André Weil Doubted the Hodge Conjecture—and Almost Built a Counterexample
André Weil and the Hodge Conjecture
André Weil was among the first to doubt the Hodge conjecture, and he came closer than most to constructing a counterexample. After Mumford found exceptional Hodge classes in a CM abelian fourfold, Weil generalized the example into a 4-dimensional family and identified the Weil classes, which arise from symmetry without any known algebraic cycle. He then tried to show that no such cycle exists. His failure foreshadowed fifty years of stalled progress, and his doubts may now be vindicated by OpenAI's possible counterexample.
As you and Mumford seem to believe Hodge's conjecture, it is now up to you to exhibit algebraic cycles corresponding to these abnormal classes. As I incline to disbelieve it, I shall rather attempt to show that there is no such cycle.
- bee_rider
I wonder if we’ll end up in a weird situation a decade from now, where every conjecture will collapse into one of two states: counterexample found by AI, or assumed to be true.
- glimshe
How does AI's mathematical abilities affect the commercial future of software like Mathematica?
- QuesnayJr
I wonder if this is a hint to a more-specific rumor. Weil constructed what are now known as "abelian varieties of Weil type". In low dimensions the Hodge conjecture has been proven for abelian varieties of Weil type, but it's open in higher dimensions. Maybe that's where they found their counterexample.