On the asymptotic behavior of isoperimetric planar partitions

What is the best way to enclose and separate N regions of volume 1 in the plane with the minimum amount of perimeter?

The solutions to this particular problem are called "minimizers planar N-clusters" and, so far, their structure is still mostly unknown except for the case N=1,2,3. In 2001 Thomas Hales proved the so-called '' hexagonal honeycomb conjecture '': the regular hexagonal tiling provides the partition of the plane in equal-area chambers having the minimum amount of localized perimeter. His result somehow provides an answer for the case N=infinity and it turns out to be a powerful instrument for the study of the asymptotic behavior (in N) of minimizing planar N-clusters. After a brief overview of the problem and of the main open questions I will discuss the consequence of Hales's result in the study of such objects. In the second part of the seminar I will give a sketch proof of a result obtained in collaboration with prof. Alberti about the asymptotic behavior of some classes of minimizing planar N-cluster from an energetic point of view.