Speaker: Mircea Petrache, PUC
Title: Building examples of graphs that allow infinitely many sharp isoperimetric shapes
Place: Pontificia Universidad Católica, Facultad de Matemáticas, Campus San Joaquin, Sala 1
Discrete isoperimetric shapes are configurations that reach equality in the edge-isoperimetric inequality among subsets of a fixed graph. Equivalently, the question is to find the shape of the best crystal grain, given the crystal structure of a material. Equality in the discrete edge-isoperimetric inequality is hard to achieve, and the ambient graphs which have infinite families of discrete isoperimetric shapes are rare: Our goal is to build a large class of examples. We start from the “macroscopic” or “continuum” isoperimetric problem, with two approaches, one via PDE and one via Optimal Transport. We build a new discrete strategy which combines the two approaches. Our strategy poses several nice new challenges, and it highlights the close link between semidiscrete optimal transport and convexity. In this introductory talk, I describe what new classes of examples we find, and also some mysterious directions still to be explored.