Differential Equations Seminar: The geometry of quantum diffusion

The phenomenon of wave propagation in random environments appears in many physical situations of practical interest. The simplest model for this phenomenon is the Schrodinger equation coupled to a weak random potential, which describes the evolution of an electron in disordered media. The effect of the disorder is to scatter the wave into random directions. The long-time behavior is described by an effective diffusion equation, which was first established by Erdös, Salmhofer, and Yau using sophisticated diagrammatic arguments. In this talk I will describe a new approach to proving this effective limit which uses a wavepacket decomposition of the solution to give a geometric meaning to the diagrams. I will focus on the geometry of the diagrams and state some elementary open problems concerning Euclidean geometry which suggest a path to simpler proofs and stronger results.