1,365 problems

Let $D(3,n)$ denote the dynamical system governing the relevant tripod Nim arrays, and consider its periodic orbits. Period-divisor conjecture. Every periodic orbit of $D(3,n)$ has…

Manifold decomposition conjecture. For every graph, the set of equilibria of the Kuramoto network is a finite union of manifolds. The question arises because the equilibrium set is…

For every $\varepsilon>0$, there exists a finite simple graph $G$ on $n$ vertices such that $\delta(G)\geq (3/4-\varepsilon)n$ and $G$ is not globally synchronizing for the Kuramot…

For every integer $k\ge 1$ and every collection of distinct natural numbers $h_1,\ldots,h_k$, where $\lambda(n)=(-1)^{\Omega(n)}$ is the Liouville function, one has…

For a dissipative dynamical system undergoing a Shilnikov–Hopf bifurcation, the Bosch–Simó conjecture predicts that, in the parameter unfolding, chaotic and periodic regimes are in…

Let … For the piecewise-defined system, call an equilibrium regular if it lies in its associated domain, boundary if it lies on the boundary between domains and is an equilibrium o…

Let $T_0$ be the critical temperature and let $T_1(t)$ denote the other time-dependent critical temperature in the model. Skipped-obliquity conjecture. When $|T_0-T_1(t)|$ is suffi…

Consider the autonomous system of equations and its equilibria, classified as regular, boundary, or virtual according to their location relative to the switching boundary. Triggeri…

Let the system of equations be forced by time-dependent functions $Q=Q(t)$ and $s_2=s_2(t)$, and let $T_0(t)$ be the specified ramp function. MPT-like transition conjecture. The so…

Let $M$ be the spatial manifold, let $[0,\tau]\times M$ be the spacetime manifold $\mathbb{M}_0$, and for each $a\geq 0$ let $S_a$ and $H_a$ denote the Sobolev and Cheeger constant…

Let $(u,v)$ solve the reaction-diffusion system of allelopathic phytoplankton with fear function $k(x)$, subject to the parameter restrictions … where $\mathbf{k}$ is either…

Let the predator-prey model described in the paper be given, with its parameters and equilibria as defined there. Cyclical-fold conjecture. There exist parameter regimes under whic…

Let the predator-prey model described in the paper be given, with its interior equilibrium point and model parameters as defined there. Multiple-limit-cycle conjecture. There exist…

Generic unbounded-distortion conjecture. For every non-Dini-integrable $\omega\in K$, unbounded distortion is $C^1$-generic in $\Gamma_{\omega}^{1}(\mathbb S^1)$.

Fix a degree $d$ and consider the torus $\mathcal H_d$ of hedgehogs whose support functions are trigonometric polynomials of degree $d$, with the amplitudes of each harmonic fixed.…

Let $T\colon X \to X$ be a smooth diffeomorphism of a compact Riemannian manifold $X$ with a natural measure $\mu$ of information dimension $\operatorname{ID}(\mu)=D$. An observabl…

Let $\Phi$ be a transitive Anosov flow on a $3$-manifold $M$, let $\widetilde{\Phi}$ be its lift to $\widetilde{M}=\mathbb{R}^3$, and let $\mathcal{P}$ be the bifoliated plane asso…

Let $\phi:[0,1]\rightarrow [0,1]$ be an $\alpha$-Hölder continuous map for some $\alpha\in(0,1)$. The metric mean dimension…

Let $\phi:[0,1]\rightarrow [0,1]$ be an $\alpha$-Hölder continuous map, with $\alpha>0$. The metric mean dimension $\operatorname{mdim}_{\operatorname{M}}([0,1],|\cdot|,\phi)$ is t…

Let $\phi:[0,1]\rightarrow [0,1]$ be an $\alpha$-Hölder continuous map. The metric mean dimension $\operatorname{mdim}_{\operatorname{M}}([0,1],|\cdot|,\phi)$ is defined using the…

Index divisor free prime conjecture. Except when $(d,k)=(27,3)$, there exists an index divisor free prime for $(d,k)$.

Throughout, let $K$ be the middle-third Cantor set, let $\mu$ be its natural probability measure, and for $\psi:\mathbb R\to[0,\infty)$ and $y\in\mathbb R$ define … Here $\|x\|$ de…

Let $K\subset\mathbb E^n$ be a smooth manifold of codimension at least $2$ with a regular embedding, and let $\mathcal W(\epsilon)$ denote the volume of the trapping locus…

Potential density under int-amplified endomorphisms. If $X$ admits an int-amplified endomorphism, then $X$ satisfies potential density.

Veech's conjecture. Condition $\pi_0\perp L^2(\Pi(\kappa))$ for each Furstenberg system $\kappa\in V_S(\boldsymbol{\mu})$ is equivalent to Sarnak's conjecture.