21 problems

Non-regular-dessin conjecture. Every such uniform passport admits a non-regular dessin d'enfant with nontrivial automorphism group, that is, a dessin $\mathscr{D}$ satisfying

Tricolour graph construction. For any tricolour graph there exists a function

Let $t\geq 2$, and consider dessins with passport … A dessin is nontrivial-automorphism when its automorphism group is nontrivial. Nontrivial-automorphism conjecture. Every dessin…

A uniform passport is a passport in which all partitions have the same part size, and a dessin d'enfant is a dessin associated with such a passport. The automorphism group of a des…

Uniqueness conjecture. For each such passport, its realization as a dessin d'enfant is unique.

Dessins tau-function convergence conjecture.

Projective-prime dessin conjecture. For each projective prime $p=1+q+\cdots+q^{n-1}>3$, there exists at least one dessin of type $(3,2,p)$ and degree $p$ with monodromy group…

General dessin conformal-block conjecture. For a dessin satisfying the proposition's conditions, it corresponds to a family of 4-point CBs whose states follow the stated table.

Dessins' conformal-block information conjecture. The dessins in question should contain the full information for certain CBs.

The magma operad is an operad whose operations encode binary compositions, and $G=\operatorname{Gal}(\overline{\bold{Q}}/\bold{Q})$ acts on the genus-zero modular operad and its as…

Type D spectral-curve conjecture. This resulting expression is related to the spectral curve for enumeration of Grothendieck's dessins of Type D. The source supplies the generating…

Type B/C spectral-curve conjecture. The curve

Unique coherent extension conjecture. This partial choice of factors has a unique coherent extension to $\tilde{\mathcal{A}}$, meaning that the ideal generated by $Z_\psi(\psi)$ fo…

Let $D\in\mathcal{D}$ and set $\psi=\psi_D=\pi_{S}(D)-\pi_{S}(\pi_{\mathcal{G}}(D))$, where $\mathcal{G}\psi$ is the finite $\mathcal{G}$-orbit of $\psi$. Let $P_\psi(x)$ be the mi…

A dessin d'enfant is a combinatorial object associated with a trigonal curve; it is maximal when it has one singular fiber in each region and is connected. Deformation conjecture.…

Let $\mathcal{L}$ be the lattice in complex Lorentz space and let $\Gamma_{\mathrm{DM}}$ be the automorphism group from Thurston's correspondence, with $\mathcal{L}_+$ the positive…

Let $\mathcal{L}$ be the lattice of biased dessins, let $\Gamma' = \operatorname{Aut}(\mathcal{L})$, and let $\Gamma'_0$ be the subgroup of $\Gamma'$ that fixes the three vertices…

Square-root class conjecture. One has

Let $X$ be a smooth projective algebraic curve defined over an algebraically closed field $K$ of characteristic zero and transcendence degree $h$. A finite ramified covering…

Let $D$ be a dessin described by permutations $a,b$, and let $G$ be its monodromy group. Define … Let the field of moduli of $D$ be the fixed field of its stabilizer in…

Finite fundamental group conjecture. If $B$ is not of torus type, then