7 problems

Tokushige's conjecture. Let $p_1,p_2\in(0,\frac{1}{t+1})$. If $\mathcal F_1$ and $\mathcal F_2$ are cross $t$-intersecting, then

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…

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…

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