8 problems

Let $V(I')$ be the affine variety defined by the specialized ideal $I'$ associated with the specialized spectral-invariant system, and let $q$ denote the period. A point is regular…

Let $q>3$, let $V$ be a nonzero $q\mathbb{Z}$-periodic potential of the form … and let $\Delta$ denote the discrete Laplacian. Floquet-isospectral conjugate-symmetric potential con…

Let $q>3$, let $V$ be a nonzero $q\mathbb{Z}$-periodic potential of the form … and let $\Delta$ denote the discrete Laplacian. Floquet-isospectral potential conjecture. There exist…

Commutation-of-limits conjecture. For sufficiently small $\epsilon_1$ and $\epsilon_2$, the scaling limit exists uniformly with respect to $T_{\mathrm{cr}}$; the deformation limits…

Uniform small-deformation scaling-limit conjecture. The small-deformation limit exists uniformly in $T_{\mathrm{cr}}$, and the scaling limit is

Let the combinatorial Laplacian be defined on a non-rectangular region, such as a rectilinear region in $\mathbb{Z}^2$, and let its determinant have the asymptotic expansion descri…

Let $n\geq 5$, and consider all $n$-gons with a fixed area and their first eigenvalue of the Laplace operator. Regular polygon conjecture. It is conjectured that the regular $n$-go…

Higher-dimensional rigidity conjecture. The discrete Laplace–Beltrami operator (or equivalently the discrete heat kernel) and the discrete metric of $M$ mutually determine each oth…