Explore the fascinating intersection of topology and condensed matter physics in this 1-hour 5-minute lecture by Martin Zirnbauer at the Hausdorff Center for Mathematics. Delve into the concept of Bott periodicity and its surprising application to topological insulators and superconductors. Discover how symmetry-protected ground states of gapped free-fermion systems form a periodic table-like structure, as explained through twisted equivariant K-theory. Follow the journey from physical context to new mathematical physics results, covering topics such as family constraints, time reversal, charge conservation, particle-hole conjugation, pseudosymmetries, and momentum conservation. Gain insights into the historical context and important papers in the field, while exploring pictorial representations and physical language interpretations of these complex concepts.