09:30-10:30 Monday, 31 March 2025 Auditorium, Rectorate
Abstract
Given a domain E in the plane, I consider a minimal N-partition of E, that is, a partition that minimizes the total perimeter among all partitions of E made of N cells with equal area. T.C. Hales proved in 2001 that if E is a flat 2-dimensional torus, then the only minimal N-partition is the regular hexagonal one (assuming that such partition exists).
But what happens for if E that does not admit a regular hexagonal partition?
We show that, as expected, cells look more and more hexagonal as N tends to infinity, and thus the next question concerns the “regularity” of such almost-hexagonal partition: is it rigid, in the sense that the orientation of the cells is (essentially) the same through the domain? I will present evidence that the answer is negative, and that the domain splits in large blocks (grains) where the orientation of the cells is essentially constant, separated by comparatively thin regions (grain boundaries) containing many non hexagonal cells. (The similarity with grain structures in the theory of dislocation in continuum mechanics is not accidental.)
Together with several people, including Marco Caroccia (Politecnico di Milano), Giacomo Del Nin (MPI Leipzig), Adriana Garroni and Emanuele Spadaro (Roma Sapienza), we tried for a long time to prove this statement in some form or another, but we only obtained some partial results. In this talk I will explain what we can do, and what remains to be done.