Lecture, four hours. Requisites: courses 241A, 241B, 241C, 242A, 242B. Covers advanced topics in condensed matter physics with focus on topology (both in real and momentum spaces). Systematic exposition of Berry phases and Chern numbers, along with underlying differential-geometric structure. Concrete practical examples including Su-Schrieffer-Heeger model for polyacetylene and Majorana modes in one-dimensional superconductors, quantum Hall effects and topological insulators in two and three dimensions. Insights drawn from quantum pumping and bulk-edge correspondence especially emphasized. Range of topics based on topological defects in magnetic and superconducting systems and exploration of notions of topology for quantum transport and quantum information applications. Focus on aspects whose robustness is rooted in topological characteristics. S/U or letter grading.