Abstract: 

In this talk, we shall discuss recent efforts in finding a characterization of regularity of a set by the Green function. Our goal is to obtain a characterization with minimal background assumptions on the domain and the operator. We show that for an optimal class of elliptic operators with non-smooth coefficients on a domain that provides some access to its boundary, the boundary of the domain is uniformly rectifiable if and only if the Green function behaves like a distance function to the boundary. This talk is based on joint work with Joseph Feneuil and Svitlana Mayboroda.