Geometry and Analysis of Metric Spaces via Weighted Partitions

The aim of these lecture notes is to propose a systematic framework for geometry and analysis on metric spaces. The central notion is a partition (an iterated decomposition) of a compact metric space. Via a partition, a compact metric space is associated with an infinite graph whose boundary is the original space. Metrics and measures on the space are then studied from an integrated point of view as weights of the partition. In the course of the text: These notes should interest researchers and PhD students working in conformal geometry, analysis on metric spaces, and related areas.