29 problems

Let $G$ be a semisimple group with discrete series. Let $\pi\in\widehat G$ be a cohomological representation, and consider the question of whether there is a lattice $\Gamma$ in…

Let $\widetilde{X}$ be a symmetric space with no Euclidean local factors, let $X=\Gamma\backslash\widetilde{X}$ be a finite-volume locally symmetric space, and let…

Stretching cusps conjecture. The sequence $\{(M,g_i)\}$ Gromov–Hausdorff converges, in the appropriate sense, to a disjoint union of two isometric copies of $N$.

Let $G$ be the reductive group, $\Gamma$ the arithmetic subgroup, $A$ the split torus, and $A^-$ its negative chamber. Let $\hat G_{\rm adm}$ denote the admissible dual, let…

Harder–Borel conjecture. The inclusions

Let $M$ be a closed, nonpositively curved, locally symmetric manifold with no local $\mathbb{R}$ factors. The Gromov norm of $M$, namely the simplicial volume, is positive. Gromov'…

Let $M$ be a closed manifold which admits a locally symmetric Riemannian metric $g_{loc}$ with nonpositive sectional curvature. Assume that $(M,g_{loc})$ has no local factors isome…

Fix a symmetric space $S$ of non-compact type without Euclidean de Rham factors, and let an arithmetic $S$-manifold be a quotient $M=\Gamma\backslash S$ by a torsion-free arithmeti…

Let $S$ be a symmetric space of non-compact type, and let an $S$-manifold be a complete Riemannian manifold locally isometric to $S$. An irreducible $S$-manifold has no nontrivial…

Let $M$ be a compact Riemannian manifold with negative sectional curvature and dimension at least $3$. Suppose that, for some $k\geq 2$, there exist infinitely many distinct immers…

Let $g_0$ be a locally symmetric metric, and let $D_{g_0}^*$ be the operator appearing in the paper. Denote by $\Pi_{\operatorname{Ker}(D_{g_0}^*)}$ the projection onto…

Let $(M^n,g)$ be a closed negatively curved manifold with higher hyperbolic rank, meaning that for every geodesic there is a nonvanishing Jacobi field whose span with the geodesic…

Partial Okounkov body conjecture. The collection of $\Delta_{\nu}(\hat{E})$ for linear valuations $\nu$ determines the numerical class of $E$.

Let $(M,g)$ be a closed, simply connected Riemannian manifold. Let $d40$ denote the curvature operator of the second kind, restricted as a bilinear form to the trace-free symmetric…

Let $M$ be a closed oriented manifold covered by $(\mathbb{H}^2)^n$, and let $\Theta_n$ be the bounded volume form defined in the preceding conjecture. Simplicial-volume equality c…

Finiteness conjecture. The diameter of the closed geodesic norm is finite.

Let $X$ be the universal symmetric space and let $(M_n=\Gamma_n\backslash X)$ be a uniformly discrete expander family that BS-converges toward $X$. Fix an eigenvalue $\lambda_0$ of…

Lyapunov-spectrum characterization conjecture. Let $Y$ be a closed negatively curved Riemannian manifold, $\dim Y \geq 3$. Suppose that for each periodic point $p$ of $g^{t}_{Y}$ t…

Homotopy complexity conjecture. There are $\alpha>0$ and $\delta>0$ such that every arithmetic $X$-manifold $M$ is homotopy equivalent to a $(\delta,\alpha\operatorname{vol}(M))$-s…

Let $(N,g)$ be a finite-volume Riemannian manifold satisfying property (P), meaning that every geodesic in its universal covering has rank at least two. Local symmetry conjecture.…

Let $M$ be a smooth closed, aspherical, smoothly irreducible manifold such that $pi_1(M)$ contains no nontrivial normal abelian subgroups. Then there exists $C>1$ depending only on…

Linear homotopy complexity conjecture. There are constants $\alpha$ and $d$, depending only on $X$, such that every such $M$ is homotopically equivalent to a simplicial complex…

Let $G$ be the Lie group under consideration, let $k$ be a degree, and let an IRS $\mu$ of $G$ be $\varepsilon$-discrete in the sense used in the source. Let $\nu_\mu^k$ be the spe…

Let $G$ be the ambient semisimple Lie group, let $X$ be its symmetric space, and let $\Gamma$ range over arithmetic lattices with $\operatorname{vol}(\Gamma\backslash X)\leq V$. Fo…

Farb and Weinberger's conjecture. 1. $M$ is aspherical, smoothly irreducible, $\pi_1(M)$ has no non-trivial normal abelian subgroup, and