3 problems

Homological connectivity threshold conjecture.

Let $\mathcal{C}_r$ be the random Čech complex generated by a homogeneous Poisson process on the flat torus, let $H_k(\mathcal{C}_r)$ denote its $k$-th homology, and let $\Lambda$…

Let $X(n,p)$ be the random flag complex obtained from the Erdős–Rényi random graph $G(n,p)$ by adding a $k$-simplex for every clique of $k+1$ vertices. For each $k$, the vanishing…