3,618 problems

In physics, the Painlevé conjecture is a theorem about singularities among the solutions to the n-body problem: there are noncollision singularities for n ≥ 4.

Zimmer's conjecture is a statement in mathematics "which has to do with the circumstances under which geometric spaces exhibit certain kinds of symmetries." It was named after the…

In mathematics, the Weinstein conjecture refers to a general existence problem for periodic orbits of Hamiltonian or Reeb vector flows. More specifically, the conjecture claims tha…

Outer billiards is a dynamical system based on a convex shape in the plane. Classically, this system is defined for the Euclidean plane but one can also consider the system in the…

The Mandelbrot set () is a two-dimensional set. It is defined in the complex plane as the complex numbers $c$ for which the function $f_{c}(z)=z^{2}+c$ does not diverge to infinity…

In mathematics, particularly in dynamical systems, the Hilbert–Arnold problem is an unsolved problem concerning the estimation of limit cycles of the dynamics of a flow on a sphere…

In applied mathematics, the Kaplan–Yorke conjecture concerns the dimension of an attractor, using Lyapunov exponents. By arranging the Lyapunov exponents in order from largest to s…

In mathematics, the Fatou conjecture, named after Pierre Fatou, states that a quadratic family of maps from the complex plane to itself is hyperbolic for an open dense set of param…

In the mathematics of dynamical systems, Eden's conjecture states that the supremum of the local Lyapunov dimensions on the global attractor is achieved on a stationary point or an…

The Collatz conjecture is one of the most famous unsolved problems in mathematics. The conjecture asks whether repeating two simple arithmetic operations will eventually transform…

Quantum chaos is a branch of physics focused on how chaotic classical dynamical systems can be described in terms of quantum theory. The primary question that quantum chaos seeks t…

The Arnold conjecture, named after mathematician Vladimir Arnold, is a mathematical conjecture in the field of symplectic geometry, a branch of differential geometry.

Furstenberg's ×2, ×3 conjecture. Lebesgue measure is the only $T_2,T_3$-invariant nonatomic measure on $\mathbb{R}/\mathbb{Z}$.

A diffeomorphism $f$ is a diffeomorphism of a compact manifold. Palis's conjecture. Any diffeomorphism can be $C^r$ approximated by a Morse–Smale one or by one exhibiting a transve…

Let $X$ be a quasi-projective variety defined over $\mathbb{C}$, let $\Phi$ be an endomorphism of $X$, let $V\subset X$ be a closed subvariety, and let $x\in X(\mathbb{C})$. Dynami…

Katok's conjecture. The collection of ergodic entropies contains

Consider a strictly convex closed planar curve $C$ and the billiard in the bounded domain it encloses. The billiard is Birkhoff caustic-integrable if a topological annulus adjacent…

A positive real number $x$ is a $Z$-number if the sequence $(\\{x(3/2)^n\\})_{n\in\mathbb N}$ is contained in $[0,1/2)$. Mahler's conjecture. No $Z$-number exists. Mahler proposed…

Consider a reaction-network dynamical system with a positive complex-balanced equilibrium, meaning an equilibrium at which the total flow into and out of each complex agrees. The a…

An outer length billiard is associated with a plane oval, meaning a closed strictly convex smooth curve, and acts on the exterior of the oval. Its periodic points are points lying…

Kaplan–Yorke conjecture. The quantity $d_{KY}$ coincides with the Hausdorff dimension of any attractor.

Let $\mathbb{A}=(A,B)\in\mathbf{G}^2$, where $\mathbf{G}=\{X\in\mathrm{GL}(2,\mathbb{R}):\det X>0\}$, and let $\hat{\rho}(\mathbb{A})$ be the joint spectral radius, defined by … A…

An integrable Hamiltonian system is a Hamiltonian system with more than two degrees of freedom whose dynamics is integrable. Consider such a system subjected to a small Hamiltonian…

Quantum unique ergodicity conjecture. It is not necessary to pass to a density-one subsequence: the entire sequence converges to the Liouville measure in the sense of semiclassical…

Let $(\alpha,A_E)$ be the Schrödinger cocycle associated with a one-dimensional analytic quasi-periodic Schrödinger operator, where…