**abstract:**
By minimal N-partitions I mean partitions of a given bounded 2-dimensional domain E which consist of N sets (cells) with equal area, and which minimize the length of the union of the boundaries of these sets.
Similarly, minimal N-cluster are families of N pairwise disjoint sets with equal prescribed areas, which again minimize the length of the union of the boundaries.
Among other results, T. C. Hales proved in 2001 that if E is a flat 2-dimensional torus then the regular hexagonal partition (when it exists) is the only minimal N-partition.
Apart from this result, not much is known about the structure of N-partitions when N is large. In particular when E is a planar domain we expect that minimizing N-partitions should look hexagonal at least in some asymptotic sense.
Similar issues arise in the study of the asymptotic shape of minimal N-clusters as N tends to infinity.
In this talk I will describe a few results obtained so far and future directions. This is a work, still in progress, with Marco Caroccia (University of Lisbon) and Giacomo Del Nin (University of Pisa).

Fri 18 May, 15:40 - 16:30, Aula Dini

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