7 problems

Let $q$ be prime, let $g_1,\ldots,g_k$ be independent uniformly random generators of the cyclic group ${\mathbb Z}_q$, and let $Diam(q,k)$ denote the diameter of the resulting dire…

Let $A$ be a set of size $s$ with doubling constant bounded by $K$, meaning $|A+A|\leq K|A|$. For each $ell$ let $\mathcal{F}_\ell$ be a collection of sets, and let $\tilde{O}$ and…

Let $G$ be a group of size $n$, and let $G(p)$ be its random Cayley graph, formed by selecting each equivalence class $\{x,-x\}$ independently with probability $p$. Write…

Lower-central-series comparison conjecture. If

The entropic-time conjecture. Under conditions similar to those in the paper's total-variation cutoff result, with high probability the random walk on $G_k$ exhibits cutoff in the…

Let $d$ and $n$ diverge with $n\leq 2^d$, let $G$ be a group of order $n$, and let $H=\mathbb{Z}_2^d$. For fixed $\varepsilon,\varepsilon'\in(0,1)$, write…

Cheeger lower-bound conjecture. There exists an absolute constant $c$ such that, for all $\varepsilon\in(0,1)$, there are constants $N(\varepsilon)$ and $M(\varepsilon)$ such that,…