30 problems

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…

Let $(M,g)$ be a compact Riemannian surface without boundary, let $\Delta_g$ be its Laplace–Beltrami operator, and let $f_{\lambda}$ be a Laplace eigenfunction with eigenvalue…

Let $M$ be a smooth compact $n$-dimensional manifold, let $\Delta_M$ be its Laplace operator, and let $u_\lambda$ be a corresponding eigenfunction satisfying … Write…

Let $(\mathcal{M},g)$ be an $n$-dimensional compact smooth Riemannian manifold, let $\varphi_\lambda$ be a Laplace eigenfunction with eigenvalue $\lambda>0$, and let … be its nodal…

Let $M$ be a closed Riemannian manifold and let $\varphi_\lambda$ be a Laplace eigenfunction. For a locally integrable function, $\|\cdot\|_{BMO}$ denotes its bounded mean oscillat…

Yau's nodal length conjecture. The nodal length satisfies

Let $\Omega$ be a Euclidean domain with boundary $\Gamma$, and let $q(u,v;\mathcal E)$ be the symmetric bilinear form … Here $u_n=\partial_nu$,…

Let $\mathcal{M}$ be a chaotic smooth compact $d$-manifold, and let $\{(\varphi_{j},\lambda_{j})\}_{j\ge 1}$ be the corresponding sequence of Laplace eigenfunctions and eigenvalues…

Yau's conjecture. The $(n-1)$-dimensional Hausdorff measure of the nodal set should satisfy

Let $(M,g)$ be an $n$-dimensional compact Riemannian manifold, let $w$ be an eigenfunction of the Laplace–Beltrami operator with eigenvalue $\lambda$, and let $\mathcal{H}^{n-1}$ d…

Let $(M,g)$ be a compact Riemannian manifold, and let … where each $\varphi_{\lambda_k}$ satisfies … For sufficiently small $r>0$, let $B_g(x,r)$ denote the geodesic ball centered…

Let $(M,g)$ be a compact Riemannian manifold of dimension $d$, and let $\varphi_{\lambda}\in C^{\infty}(M)$ satisfy … For sufficiently small $r>0$, write $B_g(x,r)$ for the geodesi…

Let $(\Omega,D,N)$ be a triple where $\Omega$ is simply connected, $D$ is a Euclidean line segment, and … Say that $(\Omega,D,N)$ has no hot spots when each first mixed eigenfuncti…

Let $M$ be a compact Riemannian manifold and let $psi_\lambda$ be a Laplace eigenfunction with eigenvalue $\lambda$. The volumes of its positive and negative sets are … For the con…

Let $(M,g)$ be a given Riemannian manifold, let $\psi_{\lambda}$ be a real Laplace eigenfunction, and let $1<p<\infty$. For a set $A\subseteq M$, write $\chi_A$ for its characteris…

Let $(M,g)$ be a smooth Riemannian manifold, and let $\psi_{\lambda}$ be a real Laplace eigenfunction satisfying … Write $\operatorname{vol}$ for Riemannian volume. Symmetry conjec…

Let $(\phi_j,\lambda_j)_{j\ge 1}$ be Laplace eigenfunctions on a smooth compact surface $\mathcal{M}$, with eigenvalues $\lambda_j$ ordered increasingly and counted with multiplici…

Let $F^N_{n,m}$ be the $L^2$-normalized eigenfunctions of the Neumann Laplace operator on the unit disk, with eigenvalues $\lambda^N_{n,m}$. For a subsequence with…

Dirichlet disk growth conjecture. If $1\leq\gamma<3$, then

Let $l_s$ denote the length scale associated with a Laplace eigenvalue, and consider the corresponding Laplace eigenfunction on the neuron geometry. Length-scale–wavelength conject…

Yau's conjecture. There are constants implicit in $\ll_M$ such that

Let $M$ be a closed $n$-dimensional Riemannian manifold and let $\varphi_\lambda$ be a non-constant Laplace eigenfunction. Write $\{\varphi_\lambda>0\}$ and $\{\varphi_\lambda<0\}$…

Let $M$ be an $n$-dimensional C^-smooth closed Riemannian manifold, and let $varphi_\lambda$ be a Laplace eigenfunction satisfying … Here $\mathcal{H}^{n-1}$ denotes $(n-1)$-dime…

Bourgain and Rudnick's conjecture. For all $\epsilon>0$ and $s<R^{1-\delta}$,