ICM 2026 Invited Section Lecture - Susanne Brenner

Value of the Information & Strength of the Argument

The value of the information is high: the proposed method is original and overcomes the challenge of enforcing convexity in finite element approximations for Monge-Ampère equations. The argumentation is logically structured and mathematically rigorous: the definition of the enhanced finite element space, the discrete constraints, and the error analysis are presented step-by-step. The stability lemma based on the Cordes condition and the use of enriching operators are appropriately invoked. Numerical results substantiate the theoretical predictions, showing second-order convergence and effective error estimation. The extension to other nonlinear PDEs demonstrates a broader applicability, though the treatment is concise and focuses on the main ideas.

Scientific Rigor, Source Quality, Title Accuracy

The scientific rigor is exceptionally high, with a careful derivation of all theoretical results. However, the talk lacks explicit citations of external references beyond a mention of two papers in Numerische Mathematik and Communications of the AMS, and a paper for the ICM. No URLs are provided in the description. The title accurately describes the content, and the presentation is well-organized, with a clear outline and concluding remarks.

The title accurately reflects the content: a specialized lecture on nonlinear least-squares finite element methods for Monge-Ampère equations, presented at ICM 2026.

9/10

The talk is a rigorous mathematical presentation by a leading expert, with a clear derivation of the method, error analysis, and numerical results. The approach is original and formally justified. No obvious weaknesses in scientific rigor. The only limitation is the lack of explicitly cited external references in the transcript.

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

• 01:00Introduction and outline of the talk
• 04:00Definition of Monge-Ampère equation and examples (Gaussian curvature, Minkowski problem, optimal transport)
• 12:00Description of the enhanced cubic Lagrange finite element space and degrees of freedom
• 18:00Formulation of the nonlinear least-squares discrete problem and constraint set
• 24:00Proof of element-wise convexity and a priori error analysis (order h²)
• 30:00A posteriori error estimator and numerical results for smooth solutions on square domain
• 34:00Numerical results for singular solution and adaptive mesh refinement
• 38:00Extension to smooth domains and isoparametric finite elements; numerical examples on elliptical domain
• 40:00Concluding remarks and possible extensions to other fully nonlinear PDEs

• Paper in Numerische Mathematik (unnamed in talk) — The author mentions two papers published in Numerische Mathematik and Communications of the AMS as the basis for the beginning part of the talk.
• Paper for the ICM (unnamed) — The author notes that details on uniqueness of the minimizer are in a paper prepared for the ICM.