Mathematics Colloquium

Currents mod p are a suitable generalization of classical chains mod p, i.e. of finite combinations of smooth submanifolds with coefficients in the cyclic group Zp. By the pioneering work of Federer and Fleming it is possible to minimize the area in this context and, for instance, represent mod p homology classes with area minimizers. For p > 2 typically (i.e. away from a small set of exceptional points) one would expect such minimizers to be a union of smooth minimal surfaces joining together ("in multiples of p's") at some common boundary. This is however surprisingly challenging to prove, especially for even p's, and up until recently only known for p = 3 and 4 in codimension 1. In this talk I will explain the difficulties and outline the outcome of a series of more recent works (some joint of the speaker with Hirsch, Marchese, Stuvard and Spolaor, some by Minter and Wickramasekera, and some joint of the speaker with Minter and Skorobogatova) which confirms this picture, with varying degrees of precision, for all p's, every dimension and codimension, and general ambient Riemannian manifolds.