Problems

For a pure O-sequence h = (h_0, …, h_e) of codimension three and type two, is h_i^2 ≥ h_i-1 h_i+1 for every interior index i? The stated monomial case is proved; the broader level-Hilbert-function case remains open.

Swinnerton-Dyer (1981) proved R-equivalence trivial on smooth cubic surfaces over p-adic fields with good reduction, except for three special types. The paper resolves two long-standing exceptional cases: triviality for the diagonal cubic…

Given n and 1 ≤ c ≤ n!, can n distinct group elements be chosen so that their n! ordered products take exactly c distinct values? Constructions realize every c.

The Hessian conjecture HC_n asks whether every polynomial f with det Hess(f) ∈ C^× has a polynomial gradient inverse. It is known for n ≤ 3, false for n ≥ 5, and open exactly in dimension four, where it implies the plane Jacobian…

For a finite-dimensional algebra A, finite global dimension forces HH_n(A) = 0 for all large n. Han conjectured the converse: eventual vanishing of Hochschild homology should detect homological smoothness. Disproved by an explicit…

Does the Hodge bundle Ω_g over the moduli stack of genus g ≥ 2 curves contain any nontrivial sub-bundles? Posed by Dawei Chen around 2015; the answer is no.

Is the exact nonreal spectral region of the four-cycle family of row-stochastic nonnegative matrices determined by the Karpelevich constraint, as Ran and Teng conjectured in 2024?

Cornulier asked, in a MathOverflow discussion, whether amenability of a module over an associative algebra depends on the ground field. It does not: the notion is invariant under change of base field.

For an extension G = A rtimes B of elementary abelian p-groups with a ∈ A satisfying C_B(a) = 1, must H = ⟨ a, B⟩ satisfy rank(Z(H) ∩ H') ≤ rank(B)? An explicit extension violates the bound.

Is every group sofic - does every group admit approximate finite permutation representations? A central open question of geometric group theory since Gromov introduced soficity: soficity implies Gottschalk's surjunctivity conjecture,…

Does there exist a group with more than one but only finitely many maximal locally soluble normal subgroups? An explicit group with exactly two settles it.

Is the depth of the mod-p cohomology ring of every finite group realized as the dimension of one of its associated primes? For G = SmallGroup(128, 859) over overlineF_2 the ring has depth 2 while every associated-prime quotient has…

For a perfect field k and a representation-infinite finite-dimensional k-algebra A, the Auslander–Reiten quiver of A has infinitely many connected components. This establishes a conjecture of Auslander, Reiten and Smalø, for…

The Huneke–Wiegand Conjecture: Let R be a one-dimensional Gorenstein local domain, and let M be a finitely generated, non-zero, torsion-free R-module. If the tensor product M ⊗_R M^* is torsion-free, then M is a projective (hence free)…

Dittert's conjecture asserts that among nonnegative n× n matrices whose entries sum to n, the functional φ(A)=∏_i r_i+∏_j c_j-per(A) is uniquely maximized by J_n/n. The paper proves the case n=16 which, with Pang's result for n≥17,…

Grothendieck asked whether every finite locally free group scheme of order n is killed by n (its n-th convolution power map equals the unit). The counterexample is an order-4 group scheme not killed by 4 (killed only by 8); since Deligne…

For a projective variety X with at worst Gorenstein canonical singularities whose stringy E-function E_st(X; u, v) is a polynomial, all stringy Hodge numbers h^p,q_st(X) are non-negative. (Batyrev 1998, Conjecture 3.10.)

Baker asked, as recorded by Poonen, whether a fixed smooth quasiprojective variety over a finite field must acquire a smooth rational hyperplane section after every sufficiently high-dimensional linearly nondegenerate embedding. Poonen…

Zhao's Generalized Vanishing Conjecture asks whether, for a differential operator with constant coefficients, Λ^m(P^m) = 0 for all large m forces Λ^m(P^m Q) = 0 for all large m. Refuted by an explicit five-variable counterexample.

Extending the minimal model program beyond threefolds in positive characteristic is a standing goal of birational geometry. Assuming the log resolution conjecture for all log pairs birational to X, the cone theorem holds for projective log…