7 problems

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

Let $(M,V,\varepsilon)$ be a closed connected weighted marked surface, where $\varepsilon:V\rightarrow\{0,1\}$. Consider an initial Euclidean or hyperbolic polyhedral metric induced…

Let $G$ be a locally cyclic graph with boundary and minimum degree $\delta\ge 4$ triangulating a disc. Such a graph is clique null if its clique dynamics converges to the one-verte…

Let $G$ be a locally cyclic graph of minimum degree $\delta\ge 4$ triangulating a closed surface of Euler characteristic $\chi\ge 0$; equivalently, the surface is a sphere, project…

Let $\mathbb{T}$ be a triangulation of a not necessarily orientable surface, and suppose that there is a rigid potential $W_l$ on the quiver-with-potential $\Diamond_l(\mathbb{T})$…

Let an abstract triangulated sphere be realized in $\mathbb{R}^3$, and let $\operatorname{D}$ denote its discrete Dirac operator. A triangulated-sphere realization conjecture. Ever…

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…