13 problems

Let $d\geq 3$, and let $t_k^{\mathrm{perc}}$ be the critical threshold for homological percolation in degree $k$, while $t_k^{\mathrm{ec}}$ denotes the corresponding zero of the ex…

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…

Let $X(n,p)$ be the clique complex of the Erdős–Rényi random graph $G(n,p)$, and fix an integer $k\geq 0$. A random property holds almost always if its probability tends to $1$ as…

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 $p^{\square}_c(i,d)$ and $q^{\square}_c(i,d)$ denote the critical probabilities for, respectively, the appearance of a nontrivial homological event and the failure of surjectiv…

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…

Property (T) conjecture. For $T \sim \mathcal{T}(n)$, with high probability $\pi_1(T)$ has Kazhdan's Property (T).

Let $M$ be a nice compact space of maximal possible homological degree $d$, and let $t_k^{\mathrm{perc}}$ denote the critical threshold for the appearance of giant $k$-cycles in a…

Let $M$ be a nice compact space, let $\{X_t\}$ be a random filtration with $X_t\subset M$, and fix a homological degree $k$. For the inclusion $i:X_t\hookrightarrow M$, define … He…

Let $\beta_{n,k}(u_n)$ be the $k$-th Betti number of the random excursion complex at level $u_n$, and let $\lambda_{n,k}(u_n)$ denote its associated mean-scale parameter. Asymptoti…

Let $X$ be the intersection of $k$ random, independent Kostlan quadrics in $\mathbb{R}\operatorname{P}^{n-1}$, and let the middle Betti numbers mean the Betti numbers whose indices…