16 problems

Cohomology-vanishing conjecture. If

Connectivity conjecture. If

Let $\Gamma_{n,m,p}$ and $Y_{n,k,q}$ be the distributions considered in the paper, let $z_Kx$ denote the relevant count associated with a positive $k$-pure shape $x$, and set…

Let $\beta=k-\frac{1}{n_m}$, $p=n^{-\frac{1}{\beta}}$, and $q=n^{-\frac{km}{\beta}}$, with $n=n_m$. Let $\Gamma_{n,m,p}$ and $Y_{n,k,q}$ denote the two distributions defined in the…

Cohomology approximation conjecture. If $d \geq 4(2k+1)^2$, then, as $h$ tends to infinity, the induced map $\tau^*$ on the $k$th cohomology groups is an isomorphism to…

Borsuk–Ulam conjecture. With high probability, for every continuous map $f\colon\widetilde Z\to\mathbb{R}^{2d+1}$, there exists an $x\in|\widetilde Z|$ such that

Łuczak–Peled conjecture. For every $d \geq 2$ and $p=p(n)$ such that $|np-c_d|$ is bounded away from $0$, $H_{d-1}(Y_d(n,p))$ is torsion-free asymptotically almost surely.

Let $d \in \mathbb{N}$ and let $(\Delta_n)_{n \in \mathbb{N}}$ be a sequence of finite $d$-dimensional buildings whose thickness satisfies … as $n \rightarrow \infty$. Let…

Let $\Delta_{k,n}$ be the finite spherical building of type $A_k$, let $\mathcal{P}(\Delta_{k,n})$ denote its induced subcomplexes, and let $\mathcal{S}(\Delta_{k,n})$ denote the s…

Let $K\subseteq {\mathbb R}^d$ be a convex body with smooth boundary, and let the distribution be uniform on $K$. Write $r_n$ for the radius parameter and ${{\mathcal R}}(n,r_n)$ f…

Let $Q$ be a random 2-dimensional cubical complex sampled as $Q\sim Q_2(n,p)$, where $p>1-(\tfrac 12)^{1/40}$. The fundamental group $\pi_1(Q)$ is the group associated with $Q$. To…

Let $Y_2(n,p)$ be the Linial–Meshulam random $2$-complex on $n$ vertices, where each possible $2$-simplex is included independently with probability $p$, and let $\gamma:=4^4/3^3$.…

Let $\mathcal{Y}_d(n)$ be the stochastic Linial–Meshulam process in dimension $d\geq2$. Let $LT(n)$ denote its largest torsion group, and for a $d$-complex $Y$ with complete…

Let $\mathcal{T}^2(n)$ be the set of $2$-trees on $n$ vertices, and let $X$ be drawn uniformly from $\mathcal{T}^2(n)$. For a fixed prime $q$, let the Sylow $q$-subgroup of…

Let $LT(n)$ be the largest torsion group found in the first homology during the torsion burst of the Linial–Meshulam process. For a fixed prime $q$, let the Sylow $q$-subgroup of…

Let $k$ be a row index, and consider a Betti table for which Theorem yields nonvanishing Betti numbers in row $k$. The $k$th row is viewed as a sequence of Betti numbers indexed by…