33 problems

Let $\Omega\subset\mathbb R^d$ be a bounded domain, and let $\{\lambda_k^1\}_{k\in\mathbb N}$ and $\{\mu_k^1\}_{k\in\mathbb N}$ denote respectively the Dirichlet and Neumann spectr…

The QALE Laplacian index-set and Fredholm conjecture. The displayed description of $\mathcal{I}_X$ holds, and the displayed map is Fredholm, with finite-dimensional kernel and coke…

For $n\geq 2$, let $\triangle_1(\lambda)$ be the Laplacian on 1-forms in the representation with parameter $\lambda$, and let $U_{ij}$ be the transposition operators defined from t…

Kigami–Lapidus lattice-case conjecture. The limit

Let $\Omega\subset\mathbb{R}^d$ be a domain, and let $\lambda_j(\Delta_D)$ and $\lambda_j(\Delta_N)$ denote the $j$-th Dirichlet and Neumann Laplacian eigenvalues. Define … where…

Let $\Omega\subset\mathbb{R}^d$ be a domain, and let $\lambda_j(\Delta_D)$ and $\lambda_j(\Delta_N)$ denote the $j$-th Dirichlet and Neumann Laplacian eigenvalues, respectively. Be…

Let $\Omega$ be a simply connected Lipschitz domain, and let $D$ be a line segment in the boundary. Consider the first mixed Dirichlet-Neumann eigenvalue problem, with Dirichlet co…

Let $M$ be a complete noncompact Riemannian manifold with finite exponential rate of volume growth $\gamma<\infty$. Let $\Delta$ denote the Laplacian on $k$-forms, and assume that…

Kwon–Lee resolvent conjecture. On the stripe $\mathcal{S}$, the estimate

Strict tessellation conjecture. $\Omega$ has a complete set of trigonometric eigenfunctions for the Laplace eigenvalue equation with the Dirichlet boundary condition if and only if…

Let $(M,g)$ be a Kähler manifold. It satisfies the Δ-property if, for every point $x\in M$, there is a coordinate system $z$ centered at $x$ such that every smooth function $\phi$…

Let $(X,\mathsf{d},\mathfrak{m})$ be the RCD space under discussion, let $k=\dim(X,\mathsf{d},\mathfrak{m})$, and let $\mathcal{R}_k^*$ be the full-measure regular set. Let $\phi$…

Let $(X,\mathsf{d},\mathfrak{m})$ be an $\operatorname{RCD}(K,N)$ space. For $f\in D(\Delta)$, let $\mathrm{Hess}_f$ be its Hessian, viewed as a $(0,2)$-type $L^2$ tensor, and let…

Let $IMC$ denote the infinite magic carpet, and let the countable family of bounded periodic eigenfunctions constructed in the paper be given. Spectral resolution conjecture. These…

Resolvent estimate conjecture. There exists a constant $C$, depending only on $p$, $q$ and $d$, such that for every $z\in\mathbb C\setminus[0,\infty)$,

Strichartz's conjecture. There exists a uniformly almost periodic function $g$ such that

Strichartz's conjecture. The function $A(t)$ is bounded and decays on the order of $O(t^{-1/4})$ as $t\to\infty$ in the flat or negative-curvature case. This is numerical evidence…

Let $Omegasubsetmathbb{R}^d$, $dgeq 2$, be a domain satisfying suitable regularity assumptions, and let $R^Omegaalpha$ be the Laplacian in $L^2(Omega)$ with Robin parameter…

Assume that the fractal has three boundary vertices and that all structures have full $D3$ symmetry. Let $\rho$ be the common value of $r_jbc_j$, let $bb_3$ be the common value of…

Connectedness conjecture. The essential spectrum of $-\triangle_g$ on functions is a connected subset of the positive real line $[a,\infty)$.

If there were an infinite number of directions, so that the graph is non-quasicrystalline, the Green function would still exponentially decay to $0$. Asymptotic Green-function conj…

Let $\overline{M(n,K,d)}$ denote the closure of the moduli space of compact Riemannian manifolds with dimension $n$, Ricci curvature bounded below by $K$, and diameter bounded abov…

Let $G$ be a compact subgroup of $O(n)$, and let a region of $\mathbb{R}^n$ be $G$-symmetric when it is invariant under the natural action of $G$. The Dirichlet integral defines a…

Let $X$ be a smooth manifold with a constant-rank distribution $W$, and let $\mathcal{J}$ be the Pfaffian system generated by the annihilator of $W$. Let $\mathcal{Q}$ be the ortho…

Let $M$ be a complete noncompact Riemannian manifold with Ricci curvature bounded below. Denote by the $L^2$ essential spectrum of the Laplacian on functions the spectral set assoc…