15 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 $k$…

Kitson and Power's conjecture. A graph $G$ is locally rigid in $\ell_p^d$ if and only if $G$ is $d$-tree-connected.

For real $p\geq 1$, let $\ell_p^d$ be $\mathbb R^d$ equipped with the norm … and let $\ell_\infty^d$ use the norm $\|\mathbf x\|_\infty=\max_i|x_i|$. Write $b(\ell_p^d)$ for the co…

Suppose $(X,d)$ is a metric space and $X=A\cup B$, where $c_p(A)$ and $c_p(B)$ are finite. Here $c_p(S)$ denotes the least bi-Lipschitz distortion with which the metric space $S$ e…

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…

Let $2<p<\infty$, and let $X$ be a subspace of an $L^p$ space over an arbitrary non-negative measure such that $X\not\supseteq \ell^p(\omega_1)$. The conjecture asserts that…

Let $1<p<\infty$, $p\neq 2$, and let $(\Omega,\mu)$ be a finite measure space such that $L^p(\mu)$ is not separable. The Enflo–Rosenthal conjecture asserts that $L^p(\mu)$ is not a…

Union conjecture for $\ell_p$. The answer is negative for every $p\notin\{2,\infty\}$.

Let $G$ be a locally compact second countable group. Its fixed-point spectrum on a Banach space is the set of $p\geq 1$ such that every action by affine isometries of $G$ on that s…

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…

Let $2<q<p$ and let $\theta\in(0,1)$. The $\theta$-snowflake of $L_q$ is the metric space $(L_q,\|x-y\|_q^\theta)$. Sharp snowflake exponent conjecture. If $(L_q,\|x-y\|_q^\theta)$…

Fix $p\in(2,\infty)$, and let $\mathbb Z_{4m}^n$ be the discrete torus with $m,n\in\mathbb N$ and $k\in\{1,\ldots,n\}$. For $S\subseteq\{1,\ldots,n\}$, write $\varepsilon_S$ for th…

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