94 problems

Let $\Omega\subset\mathbb{R}^{d+1}$ be the Lorentz future cone, let $T=\mathbb{R}^{d+1}+i\Omega$, and let $M\geq 1$. For every connected unipotent subgroup…

For every connected semisimple Lie group $G$ with finite centre and every degree $n\ge 0$, the comparison map from bounded continuous cohomology to continuous cohomology,…

Let $G$ be a connected real semisimple Lie group of real rank at least $2$, let $P<G$ be a minimal parabolic subgroup, and let $\Gamma<G$ be a discrete subgroup. If the action…

Let $G$ be a connected semisimple Lie group with finite center, all of whose simple factors have real rank at least $2$. Let $\Gamma\subset G$ be a lattice, $M$ be a compact manifo…

Matsuki–Gindikin conjecture. For every $K_\mathbb C$-orbit $S$ of nonholomorphic type on $X$, one has

Let $G$) be an exponential solvable Lie group, and let ${\mathcal J}_1\subseteq{\mathcal J}_2$ be closed two-sided ideals of $C^*(G)$ such that ${\mathcal J}_2/{\mathcal J}_1$ is a…

Optimal regularity conjecture. Any $K$-finite matrix coefficient of a unitary representation of $U$ is in $C^{(r,\delta)}(U_r)$, and this regularity is optimal.

Monod's isomorphism conjecture. The continuous bounded cohomology of $G$ with real coefficients is naturally isomorphic to its ordinary continuous cohomology:

Nonabelian Brunn–Minkowski conjecture. For every pair of compact sets $A,B\subseteq G$,

Let $(G,g)$ be a semi-Riemannian Lie group with a subgroup $K$ generating a left-invariant conformal foliation $\mathcal F$ on $G$ of codimension two. Semisimple subgroup minimalit…

Barbasch–Pandžić conjecture. If $\pi\in\widehat{G}$ has regular, half-integral infinitesimal character, then

Connection-blocking conjecture. The space $M$ is blockable if and only if

Let $G/H$ be a homogeneous space in the reductive setting, and let $K$ be a maximal compact subgroup of $G$. Rank inequality conjecture. If $G/H$ admits a cocompact discontinuous g…

Saturation conjecture for semisimple Lie groups. If $\lambda+\mu+\nu$ annihilates all elements of $T$ with semisimple centralizer, then

Converse to Theorem 15. The converse of Theorem 15 also holds.

Singular-triple criterion. There exists a triple $\sigma\in {\mathcal P}(G)\cap L^3$ with $Tr(\sigma)\in Q(R)$ and $\sigma\notin Tens(G)$ if and only if there exists a singular tri…

The structure conjecture. (1) If $R$ is simply-laced, then

Let $H$ be a connected semi-simple Lie group without almost simple factors of type $\mathrm{A}_1$ and without compact factors. For sufficiently large $x$, consider maximal irreduci…

Gray–Wolf conjecture. Every homogeneous strictly nearly Kähler space is a naturally reductive 3-symmetric space equipped with its canonical almost-complex structure.

Free-action quantization conjecture. A quasi-Poisson manifold $X$ for $({\mathfrak{g}},Z)$ can be quantized if the ${\mathfrak{g}}$-action on $X$ is free, i.e. if $X$ is a principa…

Let $A_n(s)$ be the alternating factorization defined in the paper, and let $\widetilde{\operatorname{SL}}(2,\mathbb{R})$ be the universal covering group of…

Let $n\in\mathbb{N}$ with $n\geq3$. Let $A_n(s)$ and $A(n;r,s,t)$ denote the alternating factorizations used in the paper, let $s_n$ be the associated switching parameters, and let…

Let $G$ be a Lie group, let $H$ be a closed subgroup, and let $d(H)$ denote the invariant used in the paper to measure the dimension relevant to tessellations of $G/H$. One-dimensi…

Nonexistence conjecture. None of these homogeneous spaces has a tessellation. These are presented as three special cases of Kobayashi's general conjecture; the supplied source give…

Let $G$ be a compact Lie group with Lie algebra $\mathfrak g$, let $G^c$ be its complexification, and let $\Omega$ be the data defining the examples associated to $G$. Denote the r…