Breakthroughs Weekly — 27 August 2026
A 1939 conjecture on polynomial maps falls above two dimensions, and reactors show a faint antineutrino afterglow long after shutdown.
A 1939 conjecture on polynomial maps falls in every dimension above two
arXiv · 31 Jul 2026 · Algebraic geometry
Mathematicians have disproved the Jacobian conjecture, the 87-year-old question of whether every polynomial self-map of complex space with constant nonzero Jacobian determinant must have a polynomial inverse, in every dimension above two. The refutation began on 19 July when Levent Alpöge posted a three-dimensional counterexample found with the help of an AI system, and a geometric explanation of why it works followed within days from other mathematicians, including Terence Tao, who wrote up his own account of the mechanism on his blog. This preprint, by Shuhong Gao, gives a rigorous, self-contained account of the underlying tangent-sweep construction and extends it to five new explicit counterexamples spanning three, four and five dimensions, with every algebraic identity checked in exact rational arithmetic. The original two-dimensional form of the conjecture remains open.
First direct detection of a shutdown reactor’s antineutrino afterglow
Max Planck Institute for Nuclear Physics · 4 Aug 2026 · Particle physics
The Double Chooz collaboration has measured antineutrinos still emitted by a nuclear reactor’s spent fuel and residual fission products months after shutdown, the first direct experimental confirmation of a long-standing theoretical prediction. Over 17.2 days of observation at the Chooz plant in France, the detector recorded roughly 100 candidate antineutrino events whose rate matched simulations of long-lived fission-product decay. Because the signal persists independent of active fission, the result was published in Physical Review Letters and points to a way of remotely checking whether a shut-down reactor’s fuel inventory matches what has been declared, relevant to nuclear-safeguards monitoring.