Problems

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…

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…

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 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…

Bajpai, Dona and Nitsche left three degree-six symplectic hypergeometric monodromy groups unclassified as arithmetic or thin. Two of the three, C-47 and C-55, are arithmetic.

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…