14 problems

Let $\mu$ be an isotropic log-concave probability measure on $\mathbb{R}^n$, and let $C_P(\mu)$ denote its Poincaré constant, namely the least constant for which the Poincaré inequ…

Cheeger-constant conjecture. There exists a constant $c$ such that

Let $K$ be a convex body equipped with its uniform probability measure. Consider divisions of $K$ into two measurable pieces of equal mass, and measure the interface between the pi…

In bond percolation on growing domains of the lattice, consider the giant component in the supercritical regime and its rescaled isoperimetric profile, namely the minimal boundary…

KLS conjecture. For every isotropic logconcave density in $n$-dimensional Euclidean space, $\psi_n$ is bounded by a universal constant independent of $n$.

Recall that the anchored Cheeger constant of a connected, locally finite graph $G$ is … where $v$ is a fixed vertex, and that $G$ has anchored expansion when $\Phi^*(G)>0$. Benjami…

Let $n$ be a positive integer, let $\gamma$ denote the standard Gaussian measure on $\mathbb R^n$, and let $K,L\subseteq\mathbb R^n$ be symmetric convex sets. For $p\geq 0$, define…

Spectral-order conjecture. The correct order of $\Phi_*(G_k)$ is $\sqrt{\gamma}$, namely, using the stated order of the spectral gap, it is of order

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

KLS conjecture. There is a universal constant $C>1$ such that

Kannan–Lovász–Simonovits conjecture. Up to a universal numerical factor independent of $n$, a minimum-surface-area cut of $K$ into two equal-volume parts can be achieved by dissect…

Improved threshold conjecture. The condition in Theorem Main can be replaced by

Extremality conjecture. The families $\widetilde{\mathcal{F}}_{t,s}$ are precisely extremal for the stability theorem for the biased Wilson theorem; equivalently, the factor…

The perturbed-metric conjecture. For every $0<s<1$, there exist constants $C_s,c_s>0$ such that, whenever $\varepsilon(A)<e^{-1/\rho}$,