The inverse Galois problem over Q
Decide whether every finite group occurs as the Galois group of a Galois extension of Q. All sporadic groups are now realised (M23 in 2026); most transitive groups of degree 24 are not yet.
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Open problems
Each problem states how progress is verified and what counts as a contribution. Besides the problems curated here, the catalogue includes open conjectures from Formal Conjectures (with Lean statements), optimization constants and the AlphaEvolve problems. Know one that belongs here? Propose a problem.
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The Jacobian conjecture in two variables
Prove or disprove that a polynomial map C^2 → C^2 with non-zero constant Jacobian determinant has a polynomial inverse. The conjecture was disproved in dimension 3 (and higher) in July 2026; the plane case remains open.
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10-point multi-point Seshadri constant on ℙ^2
Let x_1,…,x_10 be very general points of ℙ^2, and let π:X→ ℙ^2 be the blow-up of ℙ^2 at these points. Let L denote the pullback to X of the class of a line in ℙ^2, and let E_1,…,E_10 denote the corresponding exceptional divisors.
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Asymptotic essential-dimension ratio of the symmetric groups
For each integer n ≥ 1, let S_n be the symmetric group on n letters. Over a base field k, the essential dimension ed_k(S_n) is the smallest integer d such that the general degree-n polynomial x^n + a_1 x^n-1 + ⋯ + a_n can be reduced to a d-parameter form by a Tschirnhaus transformation.
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Asymptotic line-count constant for smooth degree-d surfaces in ℙ^3 in characteristic 0
For each integer d ≥ 3, let ℓ_0(d) denote the maximal number of lines contained in a smooth surface of degree d in ℙ^3_ℂ. We define C_76 := limsup_d→∞ℓ_0(d)/d^2. The constant C_76 measures the quadratic growth rate of the maximal line count on smooth complex degree-d surfaces.
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Exponent for commutators close to the identity
Let H be an infinite-dimensional complex Hilbert space and let B(H) be the Banach algebra of bounded operators on H, equipped with the operator norm.
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Algebraic consequences of the Farrell–Jones conjecture
Vanishing of the reduced projective class group for integral group rings. If G is torsion-free, that is, if its only element of finite order is 1, then every finitely generated projective module over ℤ[G] is stably free.
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Ben Green's Open Problem 4
What is the largest product-free set in the alternating group A_n?
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Casas-Alvero Conjecture
The Casas-Alvero conjecture states that in characteristic zero, if a monic polynomial P has the Casas-Alvero property, then P = (X - α)ᵈ for some α.
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Conjecture 1.35(c)
Do there exist simple pro-orderable groups?
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Conjecture 1.40
Is a group a nilgroup if it is the product of two normal nilsubgroups? Since H and K are normal, the product HK coincides with the join H sqcup K, so "G is the product of H and K" is stated as H sqcup K = G.
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Conjecture 1.74 (Minimal topological groups)
Describe all minimal topological groups, that is, all non-discrete Hausdorff topological groups whose proper closed subgroups are all discrete.
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Conjecture 19.25
Let G and H be finite groups of the same order with Σ_g ∈ G φ(|g|) = Σ_h ∈ H φ(|h|), where φ is the Euler totient function. Suppose that G is simple. Is H necessarily simple?
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Conjecture 20.76
Let G be a finite p-group and assume that all abelian normal subgroups of G have order at most p^k. Is it true that every abelian subgroup of G has order at most p^2k?
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Conjecture 8.8
Does there exist a non-cyclic finitely presented group G which contains an element a such that each element of G is conjugate to some power of a? Here a power of a means a^n for some n ∈ ℤ.
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Determinantal conjecture
Does the determinant of the sum A + B of two n × n normal complex matrices A and B always lie in the convex hull of the n! points Π_i (λ(A)_i + λ(B)_σ(i))? Here the numbers λ(A)_i and λ(B)_i are the eigenvalues of A and B, and σ is an element of the symmetric group S_n.
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Equational Theories
Equational Theories, Problem 8.1. Does Equation 677 imply Equation 255 in every finite magma? The project tentatively conjectures that the answer is no; a false answer is equivalent to the existence of a finite countermodel satisfying Equation 677 but not Equation 255.
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Erdős Problem #1150
Is there some constant c > 0 such that, for all large enough n and all polynomials P of degree n with coefficients in -1, 1, max_|z|=1 |P(z)| > (1 + c) √(n)?
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Erdős Problem #274
If G is a group, can there exist an exact covering of G by more than one coset of different sizes? (i.e. each element is contained in exactly one of the cosets.) The conjectured answer is no: in every such exact covering, two of the subgroups have the same cardinality.
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Erdős Problem #522
Let f(z)=Σ_0≤ k≤ n ε_k z^k be a random polynomial, where ε_k∈ -1,1 independently uniformly at random for 0≤ k≤ n. Is it true that, if R_n is the number of roots of f(z) in z∈ ℂ : lvert zrvert ≤ 1, then R_n/n/2→ 1 almost surely?
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Finite generation of the cohomology of finite-dimensional Hopf algebras
Finite generation conjecture (Etingof–Ostrik, Conjecture 2.18, algebra part). For every finite-dimensional Hopf algebra A over a field k, the cohomology ring H^(A, k) = Ext^_A(k, k) is a finitely generated k-algebra.
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Gap conjecture
If a finitely generated group has superpolynomial growth, then with respect to any finite generating set its growth function is at least e^sqrt n in Grigorchuk's preorder on growth functions, where the comparison is witnessed by linearly rescaling the radius.
