12 problems

Let $H_t(\mathcal{G})$ denote the entropy of the random geometric graph distribution at parameter $t$, and let $\bar{p}_{\max}$ be the corresponding edge-probability parameter at w…

Let $M\in\{\pm1\}^{n\times m}$ with $m\geq n$, and let $\operatorname{Res}(M)$ be the least number of entry-flips needed to produce from $M$ a matrix whose rank is strictly less th…

Let $\tau_{p,d}$ be chosen so that the spherical threshold connection…

PIIDG orthogonality conjecture. Then

Let $\bm{U}$ be an orthogonal matrix and let $\bm{U}_1$ be an $N\times(N-m)$ block of $\bm{U}$. Let $\bm{b}_1\in\mathbb{R}^{N-m}$ be a PIIDG vector, meaning a partially independent…

Grammians conjecture. There exists a constant $\kappa_n>0$ and an increasing function $\gamma_n:(0,\kappa_n)\to(0,1)$, both depending only on $n$, such that $\gamma_n(\kappa)\to1$…

Let $C^0$ be a cycle of length $k$ with its signed cycle statistic $\kappa_{C^0}$, and let $\mathcal{G}(n,p,d)$ denote the noisy high-dimensional random geometric graph model witho…

Let $k$ and $p$ be fixed, let $\mathcal{G}(n,p,d)$ denote the noisy high-dimensional random geometric graph model without edge subsampling, and let $\tau_{[k]}$ be the signed $k$-c…

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…

Zero-breaking record fraction conjecture. There exist constants $p_d\in(0,1)$ such that, almost surely,

Consider sufficiently spread-out oriented percolation in dimension $d>4$, with spread parameter $L$ sufficiently large, and let $\mathcal {T}_n$ denote the set of vertices connecte…

Let $A\subseteq S^{n-1}$ be a measurable subset with $\sigma_{n-1}(A)\geq e^{-c\sqrt{n}}$. For a subspace $H\subseteq\mathbb{R}^n$ of dimension $\lfloor n/2\rfloor$, let…