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{\Large המחלקה למתמטיקה, בן-גוריון}

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\textbf{ב}\emph{יום שלישי, 10 בינואר, 2023}
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\textbf{בשעה} \emph{14:30 -- 15:30}
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\textbf{ב}\emph{Math -101}

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ההרצאה

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{\Large\bfseries Multi-Bubble Isoperimetric Problems - Old and New\par}
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תינתן על-ידי
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{\large\scshape Emanuel Milman 
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  (Technion)
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\textbf{תקציר:}
  The classical isoperimetric inequality in Euclidean space \$\textbackslash{}mathbb\{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 \$\textbackslash{}mathbb\{S\}\^{}n\$ and on \$n\$-dimensional Gaussian space \$\textbackslash{}mathbb\{G\}\^{}n\$ (i.e. \$\textbackslash{}mathbb\{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 \textbackslash{}geq 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, Ritor'e and Ros resolved the double-bubble conjecture in Euclidean space \$\textbackslash{}mathbb\{R\}\^{}3\$ (and this was subsequently resolved in \$\textbackslash{}mathbb\{R\}\^{}n\$ as well) --  the boundary of a minimizing double-bubble is given by three spherical caps meeting at \$120\^{}\textbackslash{}circ\$-degree angles. A more general conjecture of J.\ensuremath{\sim}Sullivan from the 1990`s asserts that when \$p \textbackslash{}leq n+1\$, the optimal multi-bubble in \$\textbackslash{}mathbb\{R\}\^{}n\$ (as well as in \$\textbackslash{}mathbb\{S\}\^{}n\$) is obtained by taking the Voronoi cells of \$p+1\$ equidistant points in \$\textbackslash{}mathbb\{S\}\^{}\{n\}\$ and applying appropriate stereographic projections to \$\textbackslash{}mathbb\{R\}\^{}n\$ (and backwards).

In 2018, together with Joe Neeman, we resolved the analogous multi-bubble conjecture for \$p \textbackslash{}leq n\$ bubbles in Gaussian space \$\textbackslash{}mathbb\{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, we describe our approach in that work, as well as recent progress on the multi-bubble problem on \$\textbackslash{}mathbb\{R\}\^{}n\$ and \$\textbackslash{}mathbb\{S\}\^{}n\$. In particular, we show that minimizing bubbles in \$\textbackslash{}mathbb\{R\}\^{}n\$ and \$\textbackslash{}mathbb\{S\}\^{}n\$ are always spherical when \$p \textbackslash{}leq n\$, and we resolve the latter conjectures when in addition \$p \textbackslash{}leq 5\$ (e.g. the triple-bubble conjectures when \$n\textbackslash{}geq 3\$ and the quadruple-bubble conjectures when \$n\textbackslash{}geq 4\$).
  


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