ICM 2026 Invited Section Lecture - Emanuel Milman

Multi-Bubble Isoperimetric Problems The classical isoperimetric inequality in Euclidean space Rn states that among all sets of prescribed volume, the Euclidean ball (uniquely) minimizes surface area, explaining why a single soap bubble is always a round sphere. One may similarly consider isoperimetric problems on more general metric-measure spaces, such as on the n-sphere Sn and on n-dimensional Gaussian space Gn (i.e. Rn endowed with the standard Gaussian measure). More generally, we consider the multi-bubble isoperimetric problem, in which one prescribes the volumes of k greater then= 2 bubbles and minimizes their total surface area -– as any mutual interface will only be counted once, the bubbles are thus incentivized to clump together. The double-bubble problem (case k=2) on R3 was resolved by Hutchings, Morgan, Ritoré and Ros in 2000. We survey recent advancements in the characterization of multi-bubble isoperimetric minimizers on Gn, Rn and Sn, as well as the stability of soap bubble partitions. Based on joint works with Joe Neeman and Botong Xu. There will be many pictures! Uploaded 05-10-26 Learn more about your ad choices. Visit megaphone.fm/adchoices