166 problems

Let $G$ be a split real semisimple Lie group, let $X$ be its associated symmetric space, and let $\rho$ lie in a Hitchin component of representations of $\pi_1(\Sigma_g)$ into $G$.…

Let $M$ be an irreducible symmetric space of compact type and rank greater than one. A polar action on $M$ is an isometric group action admitting a section, namely a complete immer…

A Riemannian manifold $(M,\langle\,\ \rangle)$ is globally Osserman if the eigenvalues of its Jacobi operator $R_u$ are independent of both the unit tangent vector $u\in M_x$ and t…

Blaschke conjecture. The only Blaschke manifolds are these compact rank-one symmetric spaces with their standard metrics, up to rescaling each metric by a positive constant.

Margulis' conjecture. There is an isometry $\tilde f:X\longrightarrow Y$ at bounded distance from $f$.

Let $G$ be a connected semisimple Lie group with maximal compact subgroup $K$ and minimal parabolic subgroup $P$. The symmetric space is $K\backslash G$, and its Furstenberg bounda…

Lapid–Prasad conjecture. If $\pi$ is $H$-distinguished, then the $L$-packet of $\pi$ is invariant under the functor

Let $X=G/K$ be a Riemannian symmetric space of noncompact type, with fixed Iwasawa decomposition $G=KAN$, and let $\mathfrak a$ be the Lie algebra of $A$. For $f\in L^2(X)$, let…

Let $X$ be a higher-rank symmetric space, and let $G$ be a finitely generated group quasiisometric to $X$. Margulis's conjecture. Every quasiisometry of $X$ should lie at bounded d…

Let $X=G/K$ be a non-compact Riemannian symmetric space, where $G$ is a connected semisimple Lie group and $K$ is a maximal compact subgroup. A joint eigenfunction is a function on…

Large-$N$ trace conjecture. The reduced traces satisfy

Let $M$ be a rank one symmetric space, and let a quasiconformal self-homeomorphism of the boundary at infinity of $M$ be a quasiconformal homeomorphism from that boundary to itself…

Graczyk–Sawyer sharpness conjecture. The number $n+1$ is sharp for the property that every convolution product of $n+1$ continuous $K$-bi-invariant measures on $G$ is absolutely co…

Siegel's conjecture. The quotient map from $\mathcal{S}$ to $\mathcal{M}$ is a Hausdorff equivalence: there is a constant $c'$ such that, for all $x,y\in\mathcal{S}$,

In a symmetric space of non-compact type, a complex equifocal submanifold with flat section is a submanifold satisfying the complex equifocality condition and having flat sections;…

Suppose that $\tilde{\Phi}$ is not of type $\mathsf{B}\mathsf{C}$. Let $\tilde{\mathbf A}$ be the Cartan matrix whose realization is given by the vector space ${}^e\!{\mathfrak s}$…

Let $X=G/K$ be an irreducible symmetric space of rank at least two, let $F$ be a maximal flat in $X$, and let $p_0\in F$. A geodesic in $F$ through $p_0$ is singular if it has the…

Let $(\mathfrak{g},\sigma)$ be a symmetric pair with Cartan decomposition $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$, and let $E(X,Y)$ be the function defined from graphs for…

Let $(\mathfrak{g},\sigma)$ be a symmetric pair, with Cartan decomposition $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$. Let $U(\mathfrak{g})$ be the universal enveloping algebra,…

Cross-blocking conjecture. A compact Riemannian manifold $(M,g)$ has cross blocking if and only if it is isometric to a compact rank one symmetric space. In particular, it has both…

Let $D\subset \mathcal V$ be a simply connected domain of an $n$-dimensional $k$-symmetric Lorentzian manifold $(\mathcal V,g)$. A vector field is null covariantly constant if it i…

Existence conjecture. There exists a complex-valued harmonic morphism

Let $(M^m,g)$ be an irreducible Riemannian symmetric space of dimension $m\ge 2$. For each point $p\in M$, a complex valued harmonic morphism is a harmonic morphism … defined on an…

Polar–hyperpolar conjecture. The classes of polar and hyperpolar actions on compact irreducible symmetric spaces of higher rank coincide.

The quaternionic and octonionic projective spaces in question are … A second elliptic problem is the second elliptic system associated to the corresponding 4-symmetric space. Geome…