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Abstraction: Generic regularity of minimizing surfaces proved through dimension 11

Key points:

Plateau problem: for any closed curve (wire frame) in 3D space, a minimizing area surface always exists — proved by Douglas and Radó in 1930s; Douglas won first Fields Medal
Minimizing surfaces are smooth in dimensions 4-7; in dimension 8+ singularities (fold/pinch points) can occur — first example constructed by Jim Simons in 1968
"Generic regularity" means singularities can be wiggled away by perturbing the wire frame; proved for dimension 8 by Hardt-Simon in 1985; no progress for ~40 years after that
In 2023, Chodosh, Mantoulidis, and Schulze proved generic regularity for dimensions 9 and 10 using a new "separation function" technique
In 2025 (with Zhihan Wang), the team extended the proof to dimension 11; dimension 11 required handling a harder class of 3D singularities
Result enables extensions of geometry/topology conjectures and Schoen-Yau positive mass theorem from dimension 8 up to dimension 11

Connections: Minimal Surfaces · Differential Geometry · Mathematics