OpenAI research collection · 2 manuscripts · Source checked 2026-10-19
These papers study whether a region in the extended complex plane can be reshaped, preserving local angles, so its missing pieces become round disks or points. The family claims the general existence result and a rigidity result under additional boundary conditions.
Simpler geometric representations can make complicated analytic questions easier to study. The rigidity claim has conditions beyond the existence theorem.
Resolves the existence part of Koebe's circle-domain conjecture: every domain in the Riemann sphere is conformally equivalent to a domain whose complementary components are round disks or points. It also proves that circle domains with conformally removable boundary are rigid, meaning every conformal equivalence to another circle domain is Möbius, establishing this direction of the He–Schramm conjecture.
Short excerpts below come from OpenAI's abstracts. The full abstract and original PDF preserve the paper's precise scope.
We prove that a circle domain with conformally removable boundary is conformally rigid, with no restriction on the number of complementary components. This establishes the removability-to-rigidity direction of the He–Schramm Conjecture.
This exact PDF is not listed in the imported formalization catalogue. Family scope notes may contain further information.
We prove that every domain in the Riemann sphere is conformally equivalent to a circle domain, resolving Koebe's circle-domain conjecture positively.
This exact PDF is not listed in the imported formalization catalogue. Family scope notes may contain further information.
New mathematical reasoning could help AI tackle harder scientific and computational problems. How reliably these skills transfer beyond mathematics will determine how much they contribute to general intelligence.
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