13 problems

Eigenvalue bound conjecture. These eigenvalues satisfy

Let $\mathcal{S}^{(d)}_n$ be the collection of triangulations of the $d$-sphere, and let $\mu^{(d)}_{n,\beta}$ denote the median of the minimum volume of the Boltzmann random spher…

Subquadratic generation conjecture. The normalized minimal number of generators converges to zero in probability:

Kahle–Newman conjecture. For every finite abelian $p$-group $G$,

Tightness conjecture. For every prime $p$, $\Gamma_{n,p}$ is tight:

Let $T_n$ be a random $2$-dimensional determinantal hypertree on the vertex set $[n]$, and let $p$ be a prime. Write $\Gamma_{n,p}$ for the $p$-torsion subgroup of…

Fix integers $d,m$ with $d\geq m\geq 1$, and suppose that $\alpha$ satisfies … Let $\Delta\sim\Delta_m(n,n^{-\alpha})$ be the corresponding multiparameter random simplicial complex…

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…

Let $\mathbb{F}$ be a field and let $\Delta\tooltip\sim \mathcal{U}(n)$, where $n$ is even. Here $\mathcal{U}(n)$ denotes the uniform random simplicial complex on $[n]$. Top-homolog…

Let $t$ denote the intensity parameter and let $d$ tend to infinity. The Gaussian and Poisson limit theorems above require growth conditions on $t$ because of dimension-dependent f…

Let $Y_d(n,p)$ be the random $d$-dimensional simplicial complex, and let $c_d^*$ be the real number such that $p=c_d^*/n$ is a one-sided sharp threshold for the non-triviality of…

Let $Y_d(n,p)$ denote the Linial–Meshulam random $d$-dimensional simplicial complex, and let $c_2$ be the constant appearing in the homological phase transition. A sequence…

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…