7 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…

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…

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

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…

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…