9 problems

Talagrand's conjecture. For every fixed $\tau>0$, there is a constant $C_\tau$ such that, uniformly in the dimension and over all $f\geq 0$,

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…

Heat-smoothing conjecture. For any $1<p<\infty$, there exists $c_p>0$ depending only on $p$ such that, for every $n\ge 1$, $1\le d\le n$, and every $f:\{\pm 1\}^n\to\mathbb{R}$ sat…

Covariant Riesz transform conjecture. There exists a $\kappa_0=\kappa_0(A,m)>0$, depending only on $A$ and $m$, such that for all $\kappa\geq\kappa_0$, $j\in\{1,\cdots,m\}$, and…

Heat-flow decay conjecture. There exists a positive function $w:U\to\mathbb{R}$ such that

Let $K,p>1$. For a Banach space $X$, let $K(X)$ denote its $K$-convexity constant. For every $K$-convex Banach space $(X,\|\cdot\|_X)$ with $K(X)\leq K$, every $k,n\in\mathbb{N}$,…

Let $1<p<\infty$. Let $f\colon\{-1,1\}^{n}\to\mathbb{R}$ satisfy … for every $S\subseteq\{1,\ldots,n\}$ with $|S|<k$, where $P_t$ is the heat semigroup and $L$ is its generator. Me…

K-convexity heat-decay conjecture. Every $K$-convex Banach space satisfies this condition for every $k\in\mathbb N$, $p\in(1,\infty)$, and $t\in(0,\infty)$. The preceding theorem p…

Let $B$ be a magnetic field, and let $\beta$ and $\tilde{u}$ be the parameters and rescaled heat evolution appearing in Theorem; denote by $\gamma_B$ the decay exponent in that the…