8 problems

Let $A_1$, $A_2$, and $A_3$ be positive real numbers. Consider planar graphs enclosing and separating finite areas $B_1$, $B_2$, and $B_3$ satisfying … A standard triple bubble is…

Let $A_1$, $A_2$, and $A_3$ be three finite areas. A regular triple bubble complex is a planar graph with four vertices, six edges, and three connected regions. Triple bubble conje…

For $n>2$, the $n$-bubble problem in $\mathbb{R}^2$ seeks a least-perimeter graph enclosing and separating $n$ prescribed areas. General $n$-bubble conjecture. The least perimeter…

A tiling is a cluster which, after lifting to the universal cover, partitions the cover into regions of finite volume, equivalently finite diameter. A minimizing double-bubble tili…

For two prescribed volumes in $\mathbb{R}^3$, consider surfaces enclosing the two volumes and minimizing total area. Double bubble conjecture. The surface of minimal area enclosing…

The setting is that of Theorem 1: a compactly supported and asymptotically mean convex sequence of sets in Euclidean space satisfying the theorem's finite-total-curvature and almos…

Let $N$ be a positive integer, let $\boldsymbol a\in\mathbb{R}^N_+$ be a vector of prescribed positive areas, and let $\mathcal M(\boldsymbol a)$ denote the set of perimeter-minimi…

Consider clusters of four bubbles containing prescribed amounts of gas, and minimize the total area of the interfaces between the bubbles. A four-dimensional simplex is a polytope…