5 problems

Let $M=\mathbf{R}^n/\Gamma$ be a closed flat $n$-manifold, where $\Gamma$ is a discrete, torsion-free, cocompact subgroup of $\operatorname{Isom}(\mathbf{R}^n)$. A cusp cross-secti…

A cusp closing is a modification that replaces a finite-volume hyperbolic cusp by a compact tube while preserving the indicated curvature bounds. Cusp-closing impossibility conject…

Let $K$ be a hyperbolic knot in $S^3$, let $M=S^3\setminus K$, and let $\mu$ be the meridian slope on $\partial M$. Denote by $\ell(\mu)$ the length of the meridian on the maximal…

Cusp-area monotonicity conjecture. The quantity

Alternating-knot cusp bounds. The area of a maximal cusp metric on $K$ is bounded above and below by linear functions of the twist number of the diagram. Likewise, the length of th…