939 problems

Rudnick–Sarnak conjecture. If $M$ is a compact Riemannian manifold of negative curvature, then every orthonormal basis of Laplacian eigenfunctions on $M$ should be quantum unique e…

Let $M$ be a compact manifold. The analytic torsion is the invariant defined from the zeta-regularized determinants of the Laplacians on differential forms, while the R-torsion is…

Hilbert–Pólya-type conjecture. The nontrivial zeros of $\zeta(s)$ should be realized as eigenvalues of a self-adjoint operator; the coherence Hamiltonian provides an analogue whose…

Rauch's hot spots conjecture. The extrema of solutions of the Neumann heat equation should tend toward $\partial\Omega$ as time tends to infinity; equivalently, $u$ should attain i…

Let $m$ be a positive integer and consider all $m$-gons of a fixed area. The affine-orbit part of the Pólya–Szegő polygonal conjecture. The regular $m$-gon minimizes the first Diri…

Let $\O\subset\mathbb R^2$ be a bounded domain, and let $u$ be a second eigenfunction of the Dirichlet Laplacian eigenvalue problem on $\O$. A closed nodal line is a closed compone…

Let $\Omega$ range over bounded domains in $\mathbb{R}^n$ with fixed volume, and let $\beta<0$ be the Robin parameter. Bareket's conjecture. The ball maximizes the first Robin eige…

Let a planar domain have fixed area, and let its first Dirichlet eigenvalue of the Laplacian be denoted by $\lambda_1$. Rayleigh's conjecture. Among planar domains of fixed area, t…

Let $Y$ be a hyperbolic integral homology sphere, and let $\lambda_1^*$ denote the first positive eigenvalue of the Laplacian on coexact $1$-forms. Ozsváth–Szabó spectral-gap predi…

Let $\Omega \subset \mathbb{R}^d$ be a bounded domain, and consider the overdetermined elliptic problem referred to in the paper as equation, whose solution has constant forcing an…

Let $\Omega\subset\mathbb R^N$ be a bounded domain, let $B\subset\mathbb R^N$ be a Euclidean ball, and set $A_N=\lambda_2(B)/\lambda_1(B)=j_{\nu+1,1}^2/j_{\nu,1}^2$, where…

Let $\Omega\subset\mathbb{R}^N$ be a bounded domain, let $\mu_1(\Omega)\leq\mu_2(\Omega)\leq\cdots$ denote its first nontrivial Neumann eigenvalues of the Ornstein–Uhlenbeck operat…

Let $M_g$ be a closed orientable surface of genus $g$. Let $\mathrm{chr}(M_g)$ be the supremum of the natural numbers $n$ for which the complete graph on $n$ vertices can be embedd…

Let $(\Omega,g)$ be an $n$-dimensional compact $C^{\infty}$ Riemannian manifold with boundary $M$. Let ${\mathcal D}$ be the Dirichlet-to-Neumann operator on $M$, and let $u_h$ be…

Let $(\Omega^{n+1},g)$ be an $(n+1)$-dimensional smooth compact connected Riemannian manifold with smooth boundary $\Sigma$. Assume that the Ricci curvature of $\Omega$ satisfies…

Let $X$ be a typical hyperbolic surface, and consider the spectral statistics of its Laplacian. Bohigas–Giannoni–Schmidt conjecture. Because the geodesic flow is time-reversal symm…

Let $(M,g)$ be a compact $C^{\infty}$ Riemannian manifold of dimension $n$, let $\varphi_{\lambda}$ be an $L^2$-normalized eigenfunction satisfying … and let…

Let $M$ be a compact, embedded minimal hypersurface in the unit sphere $\mathbb{S}^{n+1}$, and let $\lambda_1$ denote the first non-trivial eigenvalue of its Laplacian. Yau's conje…

Let $M$ be an embedded minimal surface in the unit sphere $\mathbb{S}^n$, and let $\lambda_1(M)$ denote its first non-zero Laplacian eigenvalue. Yau's conjecture. One has … The con…

Let $\Sigma^n\subset\mathbb{R}^{n+1}$ be a complete properly embedded self-shrinker satisfying the self-shrinker equation, and let $\lambda_\rho(\Sigma)$ denote its weighted first…

Let $M$ be a compact $n$-dimensional Riemannian manifold, let $u_\lambda$ satisfy … where $\lambda$ is the eigenvalue, and write $Z(u_\lambda)=\{u_\lambda=0\}$ for its zero set. Ya…

Let $(\mathcal{M},g)$ be a manifold with a smooth metric, let $E_n$ be a Laplace eigenvalue, let $g_n$ be an associated eigenfunction, and let $\mathcal{Z}(g_n)$ denote the volume…

Buser's eigenvalue-sequence conjecture. There exist sequences of closed hyperbolic surfaces whose first eigenvalue tends to $\frac14$.

Let $(X,g)$ be a smooth compact Riemannian surface of negative curvature, and let $\psi_\lambda\in L^2$ satisfy … Here $\\|\psi_\lambda\\|_{L^\infty}$ and…

Let $M^n$ be a closed embedded minimal hypersurface of the sphere $\mathbb{S}^{n+1}$, and let $\lambda_1(M)$ denote the first positive eigenvalue of its Laplacian. Yau's conjecture…