23 problems

Critical nodal-set conjecture. The set $\{f=0\}$ has no unbounded connected components.

Let $(M,g)$ be a real analytic Riemannian manifold with ergodic geodesic flow, and let $\{\varphi_j\}$ be a density-one sequence of ergodic eigenfunctions. For a test function $f$,…

Let $D\subset\mathbb{R}^2$ be a convex domain symmetric with respect to both coordinate axes. Let $\lambda_n$ and $\varphi_n$ be the eigenvalues and eigenfunctions of the Cauchy pr…

Let $\Omega\subseteq\mathbb{R}^2$ be a connected planar domain, and let $u_3$ be an eigenfunction corresponding to a simple third eigenvalue $\lambda_3$ of the Dirichlet Laplacian…

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…

Let $B\subset\mathbb{R}^{n}$ be a unit ball with $n\geq 3$. Let $u:4B\to\mathbb{R}$ be harmonic and satisfy $u(0)=0$, and let $N_u(0,\frac{1}{2})$ denote its doubling index at the…

Let $u$ be a harmonic function in $B(0,1)\subset\mathbb{R}^n$ with $u(0)=0$, and let $Z(u)$ denote its nodal set. Nadirashvili's lower-bound conjecture. There exists a universal co…

Let $u$ solve the divergence-form elliptic equation … in $B(0,1)$, with … and … Let $Z(u)$ be the nodal set, $S(u)$ the singular set, and $\beta_u$ the frequency quantity used in t…

Let $B\subset\mathbb{R}^n$ be a unit ball, with $n\geq 3$, and let $u$ be harmonic in $4B$, vanish at the center of $B$, have stable growth in $B$, and satisfy…

Let $M$ be the compact manifold, let $D$ be the relevant geometric parameter, and let $\mathcal{F}_{\lambda_j}$ denote the corresponding eigenspaces. Let…

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 $\Omega \subset \mathbb R^d$ be a Lipschitz domain and let $\Sigma=B\cap\partial\Omega$ for a ball $B$ centered at a point of $\partial\Omega$. Let $u$ be harmonic in $\Omega$,…

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…

Bourgain–Rudnick conjecture. The lower bound for the number of nodal intersections is of order $\lambda$; equivalently,

Let $\Omega\subset \mathbb{R}^2$ be a bounded, simply-connected domain with real analytic boundary that is not equal to $B(x,r)$ for any $x\in \mathbb{R}^2$, $r>0$. A tunneling Ste…

Let $\Omega\subset\mathbb{R}^n$ be bounded, open, convex, and symmetric, and let $u$ be a Neumann eigenfunction with the least non-zero eigenvalue. Lipschitz nodal-set conjecture.…

Let $f_l$ be the random spherical harmonic and let $V$ be the vector field on $S^2$ considered above. Writing $DF_l$ for the derivative of the map $F_l=(f_l,Vf_l)$ and…

Let $D$ be a non-equilateral triangle, and let $u$ be a second Neumann eigenfunction on $D$. Call $D$ superequilateral if it is isosceles with aperture angle greater than…

Let $\cal H_\ell$ be the eigenspace of spherical harmonics of degree $\ell$ on the sphere, and let $\mu(u)$ denote the number of nodal domains of a spherical harmonic $u$. Leydold'…

Let $(M,g)$ be a compact smooth manifold without boundary. Let $u$ be a Laplacian eigenfunction with eigenvalue $\lambda$, and let $p_t(x)$ be globally defined with respect to its…

Let $(M^2,g)$ be a real analytic Riemannian surface with ergodic geodesic flow, and let $\gamma_{x,\xi}$ be a geodesic satisfying the QER hypothesis. Let…

Let $\mathbb T^2$ be the two-dimensional torus, let $\varphi$ be an eigenfunction with eigenvalue $E=\lambda^2$, and let $C\subset\mathcal N_\varphi$ be a regular arc. For a convex…

Let $(M,g)$ be a compact smooth Riemannian manifold with ergodic geodesic flow, and let $(\psi_j)_{j\geq1}$ be any orthonormal basis of eigenfunctions. For an eigenfunction, let…