ICM 2026 Plenary Lecture - Patrick Gérard

ICM 2026 Plenary Lecture - Patrick Gérard

by Patrick Gérard

Formal & Physical Sciences Mathematics PBMathematics
🎙 Patrick Gérard 👥 58K 📅 August 17, 2026 ⏱ 52 min 👁 107 📄 original study 🧭 2026-08-18
Available in: English (current) Français

Keywords

Hardy spacesIntegrable PDEsBenjamin-OnoSoliton resolutionLax pairs

Summary

Patrick Gérard’s ICM 2026 plenary lecture presents a new method for obtaining explicit formulas for a class of integrable partial differential equations (PDEs) with nonlocal terms, using Hardy spaces of holomorphic functions. The talk focuses on the Benjamin-Ono equation, a model for internal gravity waves in deep water. Gérard introduces a non-self-adjoint operator X* on the Hardy space of the upper half-plane, which plays a crucial role in the explicit representation of solutions. He derives a deformed Cauchy formula involving the Lax operator, leading to a proof of the soliton resolution conjecture for this equation. The theorem states that any sufficiently decaying solution asymptotically decomposes into a finite sum of solitons plus a radiating linear wave. The proof relies on the Lax pair structure and the evolution of the associated operators. The lecture is dedicated to the memory of Peter Lax and Claude Bardos.

144 words

Critical Evaluation

Value of the Information & Strength of the Argument

The lecture presents a significant original contribution: a novel explicit formula for solutions of the Benjamin-Ono equation, based on a careful analysis of Hardy spaces and a specially defined operator X*. The argumentation is rigorous and well-structured, moving from basic definitions to the main theorem and its proof sketch. The value lies in providing a new tool for tackling nonlocal integrable PDEs, potentially applicable to other equations. The proof is elegant, connecting harmonic analysis and integrable systems.

Scientific Rigor, Source Quality, Title Accuracy

The scientific rigor is high, typical of a plenary lecture at the International Congress of Mathematicians. The speaker is a leading expert, and the results are presumably peer-reviewed. The talk references key works by Peter Lax, Thomas Brooke Benjamin, and others, and mentions that full references are in the proceedings. The title accurately reflects the content. No comments were provided for analysis.

154 words

Title / Content Match

The title accurately reflects the content, focusing on Hardy spaces and explicit formulas for integrable PDEs.

Quality & Reliability

9/10

Lecture by a leading expert, presenting original research with rigorous mathematical proofs, published in conference proceedings.

Key Moments

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

Cited Sources

  • Proceedings of the ICM 2026 — Mentioned as containing full references for the talk.

Concurring Sources

Contribution & Novelties

The lecture presents a novel explicit formula for solutions of the Benjamin-Ono equation, based on a new operator X* on Hardy spaces. This provides a new approach to proving the soliton resolution conjecture for nonlocal integrable PDEs, which had resisted previous methods. The method may be applicable to other integrable systems.

Pour aller plus loin :

77 words

Radar Profile

The radar profile shows very high scores in all dimensions, indicating a technically deep, highly informative, and reliable lecture. The balance between quantity and quality of information is excellent, with a strong emphasis on rigorous mathematical exposition.

Reliability 9/10