ICM 2026 Invited Section Lecture - Otis Chodosh

Value of the Information & Strength of the Argument

The value of the lecture lies in its clear demonstration of how minimal surface techniques can be employed to prove deep theorems about curvature, providing a unified perspective on several recent advances. The argumentation is coherent and progressive: starting from simple examples, Chodosh builds up to more complex situations, carefully distinguishing proven results from conjectures. He also points out open problems and the limitations of current methods. The presentation is rigorous, with references to specific theorems and authors, and the reasoning is logical, though some steps are abbreviated for time constraints. The speaker’s own contributions (generic regularity, improvements on Rosenberg conjecture, stable Bernstein problem) are clearly highlighted, making the talk valuable for both understanding known results and identifying future research directions.

Scientific Rigor, Source Quality, Title Accuracy

The talk demonstrates high scientific rigor: Chodosh cites numerous specific results by Schoen-Yau, Hardt-Simon, Brendle-Wang, Rosenberg, Rode, Shu, and others, indicating a solid grounding in the literature. He also carefully notes the provenance of ideas (e.g., the credit for the aspherical conjecture) and often indicates when a result is folklore versus newly proven. The title is generic but accurate, conveying that it is a lecture at ICM 2026 by a named speaker; the actual content is specified in the description. There is no discordance between title and content, but a more descriptive title would have been clearer. The talk targets a specialist audience, but the technical level is appropriately high for an ICM invited lecture.

The title accurately indicates the format (ICM invited lecture) and the speaker, but does not specify the scientific topic, which is provided in the description.

8/10

The lecture is given by an expert at the ICM, presenting well-established results and recent contributions. The talk references published work and is logically structured, though it is not a peer-reviewed publication itself.

Markers derived by PSI from the transcript: the creator did not define chapters.

• 01:00Introduction of minimal surfaces and comparison geometry; definition of scalar curvature.
• 02:30Presentation of the Schoen-Yau result: no positive scalar curvature on the three-torus; outline of the area-minimization proof.
• 07:30Discussion of the positive mass theorem and its relation to scalar curvature; mention of recent progress.
• 11:40Introduction to singularities of area minimizers, the Simons cone, and Hardt-Simon smoothing.
• 16:40Generic regularity results for area minimizers up to dimension 11; explanation of the "whack-a-mole" difficulty.
• 21:40Rosenberg conjecture on product with S1; recent results by Rode, Shu, and the speaker's improvement to dimension 10.
• 26:40Stable Bernstein problem: history, recent progress in dimensions up to 6, and counterexamples in higher dimensions.
• 31:40Open questions and concluding remarks on future directions.

• Schoen and Yau, 'On the proof of the positive mass conjecture in general relativity' (1979) — Cited as the source of the result that a stable minimal surface in positive scalar curvature must be a sphere, used to prove the three-torus admits no positive scalar curvature.
• Hardt and Simon, 'Area minimizing hypersurfaces with isolated singularities' (1985) — Their work on smoothing singularities of area minimizers in eight-dimensional ambient space is mentioned.
• Brendle and Wang, recent work on excision of singular sets — Mentioned as a recent breakthrough in handling singularities via excision.
• Rosenberg conjecture (2007) — Discussed in the context of products with S1 and positive scalar curvature.
• Rode and Shu, on Rosenberg conjecture — Trimmed results: Rode proved up to dimension 7, Shu proved for zero Euler characteristic in all dimensions.
• Chodosh and collaborators, stable Bernstein problem in dimensions 4,5,6 — Mentioned as their own recent contributions.