46 problems

Let $k>4$ and let … be the Fermat–Euler surface in $\mathbf{P}^3$. Consider the rational points on $\mathcal E_k$ outside the union of its curves of genus $0$ and $1$. Bombieri–Lan…

Shioda's conjecture. The surface $X$ is unirational if and only if it is Shioda supersingular.

Let $\Sigma_{g,n}$ be a surface of genus $g$ with $n$ punctures, and let $\mathcal{A}(\Sigma_{g,n})$ denote its cluster algebra. Non-finite generation conjecture. For $g\geq 2$,…

Let $\Sigma_{g,n}$ be a punctured surface, let $\mathcal{C}(\Sigma_{g,n})_{\mathbb{C}}$ be its complexified curve algebra, and let $\mathcal{T}^{d}(\Sigma_{g,n})$ be its decorated…

Let $(\mathbb S^{2},g(t))$ be a Ricci flow on the two-sphere, and let $\lambda_{1}(t)$ denote the smallest positive eigenvalue of the Laplacian of $g(t)$. Monotonicity conjecture.…

Poincaré–Seiberg–Witten conjecture. This invariant coincides with the full Seiberg–Witten invariant computed with respect to the canonical orientation data.

Bloch's conjecture. For every $\gamma\in A^2(Y\times X)$, the induced action $Gr^*_F\gamma$ depends only on $cl(\gamma)$.

Countability conjecture. The group $\operatorname{coker}(\tau)$ is countable.

Let $S$ be a surface with cusps and geodesic boundary. Write $\operatorname{Curr}_{\mathrm{cusp}}(S)$ and $\operatorname{Curr}_{\mathrm{open}}(S)$ for the corresponding spaces of g…

Core decomposition equality. The core decomposition of $K$ satisfies

Euler characteristic conjecture. We conjecture that

Let $g\geq 1$, let $\mathcal{D}_g$ be the Dehn quandle of a closed orientable surface of genus $g$, and let $\mathbb{Z}[\mathcal{D}_g]$ be its integral quandle algebra. An element…

Le–Yu injectivity conjecture. The projected stated skein algebra equals the stated skein algebra:

Skein–cluster equality conjecture. Under the same assumption as Theorem,

Bordered-surface cluster-algebra conjecture. Such arcs are elements of the cluster algebra for bordered surfaces.

Let $M$ be a closed orientable surface of positive genus, and let $H^2_{(1)}({\rm Homeo}_0(M),{\mathbf R})$ denote the $1$-bounded cohomology of the identity component of the homeo…

González-Alonso–Torelli conjecture. Such a deformation can have rank

Let $S_k$ be the compact arithmetic hyperbolic surfaces described in the source, and let the systolic inequality compare the systole of a hyperbolic surface with its genus or area.…

Let $K_n$ be the complete graph on $n$ vertices, and let ${\rm cr}_g(K_n)$ denote its crossing number when drawn on the closed surface of genus $g$. The previously known lower boun…

Let $\Sigma$ be a surface, let $\mathcal{S}_q^{\mathrm{MRY}}(\Sigma)$ be its MRY skein algebra, and replace the ground ring $\mathbb{C}$ by $\mathbb{Z}[q^{\pm1/2}]$. Let $v_i$ deno…

Let $\Sigma_{0,n}$ be the $n$-punctured sphere, and let $\mathcal{A}(\Sigma_{0,n})$ denote its cluster algebra. Folklore finite-generation conjecture. The algebra…

A single vortex is the point-vortex dynamical system with one vortex on a compact Riemannian surface. Let the surface have constant curvature and genus greater than one. Koiller's…

Canonical-degree bound conjecture. There exist constants $A$ and $B$, depending only on $X$, such that for every integral curve $C$ on $X$,

Super-maximization conjecture. One has

Let $\Sigma_{g,n}$ be a surface of genus $g$ with $n$ punctures, let $\mathcal{C}(\Sigma_{g,n})$ be its curve algebra, let $\mathcal{C}(\Sigma_{g,n})'$ be the subalgebra generated…