humphreys–Verma conjecture. For $\lambda\in X_r$, the $G_r$-module structure on $Q_r(\lambda)$ can be lifted to $G$.

Let $G$ be a reductive group and let $H$ be a sober spherical subgroup with $H=N_G(H)$. Let $\bar X$ be the unique simple complete toroidal embedding of $G/H$, and let $X_D$ be the…

Lusztig's based-ring conjecture. There should exist a finite $F_{\mathbf c}$-set $Y$ and a bijection $\pi$ from $\mathbf c$ onto the irreducible $F_{\mathbf c}$-vector bundles on…

Let $G$ be an algebraic group, let $x,y\in G$, and let $\rho$ range over the irreducible rational representations of $G$. Steinberg's conjecture. The elements $x$ and $y$ are conju…

Let $(G,X)$ be a permutation group of finite Morley rank, with $X$ of rank $r$, and let $F$ be an algebraically closed field. A group action is generically sharply $t$-transitive w…

Let $(G,X)$ be a connected primitive permutation group of finite Morley rank, and write $\operatorname{rk}X=n$. Rank bounds and classification conjecture. … If equality holds, then…

Let $G$ be a connected compact simple real linear algebraic group, and let a word map be the map induced by a word in finitely many variables on $G$. Larsen's conjecture. Every wor…

Cherlin-Zilber Algebraicity Conjecture. The group $G$ is isomorphic to a simple algebraic group over an algebraically closed field.

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

Popov–Pommerening conjecture. Any observable regular subgroup of $G$ is a Grosshans subgroup.

Serre's congruence subgroup property conjecture. Every higher rank group has the congruence subgroup property (CSP), meaning

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 $S$ 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…