19 problems

Pu's relative 1-systole conjecture. For any such $S$ and $\hat\phi$,

Liechti–Strenner's conjecture. For all $k\geq 2$,

Unknottedness conjecture. Every $fZ_2$-surface in $S^4$ is topologically unknotted.

Let $M$ be a closed nonorientable surface, and let $N$ be a compact nonorientable surface with boundary. Nonorientable doubling conjecture. Every closed nonorientable surface admit…

Odd-genus Penner dilatation limit conjecture. The limit is the largest real root of

Genus 9 and 11 minimal stretch factor conjecture. The minimal stretch factors and minimal polynomials are predicted to be

Minimal polynomial conjecture. For all $k\geq 2$, $\delta^+(N_{2k})$ is the largest root of

Closure-equality conjecture. The inclusions in Theorems and are equalities, except when $\lambda$ is the bounding geodesic of the genus-three closed surface.

Rational-density conjecture.

Simple-closed-geodesic growth conjecture. There exists a constant $c>0$ such that

Sullivan-type invariant-measure conjecture. There exists a unique, up to scaling, ergodic $\operatorname{Map}(S)$-invariant Radon measure supported on $\mathcal{ML}^+(S)$, and it i…

Limit-set conjecture. The limit set of $\operatorname{Map}(S)$ in $\mathcal{PML}(S)$ is $\mathcal{PML}^+(S)$.

Let $\Sigma$ be a surface in $\mathbb R^4$ with Willmore energy $W(\Sigma)$. Kusner's conjecture. If … then $\Sigma$ must be a sphere. This remains open in the context of the paper…

The $b$-conjecture. The series $\Psi(\mathbf{x},\mathbf{y},\mathbf{z};t,1+b)$ has coefficients that are polynomials in $b$ with non-negative integer coefficients. The constant term…

Let $S$ be a closed nonorientable surface and let $c$ be a two-sided simple closed curve such that either $c$ is nonseparating and the genus of $S$ is $7$, or the genus of $S$ is…

Garoufalidis–Mariño conjecture. The constants $p_g$ satisfy

Ho–Liu's antiperfection conjecture. The Yang–Mills Morse stratification is antiperfect when the rank is $n=3$.

Let $G$ be a finite graph and let $\mathbb{N}_{\tilde{g}}$ be the nonorientable surface of genus $\tilde{g}$. A Kuratowski subgraph is a subdivision of $K_5$ or $K_{3,3}$ contained…

Let $G$ be a finite graph, and let $\mathbb{N}_{\tilde{g}}$ denote the nonorientable surface of genus $\tilde{g}$. A Kuratowski subgraph is a subgraph that is a subdivision of…