The transport properties of **topological quantum matter** are of central importance in Mathematical Physics, both from a fundamental viewpoint and for technological applications.

While we are still far from a complete, fundamental theory of **topological transport** based on a microscopic dynamical law, the last few years witnessed important progresses in the understanding of several specific questions in this context.

The course aims at providing the fundamental "mathematical tools" to investigate mathematical questions arising in** Solid State Physics**, both in the periodic case and in the non-periodic one, with a particular emphasis in the direction of** topological insulators** and **superconductors**.

A basic knowledge of Mathematical Quantum Theory will be assumed, as e.g. the one provided by the Ph.D. course *Self-adjoint operators in in Quantum Mechanics* at the first semester (LINK: https://www.math.sissa.it/course/phd-course/self-adjoint-operators-quant...)

**Topics**:

- Symmetries in Quantum Mechanics: a quick review
- Periodicity symmetry and Bloch-Floquet transform
- The Fermi projector and its properties
- Wannier bases and electrons localization
- The Bloch bundle and the Berry curvature
- The Localization Dichotomy in insulators
- Introduction to Topological Insulators