27 problems

Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be a $C^1$ map, and let $J(f)$ denote its Jacobian matrix. A point $x$ is a fixed point if $f(x)=x$. The two-fixed-point conjecture. If $f$ has…

Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be a $C^1$ map, and let $J(f)$ denote its Jacobian matrix. The $C^1$ stability conjecture. If the eigenvalues of $J(f)$ have strictly negative r…

Let $f:\mathbb{R}^n\to\mathbb{R}^n$ be a $C^1$ map, and let $J(f)$ denote its Jacobian matrix. A matrix is unipotent when all its eigenvalues are $1$. The $C^1$ unipotent Jacobian…

One-dimensional Maxwell bound. For any values of the charges, the function $V_{\alpha}^*(x)$ has at most $2l-1$ real critical points. This is a stronger one-dimensional version of…

Matkowski's conjecture. The function $f$ solves this equation if and only if there exist $a,b\in\mathbb{R}$ with $a\neq b$ such that

The $(P_n)$ conjecture. $(P_n)$ is true for all nonnegative integers $n$.

Single-base representation conjecture. For every $a_n,b,c_n,d\in\mathbb{R}$ and $n\in\mathbb{N}$, there exist $u_n,{\Greekmath 010D}\in\mathbb{R}$ such that

Let $\alpha^{(p)}$ be a quadratic irrational in $\mathbb{Q}_p$ that has an image in $\mathbb{R}$. Consider its continued fraction obtained by Browkin I, Browkin II or Algorithm MR.…

For $X\subseteq\mathbb{R}$, call $X$ strongly meager if $X+M\neq\mathbb{R}$ for every Lebesgue measure zero set $M\subseteq\mathbb{R}$. Full measure universality characterization c…

A subset $X\subseteq\mathbb{R}$ is full measure universal if every Lebesgue measurable subset of $\mathbb{R}$ whose complement has Lebesgue measure zero contains an affine copy of…

Let $X\subseteq\mathbb{R}$ be topologically universal if every dense $G_\delta$ subset of $\mathbb{R}$ contains an affine copy of $X$. Gallagher–Lai–Weber's conjecture. No uncounta…

Let $X\subseteq\mathbb{R}$ be a measure universal set, meaning that every Lebesgue measurable subset of $\mathbb{R}$ with positive Lebesgue measure contains an affine copy of $X$.…

Let $K$ be a positive natural number, and let $c:[\mathbb R]^2 \rightarrow K$ be a colouring of pairs of reals. Galvin's conjecture. There is a set of reals $Y$ homeomorphic to…

Let $h:(0,\infty)\rightarrow(0,\infty)$ be a log-convex function satisfying … Assume that $h^2(x)/h(2x)$ is integrable on $(0,\infty)$. Handelman's integrability conjecture. Then…

Let $f_b(x):=\frac{1}{1-bx+x^2}$, and let $\beta^+=\frac{3+\sqrt{13}}{4}$. Monotonicity conjecture. 1. $L_n(f_b)$ is neither increasing nor decreasing for all…

Maximality conjecture for Gaussian Riemann derivatives. For all functions $f$ and points $x$:

Exceptional-set conjecture.

Ash–Catoiu conjecture. The set $F$ is a null set.

Let $F:\mathbb{R}^n\to\mathbb{R}^n$ be a $C^1$ map. Chamberland–Meisters' injectivity conjecture. Suppose there exists an $\varepsilon>0$ such that … for every eigenvalue $\lambda$…

$C^1$ fixed point conjecture. The origin $0$ is the unique fixed point of $f$.

Let $\mu$ be an absolutely continuous probability measure supported on $[0,1]$ with density $f$. Assume there exist $C>0$ and $p>1$ such that, for all $0<r<R<1$ and $x$ in the supp…

Chamberland–Meisters conjecture. Then $F$ is injective. This conjecture is presented as a sufficient condition for injectivity after the real Jacobian conjecture was disproved; its…

For $r>1$, let $C_r$ denote the number of connected components of the closure of the image of the regular divisor function $[?]$. Divisor function monotonicity conjecture. $C_r$ is…

Giusto–Simpson conjecture. The following are equivalent over $\mathsf{RCA}_0$:

Let $e=(e_1,\ldots,e_n)\in\mathbb{R}_+^n$, and let $f\circ\varphi$ be the composition defined in the paper, where $\varphi$ parametrizes the roots associated with $e$. For…