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Generic and maximal rank of 3-tensors
Friedland's conjecture. In the critical range m_3 ≤ (m_1 - 1)(m_2 - 1), and away from the formats (3, 2p+1, 2p+1), the generic rank of a tensor of format (m_1, m_2, m_3) is the value ⌈ m_1m_2m_3 / (m_1 + m_2 + m_3 - 2) ⌉ predicted by a dimension count [Fri12, Conjecture 5.1].
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Gottschalk's surjunctivity conjecture
Gottschalk's surjunctivity conjecture (1973): every group is surjunctive. That is, for every group G and every finite alphabet A, every injective cellular automaton on A^G is surjective.
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Hartshorne's conjecture on Vector Bundles
There are no indecomposable vector bundles of rank 2 on ℙ^n for n ≥ 7. This is Conjecture 6.3 in [Har1974].
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Jacobson Conjecture
The Jacobson conjecture (in its modern form): In a (noncommutative) ring which is left and right Noetherian, the intersection of the powers of the Jacobson ideal is trivial
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Kaplansky's Conjectures
The zero-divisor conjecture If G is torsion-free, then the group algebra K[G] has no non-trivial zero divisors.
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Köthe conjecture
The Köthe conjecture: In any ring, the sum of two nil left ideals is nil.
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Leinster Groups
Conjecture: Are there infinitely many Leinster groups? This asks whether there exist infinitely many (non-isomorphic) finite groups that are Leinster groups. Formalized via the negation of "Does there exist an n such that all Leinster groups have order less than n".
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Local uniformization
Local uniformization in positive characteristic. Let k be a field of characteristic p > 0, let F be a finitely generated field extension of k, and let O be a valuation ring of F containing k. Then O admits local uniformization over k.
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Mathoverflow 21003
Is there any polynomial f(x, y) ∈ ℚ[x, y] such that f : ℚ × ℚ → ℚ is a bijection?
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Mathoverflow 339137
Let P(x), Q(x) ∈ ℝ[x] be two monic polynomials with non-negative coefficients. If R(x) = P(x)Q(x) is a 0,1 polynomial (coefficients only from 0,1), then P(x) and Q(x) are also 0, 1 polynomials.
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Mean value problem
Given a complex polynomial p of degree d ≥ 2 and a complex number z there is a critical point c of p, such that |p(z)-p(c)|/|z-c| ≤ |p'(z)|.
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Pierce–Birkhoff conjecture
The Pierce-Birkhoff conjecture states that for every real piecewise-polynomial function f : ℝⁿ → ℝ, there exists a finite set of polynomials gᵢⱼ ∈ ℝ[x₁, ..., xₙ] such that f = supᵢ infⱼ(gᵢⱼ).
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Resolution of singularities
Resolution of singularities in positive characteristic. Let k be a perfect field of characteristic p > 0 and let X be an integral scheme that is separated and of finite type over k. Then there is an integral scheme Y that is smooth over k together with a proper birational morphism Y → X.
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Serre's multiplicity conjectures
Positivity conjecture. Let R be a regular local ring and let M, N be finitely generated R-modules such that M otimes_R N has finite length. If dim M + dim N = dim R, then χ(M, N) > 0. The hypothesis on dimensions forces M and N to be nonzero, since the dimension of the zero module is bot.
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The Andrews-Curtis conjecture
The Andrews-Curtis conjecture. Every normally generating n-tuple in the free group of rank n is Andrews-Curtis equivalent to the standard tuple of free generators.
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The Auslander-Reiten conjecture
The Auslander-Reiten conjecture [AR75]. Let Λ be an Artin algebra and M a finitely generated Λ-module with Ext^i_Λ(M, Λ) = 0 and Ext^i_Λ(M, M) = 0 for all i > 0. Then M is projective.
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The circulant Hadamard conjecture
Every circulant Hadamard matrix has order at most four.
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The Eisenbud-Green-Harris conjecture
The Eisenbud-Green-Harris conjecture. Let I ⊆ k[x_1, …, x_n] be a homogeneous ideal containing a regular sequence of forms of degrees d_1 ≤ … ≤ d_c. Then there is a lex ideal L such that I has the same Hilbert function as L + (x_1^d_1, …, x_c^d_c).
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The Gerstenhaber problem for three commuting matrices
The Gerstenhaber problem: if A, B, and C are pairwise commuting n × n matrices over a field K, is the dimension of the unital K-algebra K[A, B, C] they generate always at most n?
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The S_3-conjecture (conjugacy classes of distinct sizes)
Markel's S_3-conjecture (1973): any nontrivial finite ah-group is isomorphic to S_3. The conjecture is open in general; it is known to be true for solvable groups.
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The small Cohen-Macaulay modules conjecture
The small Cohen-Macaulay modules conjecture. If R is a complete Noetherian local ring, then there is a finitely generated R-module M ≠ 0 such that some system of parameters of R is a regular sequence on M. Hochster stated the conjecture for complete local domains [Ho17, Conjecture 2.1].
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The symbol length of K^M_n(ℂ(x_1, …, x_m))/p
The symbol length problem for complex rational function fields [Krashen2024, Problem 2.1.3.12 and §2.1.3.4]: determine, as a function of m, n and the prime p, the symbol length of K^M_n(ℂ(x_1, …, x_m))/p, that is the least k such that every class is a sum of at most k symbols, or ∞ if there is no…
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Zariski Cancellation
The Zariski Cancellation Problem: every polynomial ring over a field k of characteristic 0 is cancellative.
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