93 problems

Berry's conjecture. The rescaled restriction of $\varphi_i$ to that wavelength-scale ball should behave like $\phi^{\mathbb{R}^n}_1$. This conjecture proposes a universal local mod…

Let $\phi_\lambda$ be an $L^2$-normalized Laplace eigenfunction on an arithmetic surface, with eigenvalue parameter $\lambda$. Iwaniec's conjecture. For every $\varepsilon>0$, one…

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 $(M,g)$ be a compact hyperbolic surface, let $e_\lambda$ be an eigenfunction of $\sqrt{-\Delta_g}$ with eigenvalue $\lambda$, and let $\gamma$ be a periodic geodesic or a geode…

Let $X$ be a compact negatively curved manifold, let $\Delta_g$ be its Laplace operator, and let $\{\psi_{\lambda_j}\}_{j=1}^\infty$ be an orthonormal basis of eigenfunctions. For…

Let $(M,g)$ be a smooth boundaryless compact connected Riemannian manifold of dimension $d$, let $\psi_\lambda$ be an eigenfunction with eigenvalue parameter $\lambda$, and let…

Let $M$ be a closed compact Riemannian manifold, and let $u_\lambda$ be a $\lambda$-eigenfunction on $M$. Denote its nodal set by $\{u_\lambda=0\}$ and its $(n-1)$-dimensional volu…

Let $X$ be a compact hyperbolic surface, and let $(f_i)$ be an orthonormal sequence of Laplace eigenfunctions on $X$, with eigenvalues satisfying $(\Delta+\lambda_i^2)f_i=0$. Call…

Steinerberger's conjecture. There exists a constant $C$, depending only on $p$ and on the manifold, such that

Let $X=G/K$ be a non-compact Riemannian symmetric space, where $G$ is a connected semisimple Lie group and $K$ is a maximal compact subgroup. A joint eigenfunction is a function on…

Let $\Omega\subset\mathbb{R}^n$ be a bounded convex domain, and let $u$ be a first Dirichlet eigenfunction of $\Omega$. Write $\operatorname{diam}(\Omega)$ for its diameter and…

Let $(M,g)$ be a compact $C^{\infty}$ Riemannian manifold without boundary, and let $u$ be an eigenfunction of the Laplace–Beltrami operator with eigenvalue $\lambda$. Its nodal se…

Yau's conjecture. There are positive constants $c_1,c_2$, depending only on $(M,g)$, such that

Let $(M,g)$ be a compact Riemannian manifold of dimension $d$, and let $e_\lambda$ be a real eigenfunction with eigenvalue parameter $\lambda$. Define its nodal set by … and write…

Let $M$ be a closed Riemannian manifold, let $e(x,y,\lambda)$ denote the spectral function of its Laplace–Beltrami operator, let $\alpha$ and $\beta$ be multi-indices, and let…

Weyl-antisymmetry conjecture. This polynomial satisfies

Let $Y$ be a compact hyperbolic surface, let $\phi_i$ be a Laplace eigenfunction with eigenvalue $\mu_i$, and let $a_n^{\sigma/\ell}(\phi_i)$ denote the Fourier coefficient associa…

Let $Y$ be a compact hyperbolic surface, let $\phi_i$ be a Laplace eigenfunction on $Y$ with eigenvalue $\mu_i$, and let $\ell$ be a geodesic on $Y$. Write…

Let $(M,g)$ be a compact Riemannian manifold of dimension $n$ with a stable elliptic orbit, and let \\{\varphi_k\} be a sequence of eigenfunctions with eigenvalues denoted by…

Let $M$ be a smooth, compact Riemannian manifold, let $\phi$ be a Laplace eigenfunction with eigenvalue $\lambda$, and let $Z(\phi)=\{x\in M:\phi(x)=0\}$ denote its nodal set. Its…

Fix a prime $p$, choose a standard observable $\xi$, and let $\Psi_\chi\in \mathrm{L}^{2}(\mathbb{F}_{p},\mathbb{C})$ be a Hecke eigenvector in the standard realization of the Weil…

Let the first eigenfunction be normalized in $L_p$ for some $p>1$, and consider its associated limit measures in the setting of the paper. Normalization conjecture. Under $L_p$ nor…

Local symmetry conjecture. For any points $P_1,P_2\in C_{\bar n}$, there exist open neighborhoods $\Omega_1,\Omega_2$ and an isometry

Local isometry conjecture. Any small neighborhood $V_1$ of $P_1$ is conjugated to a neighborhood $V_2$ of $P_2$ by an isometry.

Uniqueness and global-maximum conjecture. The coefficients $c_k$ are unique, and whenever the degeneracy cannot be removed, the charged points are charged by a global maximum seque…