23 problems

Let $\mathcal{V}\in\mathbb{R}^N$ be the vector of intrinsic velocities, and let $\kappa_c(\mathcal{V})$ denote the critical coupling strength of the Winfree system. Let initial dat…

Let $N$ oscillators have initial data $(\{\theta_i^0\}_{i=1}^N,\{\omega_i^0\}_{i=1}^N)$ and system parameters $(\{\nu_i\}_{i=1}^N,\kappa,m)$. Write $\mathcal{V}=(\nu_i)_{i=1}^N$ an…

Optimal condition-number threshold conjecture. The optimization landscape of the low-rank factorization problem is benign if

Let $\mathcal{V}=(\nu_1,\ldots,\nu_N)$ be the natural-frequency vector, let $\mathcal{D}(\mathcal{V})$ denote its frequency-diameter quantity, and let $\kappa_c(\mathcal{V})$ be th…

Let $u1,\ldots, uN$ be fixed natural frequencies, and let $u=( u1,\ldots, uN)$. Consider the inertial Kuramoto system $$, with coupling strength$kappa$, phase initial data$Theta…

A graph is globally synchronizing when its adjacency matrix has only synchronized global minima of the Kuramoto energy. Density-threshold conjecture. For every $\varepsilon>0$, the…

Let $A$ be an $n\times n$ random matrix with zero diagonal and independent off-diagonal entries satisfying … A matrix is globally synchronizing when the only local minima of its Ku…

Let a graph be globally synchronizing when its adjacency matrix is globally synchronizing, meaning that the only local minima of the Kuramoto energy are the global synchronized min…

Generic-data phase-locking conjecture. (Weak form) For every $\varepsilon>0$, there exists $\delta=\delta(\varepsilon)$ such that, if $\mathcal{D}(\mathcal{V})/\kappa<\delta$, then…

Weak-Lyapunov-functional conjecture. Either there is a constant $c\ge\frac12$ such that, whenever $\kappa>c\mathcal{D}(\mathcal{V})$, both models admit weak Lyapunov functionals, o…

Let $p$ denote the factorization rank in the Burer–Monteiro factorization … is benign with high probability in each of the following settings: the high-dimensional Kuramoto model,…

Let $\tau_{\rm sync}$ denote the synchronization time of the random DFA, and retain the notation in which $\mathbf Q$ is the uniform distribution on words of the relevant length. R…

Let $(V,\mathcal C,\{f_x\}_{x\in V})$ be a random DFA with $|V|=n$. For a fixed $t\in\mathbb N$, let $\mathbf Q=\mathbf Q_t$ be the uniform distribution over words in…

Consider the finite-particle all-to-all Kuramoto model with natural-frequency vector $\Omega$ and frequency diameter $D(\Omega)$. Let $\Theta^0$ be an initial phase vector, and let…

Let $N$ be the number of particles, let $\Theta^0$ be an initial phase vector, and let $\Omega=(\nu_1,\ldots,\nu_N)$ be the vector of natural frequencies. Write $D(\Omega)$ for the…

Let $\mathcal{A}_n(\mathbf{a},\mathbf{b})$ be a circular automaton of order $n$, where $\mathbf{a}:\mathbb{Z}_n\to\mathbb{Z}_n$ is fixed and $\mathbf{b}\in\mathcal{M}_n$ is chosen…

Let $\mu$ be the common distribution of natural frequencies in the two communities, let $\psi=\psi_{2}-\psi_{1}\in\{0,\pi\}$, and let $r_{1},r_{2}$ be the two synchronization order…

Let $\theta$ denote the phase on the circle, let $\theta_{1}$ and $\theta_{2}$ be the average phases of the two communities, and write their phase difference as … In the disorder-f…

Large-set conjecture. There is a function $L$ such that, if $n\geqslant L(k)$, then the Johnson scheme $J(n,k)$ is non-synchronizing if and only if there exists a large set of Stei…

Let $G$ be a primitive permutation group of degree $n$, and define … to be its set of non-synchronizing ranks. Non-synchronizing-rank conjecture. … This predicts that primitive gro…

Let $G$ be a primitive permutation group of degree $n$, and let a map have rank $r$, meaning that its image has cardinality $r$. Large-rank almost synchronizing conjecture. If … th…

Synchronization conjecture. Every primitive permutation group synchronizes every non-uniform map.

Let $N$ phase oscillators evolve according to the homogeneous Kuramoto system … where $\theta_0\in\mathbb{R}^N$ and $R(\theta)$ is the magnitude of the order parameter. Global phas…