ICM 2026 Invited Section Lecture - Emanuel Milman

Value of the Information & Strength of the Argument

The lecture provides a clear, well-structured overview of recent breakthroughs in the multi-bubble isoperimetric problem, a central topic in geometric analysis. Milman effectively conveys the mathematical ideas without drowning in technicalities, using many pictures to illustrate concepts. The argumentation is solid: he justifies each result, contrasts with known classical cases, and honestly states limitations (e.g., the hyperbolic case is unresolved, the quadruple bubble in dimension 3 is open, uniqueness is lost for the quintuple in R^n). The progression from classical results to state-of-the-art improvements is logical, and the stability results offer a complementary perspective. The presentation is authoritative and rigorous, with references to joint works and contributions by other researchers.

Scientific Rigor, Source Quality, Title Accuracy

The lecture is scientifically rigorous: it cites classical and recent literature accurately (e.g., Hutchings-Morgan-Ritoré-Ros for double bubble, Sullivan’s conjecture, Brascamp-Lieb inequalities). The mathematical statements are precise, with appropriate technical caveats (e.g., reduced boundary). The title is perfectly matched to the content, as it indeed is an invited section lecture on multi-bubble isoperimetric problems. No comments are provided, so no audience trends can be analyzed.

The title accurately describes the content: an invited lecture on multi-bubble isoperimetric problems at ICM 2026.

9/10

The lecture is by a leading expert in geometric analysis, presenting recent results from joint works published in high-level venues. The methods are rigorous, based on advanced mathematics, and the presentation is precise.

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

• 00:40Introduction and motivation with soap bubble photographs.
• 02:10Review of classical isoperimetric inequality in Euclidean space and its history.
• 04:00Generalization to weighted Riemannian manifolds and definition of weighted surface area.
• 06:00Model spaces: constant curvature spaces and Gaussian space; known isoperimetric minimizers.
• 08:00Setting up the multi-bubble isoperimetric problem: partitions, interfaces, and total surface area.
• 10:00Sullivan's conjecture and construction via Voronoi cells on spheres via stereographic projection.
• 12:00Gaussian conjecture for standard Gaussian bubbles with flat interfaces.
• 15:00Known results: double bubble in R^3 (Hutchings et al.), extensions to higher dimensions, and partial progress in Gaussian space.
• 18:00New results: full resolution of Gaussian multi-bubble conjecture; results for Euclidean and spherical spaces for double, triple, quadruple, quintuple bubbles.
• 25:00Stability results: standard and Möbius flat partitions are stable; Gaussian stable partitions characterized by flat interfaces with 120° angles.

• The classical isoperimetric inequality (Steiner, Schwarz) — Mentioned as known to ancients, formal proofs by Steiner and Schwarz in the 19th century.
• Isoperimetric inequality on model spaces (Polyaev, Schmidt, De Giorgi) — Stated that geodesic balls are minimizers in spaces of constant curvature.
• Gaussian isoperimetric inequality (Sudakov-Tsirelson, Borel, Ehrhard, Carlin-Curzon) — Half-spaces minimize Gaussian surface area.
• Sullivan's multi-bubble conjecture — Conjectured standard K-bubbles minimize surface area for K ≤ n+1 in Euclidean space.
• Double bubble theorem in R^3 (Hutchings, Morgan, Ritoré, Ros, 2000) — Resolved double bubble conjecture in R^3 for general volumes.
• Works by Milman, Neeman, and Xu on multi-bubble isoperimetric and stability — Recent joint works presenting the results described in the talk.

• Millman and Neeman, Gaussian multi-bubble conjecture — The talk reports a resolution of this conjecture, which is not listed here but is claimed in the lecture.
• Isoperimetric problems on constant curvature spaces — Known results from literature supporting the context.

• None provided — No conflicting sources are discussed.