7 problems

For every admissible random point-cloud sampling model, ambient dimension $d$, homological degree $k$, and filtration type, let $(b_{n,i},d_{n,i})_{i=1}^{N_n}$ be the persistence p…

Let $X\sim X(n,p)$ be a random simplicial complex, let $d\geq 3$, and fix … Set $p=n^{-\alpha}$. Bouquet-of-spheres conjecture. With high probability, $X$ is homotopy equivalent to…

Homological connectivity threshold conjecture.

Let $G(T,\delta,m)$ be the preferential attachment graph and let its Rips complex of radius $r$ be formed using the graph metric. Diameter-dominance conjecture. The Betti numbers o…

For $q\geq 2$ and $m\geq 2q$, let $X(T,\delta,m)$ denote the affine preferential attachment clique complex and let $\beta_q(X(T,\delta,m))$ be its $q$-th Betti number. If…

Let $X_\bullet$ be a random simplicial complex, and let $H_\ast(X_\bullet)_{\operatorname{tors}}$ denote the torsion subgroup of its homology. A Cohen–Lenstra distribution conjectu…

Let $d$ be the dimension of the torus and let $\lambda_{0,k}$ and $\lambda_{1,k}$ denote the lower and upper thresholds for the appearance of giant $k$-cycles, while $\lambda_c$ an…