54 problems

Generalized Liu–Terng conjecture. (i) If $A(t)$ and $H(t)$ satisfy the inequality referred to as for some $\epsilon\in(0,1/(2\kappa))$ instead of $\epsilon_F$, then $M$ is an equif…

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 $D\subset \mathcal V$ 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…

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…

Let $M$ be a compact symmetric space of rank bigger than one, and let a compact Lie group act polarly on $M$. Hyperpolarity conjecture. The action is hyperpolar. This extends the p…

Blaschke manifold classification conjecture. If $M$ is a Riemannian manifold such that

Unfoldability conjecture. A billiard triangle in $\Delta$ with sides $\overline{ox_0}$, $\overline{ox_k}$ and $x_0x_1\ldots x_{k-1}x_k$, and with the special vertices $0,x_0,x_k$,…

Let $n\geq 0$, let $\mu\overline{\nu}$ be a dominant weight of $U(2n)$, and let $Z$ normalize the integral so that the trivial polynomial integrates to $1$. Nonstandard $U(2n)$ van…

Let $n\geq 0$, let $\mu\overline{\nu}$ be a dominant weight of $U(2n)$, and let $Z$ normalize the integral so that the trivial polynomial integrates to $1$. $U(2n)$ Grassmannian co…

Let $m$ and $n$ be integers with $0\leq m\leq n$, let $\mu\overline{\nu}$ be a dominant weight of $U(m+n)$, and let $P_{\mu\overline{\nu}}$ be the corresponding Macdonald polynomia…

Type-II spectral decomposition conjecture. There exist a topological space $\widehat G_H^L\supset\widehat G_H$, a continuous finite-to-one map

Group-ring resolution conjecture. There is a function $f(n)$ such that every $n$-generated ideal in $\mathbb K[G]$ has a free resolution of length $r-1$.

Let $G$ be a reductive group with involution $\theta$, let $X_\theta=G^{\operatorname{inv}\circ\theta}=\{g\in G\mid\theta(g)=g^{-1}\}$ with its $\theta$-twisted conjugation action,…

Let $K_0\backslash\mathcal{G}_0$ be the quotient under consideration, let $\lambda$ be a dominant coweight, let $\omega_i$ be the fundamental coweights for $1\leq i\leq n-1$, and s…

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

Let $M$ be a connected compact symmetric space with a smooth isometric embedding into a Euclidean space that is unique up to Euclidean isometries. Let the diffusion mean set be the…

Let $G$ be the group acting on the set $X$ of invertible symmetric matrices by $x\cdot g=\,{}^tgxg$, and write the Plancherel decomposition as … Let $G'$ be a two-fold Kazhdan–Patt…

Asymptotically harmonic manifold conjecture. If $(M,g)$ is asymptotically harmonic, then $M$ is either flat or a rank-one symmetric space of noncompact type.

Gudmundsson's conjecture. For each point $p\in M$, there exists a complex-valued harmonic morphism

Rigidity conjecture. Any Riemannian metric $g$ which coincides with $g_0$ outside of $D$ is isometric to $g_0$.

Let $G$ be a semisimple Lie group, let $K$ be a maximal compact subgroup, and let $G/K$ be the associated homogeneous space. Write $\operatorname{rank}_{\mathbb{R}}(G)$ for the rea…

Consider the horospherical tangent cones at infinity of symmetric spaces, with restricted root systems whose factors determine the relevant types. Regularity conjecture. The horosp…