Topological principles in the theory of Anderson localization

Scaling ideas in the theory of the quantum Hall effect are fundamentally based on topological principles in Anderson localization theory. These concepts have a very general significance and are not limited to replica field theory or disordered systems alone. We will discuss these ideas in several distinctly different physical contexts. We start with a brief overview that spans two and a half decades of experimental research on quantum criticality in strong magnetic fields. Secondly, we address the new understanding of universality that has emerged from the theory of Anderson localization and interaction phenomena. In the last part we show how the experimentally observed quantum phenomena fundamentally alter the way in which strong coupling problems in theoretical physics are perceived.