Pure Math
A 76-year-old mathematics conjecture may have a proof, built from geometric cycles that turn invisible cohomology classes into algebraic objects.
The Hodge conjecture asks whether a particular kind of topological signal on a smooth complex projective variety always comes from algebraic geometry. Deep Bhattacharjee and Ushashi Bhattacharya claim that the answer is yes. Their version 5 preprint introduces relative secant cycles on abelian varieties, moves those cycles through a Fourier-Mukai transform, and then routes the general problem through five cases. The claim reaches one of mathematics's most famous open problems, and the decisive work now belongs to specialists checking every construction and reduction. If the proof survives that process, an abstract shadow in cohomology will finally have an algebraic source.
Does the relative secant construction really close every case of the Hodge conjecture? Follow the five-part argument and the exact points that specialists must now test: