19 problems

Let a spacetime be a constant scalar invariant (CSI) spacetime, meaning that all scalar polynomial curvature invariants are constant. A VSI spacetime is one for which all scalar po…

Let a spacetime be a constant scalar invariant (CSI) spacetime, meaning that all scalar polynomial curvature invariants are constant. The higher-dimensional Kundt CSI class consist…

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…

Let a Lorentzian spacetime $\mathcal{M}$ be a constant scalar invariant (CSI) spacetime, with curvature invariants $\mathcal{I}$. A spacetime is CSI when all scalar polynomial curv…

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 $I$ be a natural pseudohermitian scalar invariant whose total integral is a secondary CR invariant. Here, a secondary CR invariant is a total integral invariant under changes o…

Characterization conjecture. If a spacetime is characterised by its invariants, weakly or strongly, then it is either $\mathcal{I}$-non-degenerate or of type D to all orders; that…