16 problems

Let $H$ be a higher-rank semisimple group of the form … where each $F_i$ is a local field, each $\mathbf{G}_i$ is an absolutely almost simple $F_i$-group, and … For a group…

Margulis's conjecture. The group $\Gamma$ is an arithmetic lattice in $G$.

General semisimple-group conjecture. Theorems on the contractions and product structure—and consequently the orientability theorem and the relative-homology corollary—hold for any…

Let … where each $K_i$ is a local field, each $G_i$ is an absolutely almost simple $K_i$-group, and … Let $\rho(\Gamma)$ denote the abscissa of convergence of the representation ze…

Generalized convergence conjecture. The same convergence assertion should hold for any semisimple group $G$ with finite center and generic $n_0\in N$. This extension would generali…

Let $G$ be a semisimple Lie group and let $V$ be a uniformly bounded representation of $G$ on a Hilbert space. The uniformly bounded cohomology conjecture. The cohomology with coef…

Let $G$ be a non-compact simple group over a local field $F$ with finite center and rank $r$. The twice-the-rank non-vanishing conjecture. For every sufficiently large $p$, … A sim…

Let $G$ be a non-compact semisimple group over a local field $F$ with finite center and rank $r$. The non-uniform lattice extension conjecture. The full statement of Gromov's $L^p$…

Purity conjecture. The rational cohomology of every resolved Coulomb branch associated with connected semisimple groups, in the sense of Braverman–Finkelberg–Nakajima, has pure Hod…

Let $G$ be a complex semisimple group, let $e$ be a nilpotent element, and let $S_e$ be a slice for $e$. Generalized S-duality conjecture. The same result also holds for more gener…

Let $G$ be a semisimple Lie group without compact factors and of rank at least two. A discrete subgroup $\Lambda\le G$ is co-amenable when the Hilbert space $L^2(G/\Lambda)$ admits…

Let $G$ be a product of non-compact almost simple groups over local fields, or a connected semisimple Lie group with finite center and without compact factors, and let $\Gamma<G$ b…

Let $G$ be a semisimple linear algebraic group over an infinite field $k$. Write $G^+(k)$ for the subgroup generated by the subgroups $R_u(Q)(k)$, where $Q$ ranges over all minimal…

Let $k$ be the ground field, let $G$ be a non-special semisimple $k$-group scheme, and let $X$ be a smooth affine $k$-variety. Purity failure conjecture. There exists a smooth affi…

Let $k$ be an algebraically closed field of characteristic $0$, and let $G$ be a semisimple $k$-group. The source refers to Theorem, whose hypotheses are not reproduced in the conj…

Let … where each $K_i$ is a local field, each $G_i$ is an absolutely almost simple $K_i$-group, and … For a finitely generated group $\Gamma$, let $r_n(\Gamma)$ be the number of ir…