Problems

Djament asked whether a Grothendieck category satisfying suitable finiteness and exactness conditions must be equivalent to a module category. In the locally noetherian case the answer is no: there is a Grothendieck category with a…

Bondal and Polishchuk conjectured in 1993 that the braid group acts transitively on the set of full exceptional collections in a triangulated category. Chang, Haiden and Schroll disproved it for partially wrapped Fukaya categories, but no…

Is every weakly quasi-complete Noetherian local ring quasi-complete? Asked by D. D. Anderson in 2014. The ring A = k^p[[X, Y]][k] with k = F_p(u_1, u_2, …) is weakly quasi-complete but not quasi-complete.

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…

Do a finite group's order together with ∑_g ∈ G φ(|g|) determine whether the group is simple? A simple and a non-simple group of order 6048 share the statistic 23984.

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?

Must the right-relatively convex subgroups of a right-orderable nonabelian group form a sublattice of its subgroup lattice? A construction shows they need not.

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.

Adjamagbo's positive-characteristic refinement of the Jacobian conjecture asks that a polynomial endomorphism with unit Jacobian determinant whose induced function-field extension has degree prime to the characteristic be an automorphism.…

Twisted Deligne products categorify the tensor product of two Grothendieck rings. Classifying them leads to categorical n-cocycles, and Johnson-Freyd, Ostrik and Yu asked whether these are always pullbacks of ordinary group cocycles on the…

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?

Every purely-maximal ideal of a commutative ring is purely-prime, and the converse holds for several important classes of rings; Tarizadeh conjectured (Conjecture 5.8 of his earlier published paper) that in a commutative ring every…

For a Brauer class on a variety, the period-index conjecture bounds the index in terms of the period and the dimension. Disproved: for any uncountable algebraically closed field k of characteristic 0 and any d ≥ 3 there is a d-dimensional…

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

The BCHM theorem makes the log canonical ring of a projective klt pair finitely generated. Generalized pairs, introduced by Birkar and Zhang, add an auxiliary nef part and have become a central tool in birational geometry, so it is natural…

A group is Howson if the intersection of any two finitely generated subgroups is finitely generated, and strongly Howson if the rank of that intersection is bounded in terms of the two ranks. Zhang asked whether the two coincide for…

Baumslag asked whether a one-relator group G=F/⟨⟨ r⟩⟩ with r a commutator is Hopfian, residually finite or automatic. The paper constructs a family G_m=⟨ a,t | [t,a[a,t]^-m]⟩ answering all three negatively.

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.

Every polynomial map C^n → C^n with constant nonzero Jacobian determinant is invertible, with a polynomial inverse.