12 problems

Translation-invariance conjecture. For every $c\in\mathbb{R}$,

Let $r\ge 3$ and let $1>p_1\ge p_2\ge\cdots\ge p_n>0$. Let $\mu_{\mathbf p}$ be the product measure on $2^{[n]}$ with coordinate $i$ included independently with probability $p_i$.…

Let $\mu_{1}$ and $\mu_{2}$ be log-concave probability measures on Riemannian manifolds $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$, respectively. For $a$ in the domain of the isoperim…

Dimension conjecture. For all $m$, the dimension of $P_m$ is exactly $C(m)$.

Let $\Omega$ be a product space, let $\mu$ be a product measure on $\Omega$, and let ${\mathcal A},{\mathcal B},{\mathcal C}\subseteq\Omega$ be increasing events. Sahi's conjecture…

Let $\mu$ be a product measure satisfying the assumptions that the full product measure exists and that the scaled marginals satisfy the stated conditions. Let $E_{\gamma,m}$ be th…

Ergodic product-disintegration conjecture. A $T$-invariant measure on $S^{\mathbb N}$ is $T$-ergodic exactly when it admits a representation as a mixture of product measures whose…

Let $n\in\mathbb{N}$, let $\|\cdot\|_K$ be any norm on $\mathbb{R}^n$, and let $X_1,\dots,X_n$ be independent, non-compactly supported random variables on $\mathbb{R}_+$. For a vec…

Coordinate-star extremal conjecture. Assume

Stability conjecture. There exists a constant $c=c(p_1,p_2)$ such that, for every pair of cross-intersecting families $U_1\subset\Omega_1$ and $U_2\subset\Omega_2$ satisfying

Relaxed product-measure conjecture. Theorem still holds if the condition on the product measures is replaced by

Higher-dimensional necessity conjecture. This condition is necessary for the existence of product-measure steady states in all such cases, not only under the additional conditions…