13 problems

Grothendieck conjecture. The geometry of anabelian schemes over fields finitely generated over $\mathbb{Q}$ is completely determined by their étale fundamental groups; in particula…

Uniform finiteness conjecture. There exists a constant $C$, depending only on $K$ and $g$, such that

Let $K$ be a field finitely generated over $\mathbb{Q}$, let $X$ be a smooth variety over $K$, and let $Y$ be a hyperbolic polycurve over $K$. Write…

Converse conjecture. If $X$ is Kollár-hyperbolic with respect to $H$, then $X$ is V-hyperbolic with respect to $H$.

Let $X$ be a curve over $\mathbb{Q}$, let $b$ be a basepoint, and let $G_{\mathbb{Q}}$ denote the absolute Galois group of $\mathbb{Q}$. The étale fundamental group of the base cha…

Let $k$ be a field with absolute Galois group $G_k$, let $U_1,U_2$ be the curves under consideration, and let $m,n$ be non-negative integers. For each $r$, write…

Let $k$ be a field of characteristic $p\geq 0$, finitely generated over the prime field, with absolute Galois group $G_k$. Let $U$ be an anabelian variety over $k$ with a geometric…

Grothendieck's conjecture. The following statements hold:

Normalized-volume conjecture. The analogous statements to the transformation theorem and the bound for the local étale fundamental group should hold with normalized volume in place…

Let $X$ be a smooth variety over $\operatorname{\mathbf{Q}}$ with a rational base point $b$, let…

Finite-determinant conjecture. There exist elements $u\in\mathbb{Z}_{q'}^*$ such that $(\det V,\det\nabla,u\det\Phi)$ corresponds to a finite character of

Let $X$ be a geometrically connected curve over $\mathbb{R}$, and let $\kappa$ be the map from real points to conjugacy classes of sections of the étale fundamental-group sequence.…

Let $k$ be a number field, let $X$ be a smooth projective curve over $k$ of genus at least $2$, and fix a geometric point $\bar x$ of $X$. Consider the natural exact sequence … A c…