6 problems

Duke's torus-realization conjecture. Every triangulated torus can be realized as a polyhedron in $\mathbb{R}^3$.

Hougardy–Lutz–Zelke conjecture. Every triangulation of an orientable surface of genus $g$ with

Equivalence conjecture. After restricting to sufficiently nice spaces and twists, the construction $f:Y\to |\mathscr{X}|\mapsto\omega_f$ should induce an equivalence between the ca…

A path triangulation is a triangulation obtained from paths in a planar cell complex, and a triangulated digital image is a digital image equipped with such a triangulation. Geomet…

Let $C$ be a precubical set, let $\fundtgpd(C)$ and $\fundtcat(C)$ denote the corresponding algebraic structures of formal 2-cells, and let $\dgreal C$ be the geometric realization…

Geometric realization conjecture. Every triangulation of a closed orientable surface of genus at most $4$ is realizable in $\mathbb{R}^3$ by straight edges, flat triangles, and wit…