11 problems

Let $G$ be a connected reductive complex algebraic group, let $K$ be a maximal compact subgroup of $G$, and let $X$ be a spherical homogeneous algebraic variety. Let…

For quantized gravitational fluctuations around a symmetric background spacetime, quadratic constraints suppress transitions to configurations of lower symmetry. Enhancement–suppre…

The orbit-space homeomorphism conjecture. Consequently,

Let $phi$ be a transitive pseudo-Anosov flow on an oriented closed $3$-manifold. Let $\operatorname{sing}(phi)$ denote its singular orbits, and let positive and negative lozenges b…

Let $G$ be a connected reductive complex algebraic group, let $K$ be a maximal compact subgroup of $G$, and let $X$ be a spherical homogeneous algebraic variety. Denote by…

Let $V$ be a finite-dimensional Hilbert space, and let $\Gamma\leq \operatorname{E}(V)$ be an isometry group with closed orbits. The orbit space $V/\Gamma$ is equipped with its quo…

Let $\mathbb{T}^n$ be a compact torus, let $\mathcal{G}$ act on it, and let $\mathcal{T}$ denote the image of $\mathbb{T}^n$ under the orbit-map parametrization…

Let an orbit space be the quotient of a Riemannian manifold by an isometric group action, equipped with its metric structure and its smooth structure. Myers–Steenrod conjecture for…

Orbit-space parametrization conjecture. The set of all $G$-orbits in $X$ does not, in general, have an explicit or even a small system of parametrization by such algebraic circuits…

Let $M$ be a Hausdorff manifold with a countable basis and empty boundary, and let $v$ be an arbitrary smooth vector field on $M$. Let $M/{\sim}$ denote the orbit space of $v$. Orb…

Paracompactness conjecture. If $X$ is a paracompact proper $G$-space, then the orbit space $X/G$ is paracompact.