**Bulk-boundary correspondence in topological phases of matter ** - Abhijeet Alase

A topological insulator is a material that behaves as an insulator in its bulk, i.e. its interior, but whose surface contains conducting states, meaning that electrons can only move along the surface of the material. The existence and characteristics of such conducting states on the surface are intricately connected to certain topological properties of the matrix functions that describe the bulk. This connection is known as the bulk-boundary correspondence. In this talk, I will first provide an overview of the conventional approaches to establishing the bulk-boundary correspondence, while arguing that these approaches focus only on proving the existence of the conducting states on the surface. I will then present some results that connect other properties of these states, such as their stability and sensitivity to disorder in the material, to the properties of the bulk. These results are derived by leveraging two factorizations of matrix polynomials, namely the Smith normal form and the Wiener-Hopf factorization.