Short··0:21·Number theory
Bogomolov–Pop Conjecture: OpenAI Claims a Proof
Watch on YouTubeAbout the problem →
About this video
The Bogomolov–Pop conjecture asks whether a function field of dimension at least two over an algebraically closed field can be rebuilt from a tiny shadow of itself: its multiplicative structure modulo a prime ℓ (Milnor K-theory K1/ℓ and K2/ℓ with their product), or the matching abelian-by-central pro-ℓ Galois group. OpenAI claims a proof: these data recover the field (its perfect closure and constants) up to one scalar and Frobenius, and in characteristic p the mod-p data recover the field itself. These are OpenAI's preprints, not yet peer reviewed.
Papers (OpenAI, catalogue result 009):
- Reconstruction of Function Fields from Mod-ℓ Milnor K-Theory: github.com/openai/math/blob/main/preprints/Reconstruction-of-…
- Reconstruction from Milnor K-theory modulo the characteristic: github.com/openai/math/blob/main/preprints/Reconstruction-fro…
- The Bogomolov-Pop reconstruction theorem: github.com/openai/math/blob/main/preprints/The-Bogomolov-Pop-…
The animation illustrates the result; it is not a proof. The spheres are projective lines of the functions x, y and x + y (then x times y), carrying their values at one moving point.
Gradient Damage: mathematics you can see, equations you can hear.
youtube.com/@gradientdamage
Every result explained: gradientdamage.com/p/009
About the problem
Number theory·OpenAI Math·Result 009
Function-field reconstruction from Milnor K-theory and Galois data
Reconstructs function fields of transcendence degree at least two over algebraically closed constants from KM1/ℓ, KM2/ℓ, and their product. These data recover the perfect closure and constants when ℓ differs from the characteristic, and the original field and its named base in equal characteristic. Also proves Bogomolov–Pop reconstruction from abelian-by-central pro-ℓ Galois data away from the characteristic. (Lean)
Topics
Bogomolov–Pop conjectureBogomolov-Pop reconstructionBogomolov programbirational anabelian geometryanabelian geometryfunction field reconstructionMilnor K-theoryK-theory mod lGalois grouppro-l Galois groupSteinberg relationprojective lineRiemann spherealgebraic geometrynumber theoryOpenAIOpenAI mathAI mathematicsAI proofmathsmathematicsmath shorts3D animationBogomolov-Pop conjecture
