Explore the intricacies of topological phases and invertible phases in condensed matter physics through this comprehensive lecture. Delve into the general definition of these concepts before examining a practical illustration using a 1+1d Majorana chain and its connection to continuum field theory. Investigate bordism invariance and the classification of invertible phases, with a potential exploration of invertible phases with Z classification, focusing on the 2+1d p+ip superconductor as an example. Discover phases protected by symmetry and their implementation on lattices, including in-depth discussions on cohomology SPT and Dijkgraaf-Witten theories. If time permits, engage in a detailed analysis of the 2+1D Levin-Gu bosonic Z 2 SPT. Presented by Max Metlitski from MIT, this 1 hour and 40 minute lecture is part of a series hosted by the Institut des Hautes Etudes Scientifiques (IHES), offering valuable insights into anomalies in condensed matter physics.