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CNA Seminar/Colloquium/Joint Pitt-CNA Colloquium
Marco Caroccia
Carnegie Melllon University
Title: On the asymptotic behavior of isoperimetric planar partitions

Abstract: 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.

Recording: http://mm.math.cmu.edu/recordings/cna/marco_caroccia_small.mp4
Date: Tuesday, October 27, 2015
Time: 1:30 pm
Location: Wean Hall 7218
Submitted by:  Gautam Iyer