17 problems

Endpoint sharp-constant conjecture. In the endpoint case $p=1$,

Weissler's conjecture. Let $1\leq p\leq q<\infty$ and $z\in\mathbb{C}$. Then

Let $Q_d$ be the $d$-dimensional Hamming cube, and let a $\mathbb{Z}$-homomorphism be an integer-valued map on the vertices of $Q_d$ whose values differ by exactly $1$ along every…

Let $f:\{0,1\}^n\to\{\pm 1\}$ be balanced, meaning that $\mathbf{E}f=0$. For $\rho\in[-1,1]$, let $T_\rho f$ denote the standard noise operator applied to $f$. Hellinger conjecture…

Let $Q_n$ be the $n$-dimensional Hamming cube, and let $p_c(Q_n,n^a)$ denote the critical probability for the corresponding bootstrap percolation process with threshold $n^a$, wher…

Half-cube extremality conjecture. Indicator functions of half-cubes extremize this inequality, equivalently

Subcube extremality conjecture. For all $\beta\ge\frac12$ and all $k\ge 1$,

Let $(X,\|\cdot\|_X)$ be a $K$-convex Banach space. For $n\in\mathbb{N}$, let $\operatorname{Deg}_{\le d}$ denote the space of functions $f:\{-1,1\}^n\to X$ whose Hamming cube Four…

Hamming-cube decay conjecture. For every fixed $c>2$, the quantity

Let $X$ be an affinely independent subset of the Hamming cube $H_n=\{0,1\}^n$ with $n+1$ points, and let $D$ be its distance matrix for the Hamming metric. Inverse-distance-sum con…

Let $H_n$ be the Hamming cube, and let $D$ be the distance matrix of an affinely independent subset $\{x_0,x_1,\ldots,x_m\}\subseteq H_n$, where $m\geq 1$. Let $\mathbf{1}$ denote…

All-codimension stability conjecture. The statement in Theorem $$ holds for all $k\in\mathbb P$, even with $n^\beta$ replaced by $2^n/\partial(|A|)$.

Maximal hybrid isoperimetric conjecture. If $|A|=a$, then

Fixed-error hybrid isoperimetric conjecture. There is a fixed constant $K$ such that, whenever $\mu(A)=1/2$,

Bernstein–Markov conjecture. There exists $C(p,X)\in(0,\infty)$ such that

The 5- and 6-coloring conjecture. For $q\in\{5,6\}$,

Engbers–Galvin conjecture. For each fixed $q$,