6 problems

Valette's conjecture. Property (RD) holds for uniform lattices in $\operatorname{SL}_d(\operatorname{\mathbb{R}})$; more generally, it holds for uniform lattices in any semisimple…

Algebraic classification conjecture. Every irreducible $G$-space is isomorphic to one of the known actions of $G$ arising from an algebraic construction, modulo a nowhere-dense,…

Let $\operatorname{\mathsf{G}}$ be a connected semisimple Lie group without compact factors and with finite center. Suppose $\Gamma<\operatorname{\mathsf{G}}$ is a Zariski dense di…

Bekka–Valette conjecture. There exist unitary representations of $\Gamma$ that are not weakly contained in any representation restricted from $G$.

Let $H$ be a semisimple Lie group of type $\mathrm{A}_1$ with more than one simple factor, and let $\mathrm{IL}_{H}(x)$ denote the maximal size of a family of pairwise isospectral,…

Let $G$ be a semisimple Lie group and let $bGamma$ be a lattice in $G$. Let $K_{G,\Gamma}$ be the convex set of continuous positive definite functions $\phi$ on $G$ satisfying … ……