1,031 problems

Let $L\subset\mathbb{R}^3$ range over Bravais lattices of covolume $1$. Define $\Theta(\alpha,L)=\sum_{v\in L}e^{-\pi\alpha\lvert v\rvert^2}$ for $\alpha>0$ and…

Let $X$ and $Y$ be metric surfaces, and let $f:X\to Y$ be an area-preserving Lipschitz map. Does there exist a constant $K\ge 1$ such that, for $2$-modulus-almost every rectifiable…

For $-1<\alpha\leq \tfrac12$, let $J_\alpha$ denote the Bessel function of the first kind, let $j_{\alpha,2}$ be its second positive zero, and define $\beta(\alpha)<\alpha+1$ by…

Let $H_V=-\frac{d^2}{dx^2}+V$ on $L^2(\mathbb{R})$, where $V\colon\mathbb{R}\to\mathbb{R}$ and $V\in L^{3/2}(\mathbb{R})$. If the negative eigenvalues of $H_V$ are written as…

Let $n\ge 3$. For every open set $\Omega\subset\mathbb{R}^n$, set $F:=\mathbb{R}^n\setminus\Omega$ and, for $0<\alpha\le 1$ and $x\in\Omega$, define…

Let $\mu:(0,1)\to(0,\infty)$ be the Grötzsch modulus function $\mu(r)=\frac{\pi}{2}\frac{\mathcal{K}(\sqrt{1-r^2})}{\mathcal{K}(r)}$, where $\mathcal{K}$ is the complete elliptic i…

Let $\Phi$ be the classical Jacobi-theta kernel in the Fourier representation of the Riemann $\Xi$-function. Define $s(t)=\Phi(\sqrt{t})$ and $f(t)=s'(t)^2-s(t)s''(t)$ for $t>0$. T…

For every integer $n\ge 1$, every $-1<\alpha<\beta$, and every $0<c\le 1$, let $\pi_n$ be the monic degree-$n$ polynomial orthogonal on $[-c,c]$ with respect to the Jacobi weight…

Let $n \ge 2$ be an integer and let … be a monic polynomial all of whose roots lie in the closed unit disk, i.e. $|r_{k}|\le 1$ for every $k \in \{1,\dots ,n\}$. Since…

In harmonic analysis, a branch of mathematics, the Mizohata–Takeuchi conjecture proposed a weighted $L^{2}$ inequality for the Fourier extension operator associated with a smooth h…

For $\Omega \subset \mathbb{R}^{d}$ with positive finite Lebesgue measure, say that $\Omega$ is spectral if there exists a set $\Lambda \subset \mathbb{R}^{d}$ such that … is an or…

For a polynomial $f\in\mathbb{C}[X]$, let $\operatorname{rootArgCount}(f,I)$ be the number of roots of $f$, counted with multiplicity, whose arguments lie in $I$, where each argume…

Let $x_1,x_2,\ldots \in (0,1)$ be an infinite sequence and let $A_k=\limsup_{n\to \infty}\left\lvert \sum_{j\leq n} e(kx_j)\right\rvert,$ where $e(x)=e^{2\pi ix}$. Is it true tha…

For distinct $x_1,\ldots,x_n\in[-1,1]$, let $l_k(x)=\prod_{i\ne k}\frac{x-x_i}{x_k-x_i}$ and $\lambda(x)=\sum_{k=1}^n|l_k(x)|$. Is it true that, for every fixed $-1\leq a<b\leq1$,…

Let $f(z)=\sum_{0\leq k\leq n} \epsilon_k z^k$ be a random polynomial, where $\epsilon_k\in \{-1,1\}$ independently uniformly at random for $0\leq k\leq n$. Is it true that, if…

Is the Glöckner--Neeb multiplication-growth condition automatic for every Mackey-complete continuous inverse algebra?

For a fixed $0<\alpha\leq 1/3$, let $\left(\sigma_{n,\varepsilon}\right)_{n\in\mathbb{Z},\,0<\varepsilon<1}\subset\mathbb{C}$ be a summation process satisfying … and … Does there e…

Let $(\epsilon_k)_{k\geq 0}$ be independently uniformly chosen at random from $\{-1,1\}$. If $R_n$ counts the number of real roots of $f_n(z)=\sum_{0\leq k\leq n}\epsilon_k z^k$ th…

Let $1\leq p\leq q<\infty$ and $z\in\mathbb{C}$. Weissler's conjecture asserts that the complex noise operator $T_z$ on the Hamming cube has dimension-free norm from $L^p$ to $L^q$…

Let $K$ be a Hilbert space, let $X,Y$ be commuting positive contractions on $K$, and let $P\in B(K)$ be an orthogonal projection such that $PXP$ and $PYP$ commute. Let…

Does there exist a function $F:\mathbb{C}\to\mathbb{C}$ that is complex differentiable everywhere, maps every real number to a real number, is not equal on $\mathbb{R}$ to any affi…

For every $\varepsilon>0$, for all sufficiently large integers $n$, every monic polynomial $p\in\mathbb{C}[z]$ of degree $n$ whose sublevel set … is connected satisfies, for every…

Let $F\subseteq \mathbb{C}$ be a closed infinite set, and let $\mu(F)$ be the infimum of $\lvert \{ z: \lvert f(z)\rvert < 1\}\rvert,$ as $f$ ranges over all polynomials of the s…

For any $t\in (0,1)$ let $t=\sum_{k=1}^\infty \epsilon_k(t)2^{-k}$ (where $\epsilon_k(t)\in \{0,1\}$). What is the correct order of magnitude (for almost all $t\in(0,1)$) for…

For p ≥ 2, does Carbery’s proposed many-function almost-orthogonality inequality hold with the pairwise overlap coefficients raised to the power 2? If not, what is the largest poss…