Lean Proof: The Six-Sphere Has a Complex Structure
Formalization of the Solution to the Hopf Problem
A GitHub repository by Boris Alexeev formalizes the resolution of the Hopf problem in the Lean theorem prover, showing that the six-sphere admits a complex manifold structure compatible with its standard topology. The proof is based on a paper by Levent Alpöge, and the repository includes a comparator setup adapted from Google DeepMind's Formal Conjectures project. The code can be type-checked online.
The six-sphere admits a complex manifold structure compatible with its standard topology.
- nhatcher
Oh wow, I'm fairly impressed. I wouldn't have expected AI to solve a problem this hard just now.
There have been claims of complex structures on s6 (or absence of them) quite a few times over the last 15 years. Some by acclaimed mathematicians.
There was some discussions on HN a few days ago:
- GPerson
Surprisingly the (a, since multiple things have this name) Hopf conjecture was solved by human mathematicians around the same time. It’s the statement that S^2 x S^2 admits a positive sectional curvature Riemannian metric.
- VMG
`solution.lean` is a 12mb file
I looks like we are breezing past the point where unassisted humans can understand any of this