Autumn 2023 STAT 31250

Course: STAT 31250=CAAM 31250

Title: Mathematical Introduction to Topological Insulators

Instructor(s): Guillaume Bal

Class Schedule: MW 9:00 AM-10:20 AM in Jones 226

Description: The field of topological (acoustic, electromagnetic, electronic, mechanical) insulators analyzes asymmetric transport phenomena observed along interfaces that separate insulating bulks. It finds applications in many areas of physical and materials sciences. The topological nature of the asymmetric transport ensures that it persists in the presence of perturbations of the underlying model which forms its main practical appeal. This graduate level course will present several mathematical and physically motivated tools to model and quantify asymmetric transport such as: current physical observables, elliptic (pseudo-)differential operators, spectral theory of self-adjoint operators, index theory and classification, trace-class operators, computation of bulk invariants by Chern and winding numbers, computation of interface invariants by index theory and spectral flows, bulk-edge correspondence relating bulk and interface invariants, scattering theory, and computational methods to estimate transport numerically. We will in particular focus on magnetic Schroedinger, systems of Dirac equations, and linearized fluid wave models with applications to the Integer Quantum Hall Effect, the Quantum Anomalous Hall effect, topological equatorial waves, and possibly Floquet topological insulators and bilayer graphene systems.