The fundamental role of topology in solid state physics was recognised with the Nobel prizes in 2016, 1998, and 1985. A remarkable phenomenon is the "bulk-boundary correspondence", whereby the boundary of a material is able to holographically detect a seemingly invisible topological invariant which is stable to perturbations, thus paving the way for novel applications. In these lectures, I will explain how K-theory, noncommutative geometry, C*-algebras, and index theory provide the mathematical framework for the physics of topological phases. I will also outline how physical ideas of symmetry, T-duality and the bulk-boundary correspondence motivate some conjectures and generalisations for the mathematics. Prerequisites: Some basic knowledge of algebraic topology, functional analysis, or operator algebras will be helpful. No detailed solid state physics background is needed, but some familiarity with ideas from quantum mechanics is useful. Suggested lecture topics: 2.19(Mon) 1st lecture; 10:30 - 11:45 Brief history of topological phases, and mathematical preliminaries. 2nd lecture; 14::00 –15:00 First example: Toeplitz index theorem in the SSH model. Second example: Kane-Mele invariant and why "Real" mathematics is needed. 3rd lecture; 15:30 – 16:30 Wigner`s theorem, generalised symmetries and Bott-Periodic Table of topological phases. 2.20(Tues) 4th lecture; 13:30 — 14:15 T-duality, geometric Fourier transform, and the bulk-boundary correspondence. 5th lecture; 14:30 — 15:15 T-duality, geometric Fourier transform, and the bulk-boundary correspondence. 6th lecture; 15:30 - 16:15 Applications to hyperbolic and crystallographic topological phases. 2.22(Thur) 7th lecture; 10:00-11:00 Semimetals, generalised monopoles, and differential topology