Intersections and embeddings of self-similar sets

A few years ago Shmerkin and Wu independently proved a 50-year old conjecture by Furstenberg about the dimension of intersection of certain self-similar ("Cantor-like") sets in the real line. Their methods, however, apply only to the homogeneous case, and the general case remains open. I will discuss the connection of this problem to a related embedding problem (due to Feng, Wang and Rao), and will outline recent work with Algom and Wu in which we make some further progress on the problem. The talk will not require prior knowledge in the area.