Lipschitz Maps and Compact Domains

This lecture introduces Lipschitz maps between metric spaces and compact domains with Lipschitz boundaries. The Lipschitz principle and its corollary are discussed, along with the proof of the principle. The lecture also covers the concept of changing variables to identify compact domains with Lipschitz boundaries. Applications to counting problems and fundamental domains for quotients are presented, emphasizing precompactness and Lipschitz boundaries. The lecture concludes with discussions on the probability of ideals belonging to a given class and their asymptotic behavior.