The Jacobian Conjecture (two variables)

Let \(F: \mathbb{C}^2 \to \mathbb{C}^2\) be a polynomial map whose Jacobian determinant is a nonzero constant. Must \(F\) be invertible with polynomial inverse?

Keller's original 1939 question. Decades of structure theorems: injectivity criteria (Magni? no — Markus-Yamabe dynamics aside), Jung's 1942 theorem giving the plane case a canonical form, and the Abhyankar–Moh epimorphism theorem (1975) constraining polynomial embeddings enormously. Then catastrophe in the family: Alpöge's July 2026 three-variable counterexample (with substantial AI assistance) showed the phenomenon is real in higher dimension — sharpening, not settling, the plane case, where every known technique says TRUE yet no proof closes. The club watches this one closely.