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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Kaplansky's conjectures
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Köthe conjecture
if a ring has no nil ideal other than \{0\}, then it has no nil one-sided ideal other than \{0\}.
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Monomial conjecture on Noetherian local rings
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Existence of perfect cuboids and associated cuboid conjectures
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Pierce–Birkhoff conjecture
every piecewise-polynomial f:\mathbb{R}^{n}\rightarrow\mathbb{R} is the maximum of a finite set of minimums of finite collections of polynomials.
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Rota's basis conjecture
for matroids of rank n with n disjoint bases B_{i}, it is possible to create an n \times n matrix whose rows are B_{i} and whose columns are also bases.
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Serre's conjecture II
if G is a simply connected semisimple algebraic group over a perfect field of cohomological dimension at most 2, then the Galois cohomology set H^{1}(F, G) is zero.
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Serre's positivity conjecture that if R is a commutative regular local ring, and P, Q are prime ideals of R, then \dim (R/P) + \dim (R/Q) =…
Serre's positivity conjecture that if R is a commutative regular local ring, and P, Q are prime ideals of R, then \dim (R/P) + \dim (R/Q) = \dim (R) implies \chi(R/P, R/Q) > 0.
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Uniform boundedness conjecture for rational points
do algebraic curves of genus g \geq 2 over number fields K have at most some bounded number N(K, g) of K-rational points?
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Wild problems
problems involving classification of pairs of n\times n matrices under simultaneous conjugation.
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Zariski–Lipman conjecture
for a complex algebraic variety V with coordinate ring R, if the derivations of R are a free module over R, then V is smooth.
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Zauner's conjecture
do SIC-POVMs exist in all dimensions?
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Andrews–Curtis conjecture
every balanced presentation of the trivial group can be transformed into a trivial presentation by a sequence of Nielsen transformations on relators and conjugations of relators
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Bounded Burnside problem
for which positive integers m, n is the free Burnside group B(m,n) finite? In particular, is B(2, 5) finite?
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Guralnick–Thompson conjecture on the composition factors of groups in genus-0 systems
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Herzog–Schönheim conjecture
if a finite system of left cosets of subgroups of a group G form a partition of G, then the finite indices of said subgroups cannot be distinct.
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The inverse Galois problem
is every finite group the Galois group of a Galois extension of the rationals?
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Isomorphism problem of Coxeter groups
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Are there an infinite number of Leinster groups?
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Does generalized moonshine exist?
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Is every finitely presented periodic group finite?
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Is every group surjunctive?
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Problems in loop theory and quasigroup theory consider generalizations of groups
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Arthur's conjectures
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Dade's conjecture relating the numbers of characters of blocks of a finite group to the numbers of characters of blocks of local subgroups.
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Demazure conjecture on representations of algebraic groups over the integers.
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Kazhdan–Lusztig conjectures relating the values of the Kazhdan–Lusztig polynomials at 1 with representations of complex semisimple Lie…
Kazhdan–Lusztig conjectures relating the values of the Kazhdan–Lusztig polynomials at 1 with representations of complex semisimple Lie groups and Lie algebras.
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McKay conjecture
in a group G, the number of irreducible complex characters of degree not divisible by a prime number p is equal to the number of irreducible complex characters of the normalizer of any Sylow p-subgroup within G.
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The Brennan conjecture
estimating the integral powers of the moduli of the derivative of conformal maps into the open unit disk, on certain subsets of \mathbb{C}
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Fuglede's conjecture on whether nonconvex sets in \mathbb{R} and \mathbb{R}^{2} are spectral if and only if they tile by translation.
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Data last refreshed 2026-10-20 03:01 UTC.