66 problems

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 $X=\Gamma\backslash G/K$ be a locally symmetric space, where $G$ is a connected semisimple non-compact linear Lie group of real rank $1$. A finite cover of $X$ is a covering sp…

Let $X$ be an equal-rank locally symmetric space, let $X^*$ be a Satake compactification, and let $\widehat{X}$ be Zucker's reductive Borel–Serre compactification. Let … be the quo…

Let $G/\mathbb{Q}$ be a connected reductive group, let $X$ be the symmetric space for $G(\mathbb{R})$, and let $K\subset G(\mathbb{A}_f)$ be a neat compact open subgroup. Write…

Let $G$ be a reductive group with maximal compact subgroup $K$, symmetric space $X=G/K$, and let $\Gamma\subset G$ be a discrete subgroup of finite covolume. Let $A$ be the split c…

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 $M$ be a Riemannian manifold with universal cover $\widetilde{M}$, and let $\operatorname{Isom}(\widetilde{M})$ denote its isometry group. An orbibundle with locally symmetric…

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 $N$.

Finite-volume most-symmetry conjecture. Proposition 1 remains true when $M$ has finite volume and $h$ is any finite-volume Riemannian metric; that is, $\operatorname{Isom}(M,h)$ sh…

Let $X$ be the relevant locally symmetric space, let $d=\dim X$, let $\Gamma$ be an arithmetic subgroup, and let $\sigma$ be a $K_\infty$-type. Write…

Let $X$ be the relevant locally symmetric space, let $d=\dim X$, and let $\Gamma$ be an irreducible lattice with $\operatorname{rank}(X)>1$. Let…

Harder–Borel conjecture. The inclusions

Harder–Borel conjecture. The subcomplexes of the de Rham complex consisting of coclosed harmonic automorphic forms and automorphic forms are quasi-isomorphic to the full de Rham co…

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…

Simple homotopy type conjecture. There are constants $\alpha(S)$ and $d(S)$ such that any arithmetic $S$-manifold $M=S/\Gamma$ with volume $\mu(M)$ is homotopically equivalent to a…

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 $H^n$ be the real hyperbolic space of dimension $n$, and let $T^{(2)}(H^n)$ denote its $L^2$-torsion density, defined using the invariant metric on $H^n$. Nonvanishing conjectu…

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 $(M^n,g)$ 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…

Let $G$ be a semisimple Lie group, let $K$ be a maximal compact subgroup, and let $M=\Gamma\backslash G/K$ be a locally symmetric manifold associated to a lattice $\Gamma\le G$. An…

Characteristic-zero vanishing conjecture. The localized rational cohomology satisfies

Vanishing conjecture. The localized cohomology satisfies