ICM 2026 Invited Section Lecture - Joaquín Pérez

ICM 2026 Invited Section Lecture - Joaquín Pérez

by Joaquín Pérez · View the channel on PSI

Formal & Physical Sciences Mathematics PBMGeometryPBMPDifferential and Riemannian geometry
🎙 Joaquín Pérez 👥 60K 📅 September 21, 2026 ⏱ 45 min 👁 2 📄 expert opinion 🧭 2026-09-21
Available in: English (current) Français

Keywords

minimal surfacesRiemann exampleslaminationJacobi functionCalabi-Yau

Summary

The talk presents a comprehensive program, developed jointly with W.H. Meeks and A. Ros, to classify complete, properly embedded minimal surfaces of finite genus in Euclidean 3-space. The central result is that genus zero surfaces with infinite topology must be Riemann minimal examples. The classification proceeds in four steps: the singly periodic case, the quasi-periodicity of two-limit-ended surfaces, the non-existence of one-limit-ended examples, and the final classification. The speaker uses flux as a classifying tool, introduces Jacobi functions, and connects with the modified KdV equation to prove linearity of the Shiffman function. He also discusses minimal laminations, local removable singularity theorems, and rescaling methods, highlighting applications to the Hoffman–Meeks conjecture and the embedded Calabi-Yau problem. Throughout, the interaction between topology, flux, curvature estimates, and moduli space structure is emphasized.

129 words

Critical Evaluation

Value of the Information & Strength of the Argument

The value lies in offering a clear, expert-level exposition of a deep classification theorem, supported by rigorous arguments and references to prior work. The logical structure is impeccable: the four-step plan is well-motivated and each step is explained with sufficient detail. The speaker also connects abstract concepts to concrete examples, enhancing comprehension. The argumentation is solid, relying on established theorems and advanced techniques, making it a valuable resource for researchers in minimal surface theory.

Scientific Rigor, Source Quality, Title Accuracy

The scientific rigor is high: the talk is based on peer-reviewed research and presents original results. The speaker explicitly credits collaborators and prior results. The title is generic but accurately reflects the nature of an invited lecture; the description provides the precise topic. The content adheres to the theme stated in the description. No external sources are cited in the transcript, but the lecture itself stands as a reliable contribution.

159 words

Title / Content Match

Title is generic but accurately indicates an ICM lecture by Joaquín Pérez; the content matches the description and the speaker's expertise.

Quality & Reliability

9/10

Lecture from an ICM invited speaker presenting peer-reviewed results in minimal surface theory. Content is rigorous and well-structured.

Key Moments

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

Contribution & Novelties

The talk presents a complete classification of genus zero properly embedded minimal surfaces with infinite topology, a major advancement in minimal surface theory. The method integrates flux, lamination techniques, and dynamical systems, offering a novel framework. The connection to the modified KdV equation to prove linearity of the Shiffman function is particularly innovative.

Pour aller plus loin :

  • Catenoid — Classical minimal surface with finite topology.
  • Helicoid — Another classical example, limit of Riemann examples.
  • Riemann minimal surface — Specific examples with infinite ends; no verified URL available, but mentioned in the talk.

93 words

Radar Profile

The radar profile shows a very high technical level (10) and strong quantity of information (9), indicating a dense expert talk. Quality and reliability scores are also high, suggesting a trustworthy and informative presentation, though the accessibility for general audiences is low due to the advanced mathematics.

Reliability 9/10