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.

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.

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.

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…

A collection of open problems drawn from published lists, including Cahen, Fontana, Frisch and Glaz's Open Problems in Commutative Ring Theory and Erman and Sam's survey of Boij-Soderberg theory, each proved or disproved by one automated…

If CT_(k) is generated by all horizontal class transpositions with modulus at most k, is CT_(k) ≅ S_lcm(2,…,k) for every k ≥ 4?

For a semistable one-parameter family of complex projective varieties with smooth nearby fiber X_t and monodromy T, is the map H^1(X, Z) → H^1(X_t, Z)^T surjective? True in degree one, although the integral statement fails in higher degree.

Does the conjectured universal formula for normalized alternating syzygy power sums of numerical semigroup rings hold for every index?

If the power graph of a finite group contains no induced path on four vertices, must it also contain no induced cycle of length at least four - that is, is every cograph power graph chordal?

Does the analytic Bertini restriction theorem for multiplier ideals hold locally, outside a pluripolar exceptional set of fibers? Proved in full generality.

Can a noetherian ring have a local cohomology module whose support is not closed - equivalently, one with infinitely many minimal primes? Huneke and Lyubeznik asked; the paper constructs such rings, so the answer is yes.

Give an explicit profinite presentation of Gal(overlineQ_2 / Q_2). The tame local cases were settled by the early 1980s; the dyadic case was the last one missing. The new presentation has four generators, two word relations and a pro-2…

Can a nonabelian group admit a Rota-Baxter operator that is surjective but not injective? A construction shows yes.

Han's conjecture predicts that a finite-dimensional algebra with eventually vanishing Hochschild homology has finite global dimension. Within the tau-Hochschild framework it splits into persistence and survival, and the Liu-Morin extension…

For the set of n by n doubly stochastic matrices, Kim and Roush conjectured in 1981 that for odd n = 2k+1 > 1 the maximum of per(I-A) equals 3 times 2^(k-2), attained by an explicit block construction. Proved in full, and the maximizers…

Dogon, Levit and Vigdorovich asked for an explicit upper bound on the stability radius of an infinitely presented group. The lamplighter group provides the first: explicit polynomial bounds on both its Hilbert-Schmidt stability rate and…