4 problems

Limiting-distribution conjecture. The random variable

Let $X(n,p)$ be the random clique complex, let $M_k(n)$ denote the maximal persistence over all $k$-dimensional cycles in $X(n,p)$, let $f_k(n)=n^{1/k(k+1)}(\log n)^{1/(k+1)}$, and…

Let $d$ be an integer with $d\geq 3$, let $\alpha$ satisfy … and let $\Delta\sim\Delta(n,n^{-\alpha})$ be a random clique complex. Kahle's bouquet of spheres conjecture. With high…

Kahle's homology vanishing conjecture. For each positive integer $k$ and each $\varepsilon>0$, if