Title: Zoom in and enhance: Dimension theory and finer geometry of metric spaces

Abstract:
Dimension theory and fractal geometry study the local structures of geometric objects such as sets and measures. This is achieved through various 'dimensions' that encapsulate scaling behaviour. In recent years, much progress was made on the family of Assouad-type dimensions. These dimensions are linked to fine local information of geometry and have found their application in metric geometry and random geometry. I will give a tour of the last decade of research in the area and their connections to embeddability and structural theorems for random and deterministic structures.