10 problems

Let $(M^n,hyp)$ be a closed hyperbolic manifold, and let $g$ be another metric on $M$ with $Vol(M,g)<Vol(M,hyp)$. Hyperbolic volume comparison conjecture. … for all $R$. This stren…

Let $(M^n,g)$ be a complete Riemannian manifold, and let $(S^n,g_0)$ be a round sphere. Choose the radius of the round sphere so that the filling radius of $(M^n,g)$ equals that of…

Let $(X,\mathsf d,\mathfrak m, \ll, \leq, \tau)$ be a measured Lorentzian space, let $V\subset X$ be the achronal set occurring in the hypotheses of Theorem, and let…

Quantized volume comparison conjecture. For every $m\geq 1$,

Dual higher-order Bol inequality conjecture. One should have

Higher-order Bol inequality conjecture. One should have

Let $(X,\mathsf{d})$ be the metric limit under consideration, let $k=\dim_{\mathcal{H}}(X,\mathsf{d})$, and let $\mathcal{H}^k$ denote its $k$-dimensional Hausdorff measure. For…

Higher-dimensional Football conjecture. For every such manifold,

Higher-dimensional volume comparison conjecture. There exists a positive $\epsilon_0(n)<1$ such that, whenever

Let $p,q\ge 3$ with $1/p+1/q\le 1/2$ be given. Let ${\cal G}=({\cal V},{\cal E},{\cal F})$ be a locally tessellating planar graph without cut locus, meaning that it has no cut locu…