In this talk I will try to prove De Giorgi’s Theorem on the regularity of minimal Caccioppoli Sets(Theorem 8.4). He used a measure theoretic method closely related to properties of function of bounded variation, which is technical and powerful. I will focus on Chapter 5-8 in E. Giusti’s book. It is based on M. Miranda’s simplification of De Giorgi’s original proof.

This will be a continuation of the talk given on September 9th.

Given two actions of a countable, discrete group $G$ on probabilty space $X,Y$ there is a notion of when the action on $X$ is weakly contained in the action on $Y$ (analogous to weak containment of representations) due to Kechris: it roughly says that any finitary piece of the action of $G$ on $X$ can be approximated by some finitary piece of $G$ on $Y$ (equivalent the measure on $X$ is a weak* limit of the factors of the measure on $Y$). We then say that two actions are weakly equivalent when each is weakly contained in the other. We study when algebraic actions of $G$ (i.e. an action by automorphisms on a compact, metrizable, abelian group) are weakly equivalent to Bernoulli shifts and find a natural class of actions related to invertible convolution operators on $G$. As part of our work, we also give conditions under which such actions are free.