14 problems

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…

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 $d\geq 3$, and let $p:[n]\to{\mathbb R}^d$ satisfy $\|p(i)\|=1$ for every $i\in[n]$ and … Assume that the image of $p$ has size at least $3$. Second-largest eigenvalue conjectu…

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…

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

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

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…