ICM 2026 Invited Section Lecture – Xavier Cabré

Value of the Information & Strength of the Argument

The value of the information is high: it presents breakthrough results in the regularity theory of stable solutions, including the long-standing problem of smoothness up to dimension 9, and introduces novel results for boundary reactions. The argumentation is rigorous and well-structured, moving from historical context to precise statements of theorems, with clear explanations of the main ideas and test functions. The speaker carefully distinguishes proven results from open problems, enhancing the credibility of the presentation.

Scientific Rigor, Source Quality, Title Accuracy

The scientific rigor is excellent: the talk cites classical works (Crandall-Rabinowitz, Casten-Holland, Matano) and recent developments (Chodosh-Li, etc.) accurately, and the speaker is transparent about joint works and open questions. The title exactly matches the content, as it is indeed an invited section lecture at ICM 2026. The sources are well-integrated into the narrative, though no explicit external links are provided in the video description.

The title accurately describes the video as an invited section lecture at ICM 2026 by Xavier Cabré, matching the content.

9/10

Rigorous mathematical presentation by a leading researcher, with a clear logical structure and reference to key literature. Minor transcript errors do not affect the substantively accurate content.

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

• 02:10Historical introduction: Hilbert's 20th and 19th problems, Dirichlet energy, and the work of Beppo Levi and the Riesz-Fréchet theorem.
• 08:00Definition and importance of stable solutions, relation to minimal surfaces, and discussion of the Simons cone and regularity results.
• 20:00Presentation of the main regularity theorem: stable solutions are smooth up to dimension 9, joint work with Figalli, Ros-Oton, and Serra.
• 30:00Explanation of Casten-Holland-Matano theorem and counterexamples for interior reactions in convex domains.
• 35:00Introduction of boundary reactions and recent results with Cónsul and Kurzke, including new Ginzburg-Landau theory and the half-Laplacian.
• 40:00Conclusion: open problems and future directions, including fractional Laplacian and higher-dimensional minimal surfaces.

• Stable solutions to semilinear elliptic equations are smooth up to dimension 9 — Main result presented in the lecture, joint work with Figalli, Ros-Oton, and Serra, mentioned around the 20-minute mark.
• On the nonexistence of stable solutions in convex domains — Classical theorem by Casten-Holland and Matano, discussed in relation to interior reactions around the 30-minute mark.
• Recent preprint on boundary reactions with Cónsul and Kurzke — Introduced near the end of the talk as new work on boundary reactions, yielding a Ginzburg-Landau theory and half-Laplacian results.
• Crandall and Rabinowitz (1975) on continuation and variational methods — Referenced as early work providing regularity results for stable solutions for specific nonlinearities.