21 problems

Burago–Ivanov conjecture. Every simple manifold is a strict minimal filling.

Let $\operatorname{ssr}(\mathbb R^2)$ denote the Steiner subratio of the Euclidean plane, defined as the infimum of the ratios of minimal-filling weight to Steiner minimal-tree len…

Let $(M,g)$ be a Riemannian orientable surface with a single boundary component $S^1$ of length $2\pi$. For $x,y\in S^1$, let $d_{S^1}(x,y)$ denote the intrinsic distance along the…

Let $M$ be a surface with a directed Finsler semimetric $F$, filling without shortcuts a Finsler closed curve $(C,G)$. Let $C^+$ and $C^-$ denote the two orientations of $C$, with…

Let $M$ be an orientable Riemannian surface filling without shortcuts a Riemannian circle of length $2L$. Gromov's filling area conjecture. The area of $M$ is at least the area of…

Let $C=C_{2n}$ be a cycle graph of length $2n$, and let $M$ be a square-celled surface with boundary $C$. Distances between vertices are measured in the skeleton graph. Discrete FA…

Let $(M,W)$ be a walled surface whose boundary is a single closed curve of length $2n$, with area $[?]$ defined as the number of self-crossings of the wallsystem. The boundary metr…

Let $M$ be a surface with a self-reverse Finsler metric, filling isometrically a circle of length $2L$. The Holmes--Thompson area of $M$ is the area of the surface used here, norma…

Let ${\mathcal M}=(M,\rho)$ be a finite metric space, and let a tree $G$ connect $M$. For each corresponding tour, let the associated half-perimeter be defined as in the preceding…

Let ${\mathcal M}=(M,\rho)$ be a finite metric space. For a tree $G$ connecting $M$, let ${\mathcal O}(M,G)$ be the set of tours of $M$ with respect to $G$, and for each…

Let $D$ be a disc with boundary $\partial D$, and let $g$ be a simple Riemannian metric on $D$. For another Riemannian metric $g'$ on $D$, write $bd_{g'}$ and $bd_g$ for their boun…

Equal-perimeter conjecture. If there exists a tree joining a pseudo-metric space such that all the corresponding perimeters are equal to one another, then the space is additive.

Minimal-filling tour formula. For an arbitrary pseudo-metric space ${\cal M}=(M,\rho)$,

Maximal-tour conjecture. One has

Exact-tour conjecture. Every minimal filling possesses an exact tour.

General-position conjecture. Every minimal filling of a finite metric space in general position is a non-degenerate binary tree.

Let $M$ be a compact Riemannian manifold with boundary. It is a strict minimal filling if, for every compact Riemannian manifold $M'$ with $\partial M'=\partial M$, the inequalitie…

Let $M$ be a compact orientable two-dimensional Riemannian surface whose boundary is a circle of length $2\pi$, and suppose that every pair $x,y$ of opposite boundary points satisf…

Let $M$ be a compact orientable two-dimensional Riemannian surface whose boundary is a circle of length $2\pi$, and suppose that every pair $x,y$ of opposite boundary points satisf…

Let $M$ be a simple compact Riemannian manifold with boundary, and let $bd_M$ denote its boundary distance function. A filling of $(\partial M,bd_M)$ is a compact orientable Rieman…

Minimal filling conjecture. Every simple manifold is a minimal filling. This is the main conjecture of the lecture and concerns whether simple metrics minimize volume among all fil…