6 problems

A bounded planar domain is equipped with a decomposition of its boundary into Dirichlet and Neumann parts, and Dirichlet-Neumann isospectrality means that swapping these boundary c…

Let $q$ be a positive integer, let $s_1,\dots,s_n,s'_1,\dots,s'_n$ be the parameters defining the lens orbifolds $L(q;s_1,\dots,s_n)$ and $L(q;s'_1,\dots,s'_n)$, and let $d$ be a d…

The divisibility conjecture. If $L(q;s)$ and $L(q;s')$ form a pair of non-isometric $\{2\}$-isospectral $7$-dimensional lens orbifolds, then $3$ divides $q$; respectively, if they…

The five-dimensional $\{2\}$-isospectrality conjecture. The families of non-isometric $\{2\}$-isospectral $5$-dimensional lens orbifolds are precisely given by pairs

The manifold inheritance conjecture. If $L$ and $L'$ are $p$-isospectral for some $0<p\leq n-1$ and $L$ is a lens space, then $L'$ is also a lens space.

Let $L_{1;a,b,c,d}$ and $L_{2;a,b,c,d}$ be the two lattices in Conway and Sloane's real four-parameter family, with parameters $(a,b,c,d)\in\mathbb R^4$. Conway–Sloane's conjecture…