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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Fujita conjecture regarding the line bundle K_{M} \otimes L^{\otimes m} constructed from a positive holomorphic line bundle L on a compact…
Fujita conjecture regarding the line bundle K_{M} \otimes L^{\otimes m} constructed from a positive holomorphic line bundle L on a compact complex manifold M and the canonical line bundle K_{M} of M
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General elephant problem
do general elephants have at most Du Val singularities?
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Hartshorne's conjectures
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In spherical or hyperbolic geometry, must polyhedra with the same volume and Dehn invariant be scissors-congruent?
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Jacobian conjecture for two-dimensional spaces
if a two-dimensional polynomial mapping over a characteristic-0 field has a constant nonzero Jacobian determinant, then does it have a regular (i.e. with polynomial components) inverse function?
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Maulik–Nekrasov–Okounkov–Pandharipande conjecture on an equivalence between Gromov–Witten theory and Donaldson–Thomas theory
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Nagata's conjecture on curves, specifically the minimal degree required for a plane algebraic curve to pass through a collection of very…
Nagata's conjecture on curves, specifically the minimal degree required for a plane algebraic curve to pass through a collection of very general points with prescribed multiplicities.
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Nagata–Biran conjecture that if X is a smooth algebraic surface and L is an ample line bundle on X of degree d, then for sufficiently large…
Nagata–Biran conjecture that if X is a smooth algebraic surface and L is an ample line bundle on X of degree d, then for sufficiently large r, the Seshadri constant satisfies \varepsilon(p_1,\ldots,p_r;X,L) = d/\sqrt{r}.
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Nakai conjecture
if a complex algebraic variety has a ring of differential operators generated by its contained derivations, then it must be smooth.
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Parshin's conjecture
the higher algebraic K-groups of any smooth projective variety defined over a finite field must vanish up to torsion.
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Section conjecture on splittings of group homomorphisms from fundamental groups of complete smooth curves over finitely-generated fields k…
Section conjecture on splittings of group homomorphisms from fundamental groups of complete smooth curves over finitely-generated fields k to the Galois group of k.
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Standard conjectures on algebraic cycles
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Tate conjecture on the connection between algebraic cycles on algebraic varieties and Galois representations on étale cohomology groups.
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Virasoro conjecture
a certain generating function encoding the Gromov–Witten invariants of a smooth projective variety is fixed by an action of half of the Virasoro algebra.
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Zariski multiplicity conjecture on the topological equisingularity and equimultiplicity of varieties at singular points
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Are infinite sequences of flips possible in dimensions greater than 3?
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Resolution of singularities in characteristic p
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Borsuk's problem on upper and lower bounds for the number of smaller-diameter subsets needed to cover a bounded n-dimensional set.
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The covering problem of Rado
if the union of finitely many axis-parallel squares has unit area, how small can the largest area covered by a disjoint subset of squares be?
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The Erdős–Oler conjecture
when n is a triangular number, packing n-1 circles in an equilateral triangle requires a triangle of the same size as packing n circles.
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The disk covering problem about finding the smallest real number r(n) such that n disks of radius r(n) can be arranged in such a way as to…
The disk covering problem about finding the smallest real number r(n) such that n disks of radius r(n) can be arranged in such a way as to cover the unit disk.
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The kissing number problem for dimensions other than 1, 2, 3, 4, 8 and 24
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Reinhardt's conjecture
the smoothed octagon has the lowest maximum packing density of all centrally-symmetric convex plane sets
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Sphere packing problems, including the density of the densest packing in dimensions other than 1, 2, 3, 8 and 24, and its asymptotic…
Sphere packing problems, including the density of the densest packing in dimensions other than 1, 2, 3, 8 and 24, and its asymptotic behavior for high dimensions.
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Square packing in a square
what is the asymptotic growth rate of wasted space?
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Ulam's packing conjecture about the identity of the worst-packing convex solid
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The Tammes problem for numbers of nodes greater than 14 (except 24).
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For which dimensions of sphere differential Poincaré conjecture is true?
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The spherical Bernstein's problem, a generalization of Bernstein's problem
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Carathéodory conjecture
any convex, closed, and twice-differentiable surface in three-dimensional Euclidean space admits at least two umbilical points.
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Data last refreshed 2026-10-20 03:01 UTC.