353 problems

For every integer $n\ge 3$ and every convex planar $n$-gon $P$ of area $A$, let $\lambda_1^N(P)$ denote the first nonzero eigenvalue of the Laplace operator with Neumann boundary c…

For every bounded simply connected planar domain $\Omega$ with infinite-mass boundary condition, every mass $m\ge 0$, and the disk $B_{|\Omega|}$ having the same area as $\Omega$,…

For every integer $d\ge 1$, every bounded open set $\Omega\subset\mathbb{R}^d$ of finite positive volume $|\Omega|$, and every $k\in\mathbb{N}$, if $\lambda_k(\Omega)$ denotes the…

Let $T$ be a nondegenerate planar triangle, and let $0<\lambda_1^N(T)\leq\lambda_2^N(T)$ be its first two positive Neumann Laplace eigenvalues. Define the harmonic and geometric me…

Let $\Omega\subset\mathbb{C}$ be a bounded simply connected Lipschitz domain equipped with the conformal metric $g=\omega\lvert dz\rvert^2$, where…

The Schiffer conjecture asserts that if a bounded connected domain $\Omega\subset\mathbb{R}^n$ with sufficiently smooth boundary admits some $\lambda>0$ and a nonconstant function…

For the relevant semiclassical magnetic Schrödinger inverse problems, the spectral data should determine the coefficients in the following symmetry-restricted settings: (i) an even…

Let $\Omega\subset\mathbb{C}$ be a bounded simply connected Lipschitz domain, let $\omega\in C^2(\Omega)\cap C(\overline{\Omega})$ be positive on $\overline{\Omega}$, and equip…

Let $(M,g)$ be a smooth closed Riemannian manifold of dimension at least $3$, let $\Sigma\subset M$ be a fixed smooth submanifold of codimension $2$, and let $\phi_\lambda$ be an…

For every integer $n\ge 3$ and every area $A>0$, among all convex $n$-gons $P\subset\mathbb{R}^2$ with $|P|=A$, the regular $n$-gon $R_{n,A}$ uniquely minimizes the first Dirichlet…

Let $M=\Gamma\backslash\mathbb{H}_d$ be a compact Heisenberg manifold, let $N_M(\lambda)$ be the spectral counting function of its sub-Laplacian, and write…

Let $\Omega\subset\mathbb{Q}_p^d$ be an admissible bounded domain, let $D_{\Omega}^{\alpha}$ denote the Dirichlet realization of the Vladimirov–Taibleson operator, and let…

For every integer $n\ge 3$, every compact $n$-dimensional Riemannian manifold $(M^n,g)$ with nonnegative Ricci curvature, $\operatorname{Ric}_g\ge 0$, and every principal curvature…

For every integer $n\ge 2$, if $(M^{n+1},g)$ is a compact Riemannian manifold with boundary satisfying $\operatorname{Ric}_g\ge n g$ and $\operatorname{II}_g\ge 0$ on $\partial M$,…

Does there exist a universal constant $C>0$ such that, for every connected graph $G=(V,E)$ with boundary $B$, maximum degree $d_{\max}$, and genus $g$, one has…

Let $\Omega\subset\mathbb{R}^{d}$ be a bounded domain, and let $\lambda_k(\Omega)$ and $\mu_k(\Omega)$ denote its $k$-th Dirichlet and Neumann Laplacian eigenvalues, respectively.…

Let $T$ be a bounded triangle in the Euclidean plane, and let the Dirichlet eigenvalues of its Laplacian, listed with multiplicity, be … An equilateral triangle is a triangle whose…

Let $\Omega\subset\mathbb{R}^n$ be compact with Lipschitz boundary, and let $w$ be the single-term Weyl function of $\Omega$. The Dirichlet Laplace operator is superspectral to $w$…

Let $\Sigma$ be an embedded asymptotically flat surface in $R^3$ that is not totally geodesic, and suppose its Gauss curvature is integrable. Consider the quantum layer built from…

Let $(M,g)$ be a compact Riemannian manifold whose geodesic flow is completely integrable. Let $\Delta_g+V$ be a Schrödinger operator on $(M,g)$, and suppose that all of its orthon…

Selberg's eigenvalue conjecture. For every $N$,

Let $(M,g)$ be a compact Riemannian manifold without boundary, and let $\varphi$ be an eigenfunction satisfying … Write $\mathcal{L}_{\varphi}(M)$ for its nodal volume. Yau's conje…

Fix $\gamma\in(0,2)$, let $\mu_\gamma$ be the Liouville measure on a bounded domain $D$, and let $(\mathbf{f}_n)$ be eigenfunctions normalised to have unit $L^2(\mu_\gamma)$ norm.…

Let $\Omega\subset\mathbb R^2$ be a bounded domain, and consider the second eigenfunction of the Laplacian on $\Omega$ with Dirichlet boundary condition. Payne's nodal line conject…

Let $\mathscr{N\!D}_n$ be the class of completely integrable systems on a $2n$-dimensional symplectic manifold with non-degenerate singularities. Let $Q(\mathscr{N\!D}_n)$ be the c…