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.

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.

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.

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?

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.