12 problems

Small-volume triple-bubble conjecture. The minimizers will look like the double bubbles of Figure 2 with a small ball attached, and the phase diagram will look just like Figure 3.

Complete-classification conjecture. The double bubbles of Figure 1 together with the Hexagonal Honeycomb of Figure 4 comprise the complete set of area-minimizing double bubbles for…

No periodic three-volume foam conjecture. There are no least-area divisions of $T^3$ into three volumes that lift to a foam in $\mathbb{R}^3$.

Hexagonal Honeycomb conjecture. For these volumes, the Hexagonal Honeycomb prism of Figure 4 is a perimeter-minimizing double bubble.

Very-long-torus Double Slab conjecture. In the special case of a very long $T^3$, the Double Slab is optimal for most volumes.

Concavity conjecture. The least area to enclose and separate two given volumes in the three-torus is a concave function of the volumes.

Delaunay-chain transition conjecture. The first phase transition as small equal, or close to equal, volumes grow is from the standard double bubble to a chain of two bubbles bounde…

Slab Lens conjecture. For one very small volume and two moderate volumes, the Slab Lens is optimal.

Small-volume standard double-bubble conjecture. For two small volumes the standard double bubble is optimal.

Connectedness conjecture. An area-minimizing double bubble in $T^3$ has connected regions and complement.

Ritoré–Ros conjecture. The optima for the isoperimetric problem in a cubic $T^3$ are the sphere, cylinder, and slab.

Central conjecture. The ten different types of two-volume enclosures pictured in Figure 1 comprise the complete set of surface-area-minimizing double bubbles in the flat cubic thre…