9 problems

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…

Genericity conjecture. Triple intersection points are conjectured to be generic for large $k$ in spectral partitions.

Let $d\geq 2$. Let $\Gamma_d(\lambda)$ denote the maximal order of vanishing of a Laplacian eigenfunction on the $d$-dimensional torus with frequency parameter $\lambda$, and let…

Strichartz's conjecture. For these values of $\lambda$, such eigenfunctions are characterized by an $L^2$-weight norm on $X$.

Let $(M,g)$ be a smooth, compact Riemannian manifold without boundary, let $c(phi)$ be an $L^2$-normalized Laplacian eigenfunction, and let $c(lambda)$ be its eigenvalue, so that ……

Local-extrema count conjecture. The numbers of local extrema of the restrictions of $\psi_n^{(2)}$, $\psi_n^{(5)}$, and $\psi_n^{(6)}$ to the bottom edge of $\mathcal{SG}$ are, res…

Let a Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime be expressed in comoving coordinates, and let its Killing vectors have components in those coordinates. Consider also max…

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 $M$ be a smooth manifold without boundary, let $\Delta$ be its Laplacian, and let $u_2$ be a second eigenfunction of $\Delta$. Let $d_M$ denote geodesic distance on $M$. Geodes…