ICM 2026 Invited Section Lecture - Susanna Zimmermann

Value of the Information & Strength of the Argument

The talk provides substantial value by clarifying the state of the art in classifying algebraic subgroups of birational maps. The speaker builds intuition through concrete examples, then progresses to deep results, effectively explaining the motivation behind inclusions and maximality. The argumentation is rigorous: definitions are precise, and the main theorem is supported by a construction that leverages known results (Weil’s regularization, conic bundle theory) to create a contradiction to maximality. The logical flow is clear, and the proof idea is conveyed geometrically, making the advanced material accessible to an expert audience.

Scientific Rigor, Source Quality, Title Accuracy

The lecture demonstrates high scientific rigor: definitions and theorems are stated accurately, and historical attributions (Enriques, Weil, Fong, Umeimura, etc.) are appropriate. The quality of sources is strong, as it builds on established literature in birational geometry. However, no explicit URLs or paper references are provided in the video description, so external verification would require consulting the cited mathematicians’ works. The title exactly matches the content: it is an invited lecture on algebraic groups of birational maps, and the presentation stays focused on that topic throughout.

The title accurately describes the content: an ICM invited lecture on algebraic groups acting via birational maps.

8/10

The talk is a mathematically rigorous presentation of research results in algebraic geometry, with clear definitions, examples, and historical context, delivered by an expert in the field.

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

• 00:30Introduction and definition of birational maps
• 02:30Examples: Cremona map and involution at circle
• 05:00Definition of algebraic groups and rational actions
• 08:00Example of PGL2*PGL2 action on P2 and regularization
• 13:00Infinite chains and inclusion problem
• 18:00Classification history: Enriques, Fong, Umeimura
• 24:00Main result: algebraic subgroup not contained in maximal one for dimension ≥5
• 30:00Open questions and conclusion

• Weil's theorem on regularization of algebraic group actions — Classical result ensuring a birational model where the action becomes regular.
• Enriques classification of birational self-maps on P2 — First known classification of connected algebraic subgroups of the Cremona group in dimension 2.
• Fong's classification for surfaces — Fong proved that maximality holds only for rational surfaces for the property of being contained in a maximal subgroup.
• Umeimura's classification for P3 — Classification of connected algebraic subgroups acting regularly on P3 up to inclusion and conjugation.