30 problems

Let $M$ be a compact Riemannian manifold, and let $K$ be a curvature formed locally from the Riemannian curvature tensor, its covariant derivatives, and the metric. A conformal inv…

Let a CSI spacetime be a spacetime all of whose scalar curvature invariants are constant, and let a degenerate Kundt spacetime be a Kundt spacetime whose curvature tensor and all c…

An $\mathcal{I}$-non-degenerate spacetime is one for which the scalar polynomial curvature invariants characterize the spacetime locally up to the relevant equivalence. Cartan-inva…

Let a spacetime have a Riemann tensor $R$ and all of its covariant derivatives. A null frame is a frame adapted to the Lorentzian structure, and the boost order and boost weight re…

Curvature-invariant basis conjecture. The curvature invariants provide a basis for multi-energy-momentum tensor contributions also for theories which are not at large $N$.

Curvature-invariant detection conjecture. Curvature invariants are able to detect the appropriate bounding hypersurface for any black hole solution as the zero-set of some invarian…

Geometric horizon conjecture. Geometric horizons are surfaces for which the Riemann tensor, or one of its covariant derivatives, is algebraically special compared to the external s…

Positive surface curvature implies no nonpositive-Euler-characteristic surface subgroups. If $\sigma_-(X)>0$ then no subgroup of $\pi_1(X)$ is isomorphic to $\pi_1(S)$.

Negative surface curvature implies hyperbolicity. If $\sigma_+(X)<0$ then $\pi_1(X)$ is word-hyperbolic.

Non-positive surface curvature implies solvable word problem. If $\sigma_+(X)\leq 0$ then the word problem is solvable in $\pi_1(X)$.

Non-positive surface curvature implies asphericity. If $\sigma_+(X)\leq 0$ then $X$ is aspherical.

Negative irreducible curvature implies locally quasiconvex. If $\rho_+(X)<0$ then $\pi_1(X)$ is a locally quasiconvex hyperbolic group.

Non-positive irreducible curvature implies coherence. If $\rho_+(X)\leq 0$ then $\pi_1(X)$ is coherent.

Graphs of free groups with cyclic edge groups. Then $\rho_+(X)\leq 0$. Furthermore, $\sigma_+(X)<0$ unless $\pi_1(X)$ has a Baumslag--Solitar subgroup.

Surface curvatures of mapping tori of free group endomorphisms. Then $\sigma_+(X)<0$ unless $\pi_1(X)$ has a Baumslag--Solitar subgroup.

Irreducible curvatures of mapping tori of free group endomorphisms. Then $\rho_+(X)=0$.

Negative curvature for geometric complexes. If $\sigma_+(X)=0$ then either $\partial M(X)$ has a toroidal boundary component or $\pi_1(X)$ has a $\mathbb{Z}^2$ subgroup.

Non-positive curvature for geometric complexes. If $\sigma_+(X)>0$ then either $\partial M(X)$ has a spherical component or $\pi_1(X)$ splits freely.

Surface curvature of random complexes in the density model. The probability that $\sigma_+(X)<0$ tends to $1$ as $l\to\infty$.

Irreducible curvature bounds of few-relator complexes. The probability that $\rho_+(X)=\kappa(X)$ tends to $1$ as $l\to\infty$. In particular, with high probability, $\rho_+(X)<0$…

Let $M$ be a compact manifold of dimension $n$, let $\tilde{M}$ be its universal cover, and let $\dim_{mc}(\tilde{M})$ denote the macroscopic dimension of the universal cover. Let…

Let $S^n$ be the standard $n$-sphere and let $T^k$ be the $k$-torus, with $n\geq2$ and $k$ a positive integer. Let ${\rm Ein}(M)$ denote the Einstein invariant of a compact manifol…

Let $K$ be an isotropic local curvature scalar on a Riemannian manifold, and let $K_{GBC}$ denote the Gauss–Bonnet–Chern curvature. A divergence is the divergence of a vector field…

Let a CSI spacetime be a spacetime all of whose scalar curvature invariants are constant, and let a Kundt-CSI spacetime be a degenerate Kundt spacetime whose transverse metric is l…

Let $D$ be the spacetime dimension. A Lorentzian spacetime is $I$-degenerate if it admits a null frame in which all positive boost-weight terms of the curvature tensor and its cova…