ICM 2026 Plenary Lecture - Felix Otto

Value of the Information & Strength of the Argument

The lecture offers substantial value by presenting a unified geometric perspective on two distinct areas of PDE theory. The first part provides a clear and insightful connection between Arnold’s geometric mechanics and optimal transport, demonstrating how gradient flow structures can yield quantitative results for mixing in multiphase flows. The argumentation is rigorous, building from classical results to new developments, and the explicit solution for the mixing zone is a strong illustration. The second part is more advanced and speculative, but it presents a compelling framework for renormalization in singular SPDEs. Otto’s argumentation is logical and well-structured, though it assumes a high level of mathematical maturity. The lecture does not oversimplify; it acknowledges open questions and the need for further development, which enhances its credibility.

Scientific Rigor, Source Quality, Title Accuracy

The lecture demonstrates high scientific rigor, with precise mathematical statements and references to established works (Arnold, Brenier, Otto, Jordan-Kinderlehrer-Otto, Hairer, etc.). The sources are not explicitly cited with URLs, but the speaker’s authority and the context of an ICM plenary lecture ensure reliability. The title accurately reflects the content, focusing on geometric concepts in PDEs. The lecture is well-organized and the mathematical exposition is careful, though the second part is more technical and may require prior knowledge of regularity structures. No external sources are provided in the description, so the evaluation relies on the speaker’s reputation and the internal consistency of the presentation.

The title accurately reflects the content: the lecture focuses on geometric concepts (Riemannian geometry, gradient flows, manifold constructions) applied to partial differential equations.

8/10

Lecture by a leading mathematician, presenting original research and known results, with rigorous mathematical reasoning. The content is highly technical and assumes expert audience. No external sources cited within the talk, but the speaker references established works (Arnold, Brenier, Otto, Jordan-Kinderlehrer-Otto, Hairer, etc.). The lecture is a plenary at ICM, indicating high scientific standing.

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

• 00:00Introduction by Irene Fonseca, presenting Felix Otto and the topic.
• 02:00Otto begins his lecture, outlining the two vignettes: overdamped fluid dynamics and renormalization in SPDEs.
• 05:00First vignette: Arnold's geometric interpretation of Euler equations, geodesics, and sectional curvature.
• 10:00Transition to overdamped dynamics: Darcy flow, mobility, and gradient flow structure.
• 15:00Variational time discretization, relaxation, and the emergence of Wasserstein metric.
• 20:00Explicit solution for mixing zone in stratified initial data, connection to non-convexity.
• 25:00Second vignette: nonlinear PDEs with rough noise, need for renormalization.
• 30:00Geometric approach: solution manifold, charts, transition maps, and analytic parameterization.
• 35:00Use of Malliavin derivatives and spectral gap inequalities, connection to stochastic homogenization.
• 40:00Conclusion and outlook, emphasizing the potential of the geometric framework.

• Arnold's geometric interpretation of fluid dynamics — Mentioned as the basis for the first vignette.
• Brenier's relaxation of the action functional — Cited as inspiration for the variational approach.
• Jordan-Kinderlehrer-Otto (JKO) scheme — Mentioned as the origin of Wasserstein gradient flows.
• Hairer's theory of regularity structures — Basis for the second vignette's renormalization approach.
• Gloria and Otto's work on quantitative stochastic homogenization — Mentioned as inspiration for using Malliavin derivatives.

• Arnold, V. I. (1966). Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits. — Foundational work on geometric mechanics.
• Brenier, Y. (1989). The least action principle and the related concept of generalized flows for incompressible perfect fluids. — Relaxation of the action functional.
• Jordan, R., Kinderlehrer, D., & Otto, F. (1998). The variational formulation of the Fokker-Planck equation. — Introduction of Wasserstein gradient flows.
• Hairer, M. (2014). A theory of regularity structures. — Foundational work on renormalization of singular SPDEs.