7 problems

Intrinsic metric homeomorphism conjecture. The map $\mathcal{I}_-^s$ is a homeomorphism.

Let $(M,V,\varepsilon)$ be a closed connected weighted marked surface, where $\varepsilon:V\rightarrow\{0,1\}$, $\chi(M)=0$, and $d$ is a piecewise linear metric on…

Let $X_p$ be the doubled regular $p$-gon, where $p$ is an odd prime. The minimizing index $\operatorname{minind}(X_p)$ is the smallest positive integer $k$ such that $X_p$ admits a…

Let $\Delta_{sing}$ be the Laplacian on the singular polyhedral surface considered in the paper, and let $T(\lambda)$ be the associated operator-valued function in the determinant…

A regular polygon surface (RPS) is a polyhedral surface whose faces are regular polygons. Dodecahedra and truncated icosahedra realization conjecture. Any genus-zero RPS with faces…

Let $W(u)$ be the convex potential for the discrete curvature map, let $K^*=(K_1^*,\ldots,K_n^*)$ be the target curvature, and consider the gradient flow on the admissible paramete…

Weak convergence conjecture. If $\delta_2(P_h,P_h^\star)\to0$ as $h\to0$, then