The world's unsolved problems
993 open questions across 14 fields, gathered from Wikipedia. Pick one and start thinking.
488 results in Mathematics · clear · surprise me
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Banach's problem
is there an ergodic system with simple Lebesgue spectrum?
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Birkhoff conjecture
if a billiard table is strictly convex and integrable, is its boundary necessarily an ellipse?
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Collatz conjecture (also known as the 3n + 1 conjecture)
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Eden's conjecture that the supremum of the local Lyapunov dimensions on the global attractor is achieved on a stationary point or an…
Eden's conjecture that the supremum of the local Lyapunov dimensions on the global attractor is achieved on a stationary point or an unstable periodic orbit embedded into the attractor.
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Fatou conjecture that a quadratic family of maps from the complex plane to itself is hyperbolic for an open dense set of parameters.
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Furstenberg conjecture
is every invariant and ergodic measure for the \times 2,\times 3 action on the circle either Lebesgue or atomic?
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Kaplan–Yorke conjecture on the dimension of an attractor in terms of its Lyapunov exponents
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Margulis conjecture
measure classification for diagonalizable actions in higher-rank groups.
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Hilbert–Arnold problem
is there a uniform bound on limit cycles in generic finite-parameter families of vector fields on a sphere?
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MLC conjecture
is the Mandelbrot set locally connected?
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Many problems concerning an outer billiard, for example showing that outer billiards relative to almost every convex polygon have unbounded…
Many problems concerning an outer billiard, for example showing that outer billiards relative to almost every convex polygon have unbounded orbits.
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Quantum unique ergodicity conjecture on the distribution of large-frequency eigenfunctions of the Laplacian on a negatively-curved manifold
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Rokhlin's multiple mixing problem
are all strongly mixing systems also strongly 3-mixing?
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Triangular billiards
does every triangle have a periodic billiards path?
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Weinstein conjecture
does a regular compact contact type level set of a Hamiltonian on a symplectic manifold carry at least one periodic orbit of the Hamiltonian flow?
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Does every positive integer generate a juggler sequence terminating at 1?
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Lyapunov function
Lyapunov's second method for stability – For what classes of ODEs, describing dynamical systems, does Lyapunov's second method, formulated in the classical and canonically generalized forms, define the necessary and sufficient conditions for the...
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Is every reversible cellular automaton in three or more dimensions locally reversible?
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Sudoku
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Tic-tac-toe variants
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Chess
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Are the nim-sequences of all finite octal games eventually periodic?
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Is the nim-sequence of Grundy's game eventually periodic?
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Rendezvous problem
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Abundance conjecture
if the canonical bundle of a projective variety with Kawamata log terminal singularities is nef, then it is semiample.
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Bass conjecture on the finite generation of certain algebraic K-groups.
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Bass–Quillen conjecture relating vector bundles over a regular Noetherian ring and over the polynomial ring A[t_{1}, \ldots, t_{n}].
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Deligne conjecture
any one of numerous named for Pierre Deligne.
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Dixmier conjecture
any endomorphism of the Weyl algebras A_{1} and A_{2} is an automorphism.
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Fröberg conjecture on the Hilbert functions of a set of forms.
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