24 problems

Let $\Omega\subset\mathbb R^n$ have a fractal boundary $\partial\Omega$ with Hausdorff dimension $H\in[n-1,n]$. Let ${\Greekmath 011A}(x)$ be the eigenvalue counting function for t…

Let $M$ be a manifold with Riemannian metric $g$, let $K$ be a triangulation of $M$ with mesh tending to zero, and let $\beta_j(M,g)$ be the analytic invariant. Let $\beta_j(K,g)$…

Let $M$ be a manifold with Riemannian metric $g$, and let $\beta_0(M,g)$ denote the invariant in degree zero, corresponding to functions. Sunada's positivity conjecture. … Sunada p…

Let $M$ be a manifold with Riemannian metric $g$, and let $\beta_j(M,g)$ denote the corresponding bottom spectral invariant in degree $j$. Positivity conjecture. The quantity … is…

Let $X$ be a finite simplicial complex, let $csigma_{\max}(\partial_k)$ denote the largest singular value of its $k$-th boundary operator, and let $L_k^{\mathrm{up}}$ denote the…

Laplacian recurrence conjecture. The observed recurrence relation for $\Pi_n(\lambda)$ continues to hold for larger $n$.

Let $\mathrm{SG}$ be the Sierpiński gasket with Laplacian $\mathcal{L}$, and suppose $\mathcal{L}f$ is Hölder continuous on $\mathrm{SG}$. Let $\mathcal{L}_\epsilon$ and…

Let $\mathrm{SG}$ be the Sierpiński gasket with its standard Laplacian $\mathcal{L}$, and define the cellular semimetric by … Let $\mathcal{L}_\epsilon$ be the averaging operator o…

Let $p:[n]\to{\mathbb R}^d$ be injective, let $G$ be a graph with vertex set $[n]$, and write $\lambda(G,p)=\lambda_{\binom{d+1}{2}+1}(L(G,p))$. For $k\geq 1$, let…

Let $p:[n]\to{\mathbb R}^d$ be injective. Large-eigenvalue sum conjecture. The sum of the $n$ largest eigenvalues of $L(K_n,p)$ is at least … The conjecture improves the previously…

Let $\mathcal{B}_h$ be the metric graph associated with the complete rooted binary tree $B_h$ of height $h$, and let $l$ be its edge-length function. The estimates under comparison…

Let $\Delta_{sing}$ be the Laplacian on the singular polyhedral surface considered in the paper, and let $T(\lambda)$ be the associated operator-valued function in the determinant…

Boundary-value decay conjecture. For $r\neq 1$ and $r\neq\frac{\sqrt{17}-3}{4}$,

The ratio-existence conjecture. For all but at most finitely many values of $r$,

The asymptotic ratio conjecture. For $r\neq \frac{\sqrt{17}-3}{4}$,

The asymptotic ratio conjecture. For $r\neq 1$ and $r\neq \frac{\sqrt{17}-3}{4}$,

Let $q$ be the parameter in the construction of $X^n_\varnothing$, let $0\leq i\leq n-1$, and let $\Delta$ act on $C^i(X^n_\varnothing)$. The positive eigenvalues of $\Delta$ are a…

Graph quantum mechanics has walk sum formulae and gluing formulae, and Laplacians for CW-complexes provide a more general setting. The CW-complex extension conjecture. The same app…

Sign conjecture. All coefficients $x_H$ are integers of the same sign.

Consider the normalized Laplacian $\Delta=I-D^{-1}L$ associated to a regular tessellation of the hyperbolic plane, acting on $\ell^2(V,D)$, and let $\inf\sigma(\Delta)$ denote the…

Almost-commuting Laplacians conjecture. There exists a function $\delta(\epsilon)$ satisfying

Let $H$ be a compatible Laplacian on a manifold with corners of codimension $2$, and call an eigenvalue embedded when it lies in the continuous spectrum of $H$. Generic absence con…

Let $\Delta$ and $\Delta'$ be two Laplacians whose Rayleigh quotients satisfy … For each sign, let $M^{\pm}(\lambda)$ and $(M^{\pm})'(\lambda)$ denote the corresponding upper and l…

Let $Q_n$ be the $n$-dimensional cube, and let $x_i,y_i,q_i$ be commuting indeterminates. For a cubical complex, write…