14 problems

Cheeger's chart-image conjecture. If $(U,\phi)$ is an $n$-dimensional chart, then

Let $F_k$ and $G_k$ be the arithmetic Fourier series considered in the paper, and let $q_n(x)$ denote the denominators of the continued-fraction convergents of $x$. Let…

Let $X$ be a Banach space with the Radon–Nikodym property, let $Y$ be a Banach space, and let $f:X\to Y$ be a Lipschitz quotient map. Suppose that $Y^\star$ is separable, and let…

C-minimal analogue conjecture. There is some $\alpha\in\mathbb{C}_p$ and an algebraically closed subfield $M\subseteq\mathbb{C}_p$ closed under $f_\alpha$ such that the structure…

Let $\mathbb{I}=[0,1]$, let $\mathcal{C}$ denote the family of bivariate copulas, and let $\mathcal{C}_{mcd}$ be the subclass of mutually completely dependent copulas. Let…

For a copula $C$, write $\partial_1C(x,y)$ for the partial derivative with respect to the first variable, when it exists. A subclass of copulas is called typical when it contains a…

Let $(X,\mathsf{d},\mathfrak{m})$ be the RCD space under discussion, let $k=\dim(X,\mathsf{d},\mathfrak{m})$, and let $\mathcal{R}_k^*$ be the full-measure regular set. Let $\phi$…

Let $X$ be a Banach space, let $d4a5=\left\{F_t\right\}_{t\ge0}\subset C^k(\mathcal{D})$ be a semigroup on a domain $\mathcal{D}\subset X$, and let $T$-continuity mean continuity i…

Higher-order differentiability conjecture. The procedure of Proposition can be repeated arbitrarily often, provided that $psi$, $mathds{h}$, $varphi0$, $sigmaB$, and $Gamma$ a…

Let $Q^d$ be the $d$-dimensional cube, and let $f$ be a function on $Q^d$. For an integer $k\geq 2$, write $D_x^k f$ for the Peano $k$-th differential at $x$ and $\partial_x^k f$ f…

Let $mathcal D$ be the set of directions at which the relevant shape function is differentiable, and let $mathcal U$ be the direction set, with $\operatorname{ri}mathcal U$ denotin…

Tangent-field characterization conjecture. Sets of non-differentiability of real-valued Lipschitz functions may be described as precisely those sets for which there is an $(n-1)$-d…

Let $S^d$ be the $d$-dimensional sphere, and let $K$ be a continuous, positive definite, zonal kernel on $S^d$. Write its radial part as $K_r^d:[-1,1]\to\mathbb{R}$, so that…

A real $x\in[0,1]^n$ is computably random if it avoids every effectively presented sequence of open sets forming a Martin-Löf test with the stronger computable-martingale randomnes…