7 problems

General-dimensional characterization conjecture. The following conditions are equivalent:

Global rigidity and self-stress conjecture. The framework $(G,\bm p)$ is globally rigid in $\ell_p^d$ if and only if it has a $k$-th coordinated self-stress $\omega^k$ for some $k$…

Mean-bounded ergodic convergence conjecture. The sequence $(A_nx)_{n\geq1}$ converges bilaterally almost uniformly to $Px$ for every $x\in L_p(\mathcal{M})$. If additionally…

For $2<q<p$ and $m,n\in\mathbb N$, let $[m]_q^n$ denote the $n$-dimensional integer grid with its $\ell_q$ metric, and let $c_p([m]_q^n)$ be its least bi-Lipschitz distortion in…

Nica's conjecture. The mapping class group of a surface admits a proper affine isometric action on an $L_p$-space for some sufficiently large

Gromov's conjecture. The reduced $\ell_p$-cohomology of $G$ vanishes for

Let $1<p<\infty$ with $p\ne 2$, let $\Omega$ be a measure space, and let $T$ be a contraction on $L^p(\Omega)$. Let $S$ be the right shift on $\ell^p$, defined by … For a complex p…