18 problems

Let a closed hyperbolic surface be equipped with its hyperbolic structure, and let its length spectrum be the multiset of lengths of closed geodesics, counted with multiplicities.…

Okada et al.'s conjecture. The pole distributions of the scattering matrices in the exteriors of isospectral domains in $\mathbb{R}^{2}$ are different.

Let $G$ be a compact Lie group, let $g_0$ be a bi-invariant metric on $G$, and consider the class of all Riemannian metrics on $G$. A metric is spectrally isolated if some neighbor…

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 $s$ denote the fractional order, with $0<s<1$, and consider the spectra of two domains for the corresponding fractional Laplacian. Fractional-order spectral distinction conject…

Let two domains be such that their spectra for the Laplacian are identical. Fractional Laplacian spectral distinction conjecture. Their spectra for the fractional Laplacian would n…

A nodal count is the number of nodal domains of an eigenfunction, namely the connected components of the complement of its zero set. Two planar domains are isospectral when their L…

Let $M$ be the black-hole mass, $\omega$ the frequency, and $f(z)=1-1/z$. For the radial Teukolsky equation and the alternative equation … with … where $c=a\omega$, the rotation pa…

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.

The $\{0,\dots,n-2\}$-isospectrality conjecture. If $L$ and $L'$ are $p$-isospectral for all $p$ satisfying $0\leq p\leq n-2$, then they are also $(n-1)$-isospectral. Consequently,…

Nonexistence conjecture. There are no pairs of untwisted isospectral non-isometric orbifolds covered by $S^n$ with $n\leq 4$.

Nodal-domain conjecture. For some instances of isospectral pairs, the two isospectral domains produce a different number of nodal domains.

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…

Self-isospectrality conjecture. Any finite-gap system having antiperiodic singlet states admits a self-isospectral tri-supersymmetric extension with the partner potential equal to…

Let $G$ be a compact simple Lie group, let $g_0$ be a bi-invariant metric on $G$, and consider the class of left-invariant metrics on $G$. A metric is globally spectrally determine…