31 problems

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…

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 $n\geq2$, 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…

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…

Let $f:\{-1,1\}^n\to\{-1,1\}$ be balanced and have low influences, and let $\widehat f(S)$ denote its Fourier coefficients. Majority's low-level Fourier-weight conjecture. For ever…

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

MAX-$k$-CUT analogue. An analogue of the relevant MAX-CUT theorem should hold for MAX-$k$-CUT.

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…

A ranked vote is a ranking of $k$ candidates, and a voting method assigns a winning candidate to each profile of voters' rankings. A method is fair when candidates are treated symm…

Noise-stability conjecture for linear threshold functions.

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 $\rho\textsuperscript{}$ be the noise parameter, let $\boldsymbol{\operatorname{MaxStab}}_{\alpha}$ denote the maximal asymmetric $b1$-stability, and let dictator functions be…

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…

For $0<\alpha,\beta\leq\frac{1}{2}$, let $\underline{\Gamma}_{\mathrm{even,S}}(\alpha,\beta)$ and $\overline{\Gamma}_{\mathrm{even,S}}(\alpha,\beta)$ denote the lower and upper bou…

Let a Hamming sphere or Hamming ball be a subset of the discrete cube, and let its noise stability be measured at a given noise level among sets of the same cardinality. Hamming-sp…

Let $T_\epsilon$ be the noise operator on Boolean functions, let $f:\{0,1\}^n\to\{0,1\}$ be balanced, and define the $\Phi$-entropy by … For … this is the squared Hellinger distanc…

Let $T_\epsilon$ be the noise operator on Boolean functions, let $f:\{0,1\}^n\to\{0,1\}$ be balanced, and let $1\leq\alpha\leq2$. Thus $\mathbb{E}f=1/2$, where the expectation is w…

Let $\Omega\subseteq\mathbb{R}^{n}$ be a periodic set, meaning that $\Omega+v=\Omega^{c}$ for every standard basis vector $v$ and $-\Omega=\Omega^{c}$. A periodic half space is a s…

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