47 problems

Let $\gamma_1$ and $\gamma_2$ be critical invariant circles, and let $G_{\gamma_1,\gamma_2}$ and $H_{\gamma_1,\gamma_2}$ denote the associated maps between them. Let…

Let $g$, $h$, and $h^{-1}$ be the conjugacies associated with a critical invariant circle, and let $\kappa(\cdot)$ denote their regularity. A numerical characteristic is universal…

Let $R$ denote a critical invariant circle, and let $\kappa(R)$ denote its regularity. A numerical characteristic is universal when it takes the same value on an open set of functi…

Consider the renormalization operator acting on the class of twist maps under discussion, and call a property universal when it holds on an open set of functions. A fixed point is…

Let $\mu$ be the unique normalized invariant measure of a cubic critical homeomorphism of the circle, and let $HD(\mu)$ denote the Hausdorff dimension of its measure-theoretical su…

Fix a real-valued probability distribution $\mu$ with mean zero and finite moments, and a complex-valued probability distribution $\nu$ with mean zero, unit variance, finite moment…

Let $\mathscr{U}$ be the universality class used in the paper, let $\mathscr{Q}$ be a matrix class, let OAMP/VAMP denote the stated algorithms, and let SE denote their state evolut…

Let $\mu_1,\mu_2,\mu_3$ be probability measures with mean $0$ and variance $1$, and let $\Delta(\mu)$ be the discrepancy of a probability measure. Let $\beta(\mu,\infty)$ denote th…

Let $\mu$ be a probability measure with mean $0$ and variance $1$, and define its discrepancy by … Let $\beta(\mu,\infty)$ denote the scaling exponent of the runtime of $\infty$-re…

Let $\mu$ be a “sufficiently nice” probability measure with mean $0$ and variance $1$, and let $\beta(\mu,0)$ denote the scaling exponent of the runtime of $0$-reluctant, or greedy…

Fix $\alpha>0$. Let $\mu_1\leq\cdots\leq\mu_N$ be the eigenvalues of a GOE matrix, and let $\lambda_1\leq\cdots\leq\lambda_N$ be the eigenvalues of a real generalized Wigner mat…

Let $(H_N)$ be a generalized Wigner ensemble with uniform $p$-support, and let $\mu_1\leq\cdots\leq\mu_N$ be the eigenvalues of a GOE matrix while…

For a finite point configuration $P\subset\mathbb{R}^d$ and an integer $k\geq 0$, let $[P]_k$ be the set of point configurations $Q\in(\mathbb{R}^d)^X$ such that every chirotope of…

Table-profile conjecture. The profile functions $f(s)$ are the functions listed in Table for both SR and LR models in every dimension $d\ge d_{\mathrm{c},\alpha}$, including the up…

Let $r\geq 2$ and let $\beta=1$ or $2$. Consider the point process associated with the $r=2$, $\beta s=2$ joint density in Theorem 1, and the point process associated with the…

Universal fluctuation conjecture. The distribution of $Z$ is independent of $\vb{M}$ and $z$, satisfies

Let $\vb{M}$ be a rotationally invariant random matrix of size $N$ with a non-pathological limiting spectral density. For $z$ outside the spectrum of $\vb{M}$, define … Let…

Masoero–Raimondo string-equation conjecture. Under the nonvanishing assumption, $U(T,X)$ is a global solution of this equation.

Non-sub-Gaussian ESD conjecture. The conclusion of Theorem (ii) holds without assuming that the base measure $\mu$ is sub-Gaussian.

Let $X_n$) be an $n\times n$ generalized random Toeplitz, circulant, or symmetric circulant matrix with standard Gaussian entries. Let $p\geq 2$ be a positive integer and set…

Let $\nu$ be a rotationally invariant probability measure on $\mathbb{C}\backslash\{0\}$ satisfying $\int_{0}^{\varepsilon} \nu(\{|z| < r\})r^{-1}\,\operatorname{d}r<\infty$ for so…

Robust universality conjecture. For any $k\geq 2$, there exists a constant $C>0$ such that, for all $n\in\mathbb N$ and $p\geq Cp^*_{k+1}$, every graph $G$ with…

Let $\mathbb{S}$ be a sampling model in the class $\mathcal{U}$. For a complex type $\mathcal{T}$ and homological degree $k\geq 1$, let $\mathcal{L}_n$ be the empirical measure of…

Semi-Baxter and strong-Baxter universality conjecture. The permuton limit of uniform semi-Baxter or strong-Baxter permutations is the skew Brownian permuton. In particular, for str…

Let Bernoulli percolation and lattice trees be defined on transitive graphs of polynomial growth, and let the volume-growth dimension denote the dimension governing the graph's pol…