17 problems

Let $0\leq\rho\leq1$ and let $q\geq2$ be an integer. For $f:[q]^n\to[0,1]$, let $\mathbb{S}_\rho(f)$ denote its noise stability and let $\mathrm{Inf}_i(f)$ denote the influence of…

For $q\geq 1$, let $\ln_q:(0,+\infty)\to\mathbb{R}$ be the $q$-logarithm … and define $\Phi_q^{\mathrm{sym}}(t)=t\ln_q(t)+(1-t)\ln_q(1-t)$ and the associated symmetric $q$-stabilit…

Vector-valued Borell inequality. If $0<\rho<1$ and

Plurality is Stablest conjecture. If $\rho\geq0$ and $\mathbb{E}f=\frac1k\sum_{j=1}^{k}e_j$, then

Let $m\geq 3$ and $0<\rho<1$. Consider two measurable partitions $\Omega_1,\ldots,\Omega_m$ and $\Omega'_1,\ldots,\Omega'_m$ of $\mathbb{R}^{n+1}$ with…

Let $m\geq 2$, let $\rho\in[0,1]$, and let $\varepsilon>0$. For a function $f:\{1,\ldots,m\}^n\to\Delta_m$, write $\operatorname{Inf}_i(f_j)$ for the influence of coordinate $i$ on…

Let $\rho\in(0,1)$ and $n\geq k!+k-1$. Consider partitions $\Omega_1,\ldots,\Omega_k$ of $\mathbb{R}^n$ with Gaussian measure $\gamma_n(\Omega_i)=1/k$ that maximize the ranked-choi…

Let $S_k$ be the set of rankings of $k$ candidates, and let $f\colon S_k^n\to\Delta_k$ be a ranked-choice voting method, where $\Delta_k$ is the simplex of randomized outcomes on t…

Let $n$ be large, let $k\geq 3$ be the number of candidates, and let voters cast independent uniformly random votes. A voting method is balanced when each candidate has equal proba…

Let $\mathcal{C}_n=\{-1,1\}^n$ be the Boolean hypercube, let $X$ be uniformly distributed on $\mathcal{C}_n$, and let $\mathcal{N}_{X,\rho}$ denote the corresponding noisy version…

Let $\rho$ satisfy $0\leq\rho\leq1$, let $b1\in[1,9]$, and let $\mathbf{MaxStab}_{\alpha}(1/2)$ denote the maximal asymmetric $b1$-stability at mean $1/2$. A dictator function is a…

Let $m\geq3$, let $0<\rho<1$, and let $(\Omega_i)_{i=1}^m$ and $(\Omega_i')_{i=1}^m$ minimize the bilinear Gaussian noise-stability problem with equal prescribed measures, under th…

Let $f\colon\{-1,1\}^{n}\rightarrow\{-1,1\}$ be a linear threshold function, with $n$ odd. For $\rho\in[0,1]$, let $\operatorname{Stab}_{\rho}[f]$ denote the noise stability of $f$…

Quadratic Symmetric Gaussian Problem. Either $B(0,r_a)$ or $B(0,r_a')^c$ achieves

Symmetric Gaussian Problem. If $\rho>0$, then either $(B(0,r_a),B(0,r_b)^c)$ or $(B(0,r_a')^c,B(0,r_b'))$ achieves

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

Let $n\geq2$, let $\rho\in[-1,1]$, and let $3\leq k\leq n+1$. Let $\{A_i\}_{i=1}^k$ be a measurable partition of $\mathbb{R}^n$, meaning that $\bigcup_{i=1}^k A_i=\mathbb{R}^n$ and…