ICM 2026 - Fields Medal Award Lecture - Yu Deng

Value of the Information & Strength of the Argument

The lecture provides a comprehensive overview of a fundamental problem in mathematical physics, with clear explanations of the physical intuition and mathematical challenges. The argumentation is rigorous, presenting the logical progression from microscopic dynamics to macroscopic equations. The speaker emphasizes the limitations of previous perturbative results and justifies the need for non-perturbative methods. The presentation is well-structured, making complex ideas accessible while maintaining mathematical precision.

Scientific Rigor, Source Quality, Title Accuracy

The lecture demonstrates high scientific rigor, with careful attention to definitions and assumptions. The speaker cites key historical works (Lanford, Grad, etc.) and recent contributions, providing a solid foundation. The title accurately reflects the content, focusing on Hilbert’s Sixth Problem and the speaker’s contributions. The presentation is suitable for an expert audience, and the speaker appropriately acknowledges open questions and limitations.

The title accurately reflects the content: a Fields Medal lecture on Hilbert's Sixth Problem, focusing on kinetic limits for particles and waves.

9/10

Lecture by a Fields Medalist presenting rigorous mathematical results with clear methodology and references to prior work. High technical accuracy and appropriate caveats.

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

• 00:00Introduction and overview of Hilbert's Sixth Problem
• 02:00Discussion of the Grad program and scaling laws
• 05:00Explanation of the Boltzmann equation and molecular chaos
• 10:00Historical review of kinetic theory and Lanford's theorem
• 15:00Main theorem: derivation of Boltzmann equation for long times
• 20:00Introduction to wave kinetic theory and wave turbulence
• 25:00Applications of wave turbulence in oceanography and plasma physics
• 30:00Mathematical challenges in deriving wave kinetic equations
• 35:00Recent results and open problems
• 40:00Conclusion and acknowledgments

• Hilbert's Sixth Problem — Central problem discussed in the lecture.
• Lanford's theorem — First rigorous derivation of Boltzmann equation for short times.
• Grad program — Framework for deriving fluid equations from kinetic theory.
• BBGKY hierarchy — Hierarchy of equations for density functions.
• Wave kinetic equation — Analogue of Boltzmann equation for waves.

• Lanford's theorem — Supports the short-time derivation of Boltzmann equation.
• Grad program — Framework consistent with the lecture's approach.