Problems

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.

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…

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 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…

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…

If a subgroup of a product of groups of type F_k virtually surjects onto every k-tuple of factors, must it be of type F_k itself? Yes, for discrete groups, and likewise for FP_k. The homological n-(n+1)-(n+2) Conjecture follows for…

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…

Let L(x^ay^b)=a! b! on C[x,y]. The Factorial Conjecture asks whether L(f^m)=0 for every m≥ 1 forces f=0. The homogeneous two-variable case was settled by Liu and Sun; the inhomogeneous problem does not reduce to it, because radial…