Problems
Everything in the archive: the problem as it was posed, what has been attempted against it, and who checked each attempt. The mark down the left of the list says who has looked — a person, a machine, or nobody yet. Human reviews and machine checks are counted separately and are never added together.
14 problems
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…
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 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.
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…
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…
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.
The paper exhibits an explicit integer polynomial in five variables, of total degree 14 with constant Hessian determinant 128, whose gradient is not injective. Its formal Legendre transform is therefore not a polynomial, so the Hessian…
For the integral I(h) over the unit interval and the torus, the paper gives the three-term Laurent polynomial f(x,z)=(1-z^-1)((1-x)+xz) with I(f^n)=0 but I(z^-1f^n)=(-1)^n-1/(n+1)≠0. This disproves the xz-conjecture with one interval and…
Connes' rigidity conjecture asks whether an ICC group with Kazhdan's property (T) is determined by its group von Neumann algebra. Disproved for this class: two explicit countable discrete groups Γ_1 and Γ_2, both ICC and property (T), are…