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The classical isoperimetric inequality in Euclidean space R^n states that among all sets ("bubbles") of prescribed volume, the Euclidean ball minimizes surface area. One may similarly consider isoperimetric problems for more general metric-measure spaces, such as on the n-sphere S^n and on n-dimensional Gaussian space G^n (i.e. R^n endowed with the standard Gaussian measure). Furthermore, one may consider the "multi-bubble" isoperimetric problem, in which one prescribes the volume of p ≥ 2 bubbles (possibly disconnected) and minimizes their total surface area -- as any mutual interface will only be counted once, the bubbles are now incentivized to clump together. The classical case, referred to as the single-bubble isoperimetric problem, corresponds to p=1; the case p=2 is called the double-bubble problem, and so on.
In 2000, Hutchings, Morgan, Ritoré and Ros resolved the double-bubble conjecture in Euclidean space R^3 (and this was subsequently resolved in R^n as well) -- the boundary of a minimizing double-bubble is given by three spherical caps meeting at 120-degree angles. A more general conjecture of J. Sullivan from the 1990's asserts that when p ≤ n+1, the optimal multi-bubble in R^n (as well as in S^n) is obtained by taking the Voronoi cells of p+1 equidistant points in S^n and applying appropriate stereographic projections to R^n (and backwards).
In 2018, together with Joe Neeman, we resolved the analogous multi-bubble conjecture for p ≤ n bubbles in Gaussian space G^n -- the unique partition which minimizes the total Gaussian surface area is given by the Voronoi cells of (appropriately translated) p+1 equidistant points. In the talk, I will describe our approach in that work, as well as recent progress we have made on the multi-bubble problem on R^n and S^n. In particular, we show that minimizing bubbles in R^n and S^n are always spherical when p ≤ n, and we resolve the latter conjectures when in addition p ≤ 5 (e.g. the triple-bubble conjectures when n ≥ 3 and the quadruple-bubble conjectures when n ≥ 4).