Explore a 44-minute lecture on finite approximations as a tool for studying triangulated categories, presented by Amnon Neeman for the International Mathematical Union. Delve into the concept of metrics on categories, starting with a brief review of basic constructions like Cauchy completion. Discover surprising new results, including the notion of Fourier series in triangulated categories with metrics and the concept of approximable triangulated categories. Learn about theorems providing examples of approximable categories, general structure theorems, and their applications. Understand how these concepts relate to proving conjectures by Bondal and Van den Bergh, generalizing Rouquier's theorem, offering a concise proof of Serre's GAGA theorem, and addressing a conjecture by Antieau, Gepner, and Heller. Follow the lecture's progression through topics such as intrinsic subcategories, pairings, stability conditions, and more, culminating in a final theorem and its proof.