11 problems

Let $\mu$ be a distribution and let $\mathcal{F}[\mu,\psi]$ denote the relevant concentration objective for a coordinate $\psi$. Given $n$ samples, let $\hat{\psi}_n$ be a data-dri…

Let $\mathcal{P}$ be a distribution family on $(0,\infty)$ whose members have power-law tails satisfying $P(X>x)\sim x^{-\alpha}$ for some $\alpha>0$. Let the logarithmic coordinat…

Zigzag Conjecture. Set $p=\frac12$. There is a sequence of intervals of length $n^{\lambda+o(1)}$ containing $\chi(G_{n,1/2})$ with high probability. However, for every fixed…

Bobkov–Houdré–Tetali conjecture. There exists a vertex $x_0\in V(C_n)$ such that, for every $v\in V(C_n)$,

Tree characterization conjecture. There exists a vertex $r$ such that, for every vertex $u\in V(T)$,

Isoperimetric extremizer conjecture. For $n$ sufficiently large and $d,r$ in appropriate ranges,

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…

Variance conjecture—bis. There exists a constant $C$ such that for every isotropic log-concave random vector $X$,

Variance conjecture. There exists a constant $C$ such that for every isotropic log-concave random vector $X$,

Let $X$ be a centered log-concave random vector in $\mathbb R^n$, let $\lambda_X^2$ denote the largest eigenvalue of its covariance matrix, and let $|X|$ be its Euclidean norm. The…

Let $X$ be an isotropic log-concave random vector in $\mathbb R^n$, and let $|X|$ denote its Euclidean norm. The variance conjecture. There exists an absolute constant $C$ such tha…