18 problems

Let $d_{T_n}$ be the word metric on the Cayley graph $\Gamma(\operatorname{Sym}_n,T_n)$. For the simple random walk $(W_t)_{t=1}^{\infty}$ on this graph started at $W_0=1$, write…

Naor's lattice quotient conjecture. There exists a constant $c>0$ such that for every $r\in\mathbb{N}$ and every lattice $\Lambda\subset\mathbb{R}^r$,

Let $\{G_n\}_n$ be a sequence of finite groups with $\sup_n \operatorname{rank}(G_n)<\infty$. For a generating set $S$ of size $r$, let $d_S$ be the associated word metric, and let…

Let $G$ be a finitely generated group with a word metric $d_S$. Write $c_2(G,d_S)$ for the infimal bi-Lipschitz distortion of $(G,d_S)$ into Hilbert space $L_2$. Cornulier–Tessera–…

Conjecture on efficient construction of optimal separating partitions. Computing a sufficiently good approximation to $\varphi$ in oracle polynomial time, and repeating the constru…

Let $(\mathbf{E},\|\cdot\|_{\mathbf{E}})$ be a Banach space with a $1$-symmetric basis, and let $\mathsf{S}_{\mathbf{E}}$ be the corresponding unitary ideal. Assume that…

For $n\to\finity$ and $p>2$, let $\mathsf{c}_2^n(L_p)$ denote the maximum, over all $n$-point metric spaces, of the least distortion with which the space embeds into $L_p$ and then…

Let $n\in\mathbb{N}$, and let $(\mathcal{M},d)$ be an $n$-point metric space. The space is of negative type when its $\frac{1}{2}$-snowflake $(\mathcal{M},d^{1/2})$ embeds isometri…

Let $n\in\mathbb{N}$ and let $X$ be an $n$-point subset of $\ell_1$. Its Euclidean distortion is the least distortion among embeddings of $X$ into $\ell_2$. Goemans's conjecture. E…

Markov convexity threshold conjecture. $G$ is not Markov $p$-convex for every

For $p\in[2,\infty)$, let $\upalpha_p$ be the distortion parameter furnished by the average metric dimension-reduction theorem for $X=\ell_p$. The conjecture. … The source notes th…

Let $\mathsf{k}_n^\upalpha(\ell_\infty)$ denote the least dimension, in the relevant metric dimension-reduction problem, for which every $n$-point metric space admits distortion at…

Pisier's dichotomy conjecture. The $\ell_2$ (Pythagorean) direct sum

Let $f(n,d,\varepsilon)$ denote the smallest $m$ such that every $n$-point subset of $\ell_2^d$ can be embedded into $\ell_2^m$ with distortion at most $1+\varepsilon$. Dimension-r…

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

Union distortion conjecture.

Let $p,q\in(2,\infty)$ satisfy $q<p$. For $\theta\in(0,1]$, consider the metric space $(L_q,\|x-y\|_{L_q}^\theta)$. Snowflake exponent conjecture. If this metric space admits a bi-…

Let $X$ be a Banach space of cotype $2$, and let $S\subseteq X$. The subset $S$ threshold-embeds into Hilbert space if there is a threshold embedding of $S$ into a Hilbert space. C…