5 problems

Let $P$ be a Fano polygon, that is, a two-dimensional Fano polytope. Call $P$ symmetric when its symmetry group is nontrivial. Hacking–Kim conjecture. If $P$ is a Kähler–Einstein F…

Let $P$ be a smooth Fano polytope, and let $B(P)$ denote its barycentric transformation. A Fano polytope is of type $B_\infty$ when the barycentric transformation can be applied in…

Normalized-volume equality conjecture. Equality should hold in this inequality even when the metric tangent cone $C(Y)$ is irregular. Equality is known when $C(Y)$ is quasi-regular…

Let $(Z,p)$ be a singular point of a Gromov–Hausdorff limit carrying a weak Kähler–Einstein metric. Let $C(Y)$ be its unique metric tangent cone, and let…

Let $C(Y)$ be a Ricci-flat Kähler cone in dimension $k$ with an isolated singularity at the vertex. Its volume density is the ratio between the volume of its unit ball centered at…