15 problems

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 $\mathcal{H}$ be a hypergraph on $n$ vertices, and let the degree of a vertex in a triangulation be the number of incident faces. Bounded-degree sphere triangulation conjecture…

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

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…

Let a simply connected triangulated surface in space have discrete Dirac operator $\operatorname{D}$. A triangulated-surface Dirac nullity conjecture. The nullity … is invariant un…

Let $S$ be a closed, unweighted triangulated surface of genus $g$ with $n$ triangles, and consider closed curves whose edges lie in the triangulation. The relevant shortest non-con…

Let $S$ be a closed, unweighted triangulated surface of genus $g$ with $n$ triangles, and consider closed curves whose edges lie in the triangulation. Hutchinson's conjecture. The…

Right-equivalence conjecture. The QPs $\mu_i(Q(\tau),S(\tau,\mathbf{x}))$ and $(Q(\sigma),S(\sigma,\mathbf{x}))$ are right equivalent, not merely weakly right equivalent.

The convex-polytope conjecture. The map $\Phi_h\colon T_c(S)\times\mathbb{R}_{>0}^V\longrightarrow\mathbb{R}^E$ is a smooth embedding, and its image is a convex polytope.

Let $h\in\bold R$ and let $(S,T)$ be a closed triangulated surface. For the discrete curvatures $k_h$, $\phi_h$, and $\psi_h$ associated with the polyhedral, circle packing, Euclid…

Non-degeneracy and non-rigidity conjecture. If $\Sigma$ has empty boundary, then $(Q(\tau),S(\tau))$ is non-degenerate but non-rigid.

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…