10 problems

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…

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…

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 $(X,\mathsf{d},\mathfrak{m})$ 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…

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

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

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…

Let the unit disk carry mixed Dirichlet--Neumann boundary conditions, with the Dirichlet part varying among boundary arrangements; the uniform $2$--partition is the arrangement con…