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Explore the fascinating world of nonlinear optical probing and control of magnetic and electronic band topology in this comprehensive lecture by Gregory Fiete from Northeastern University. Delve into the intriguing responses of materials to light, focusing on two key aspects: the non-linear photogalvanic response of Weyl semimetals and the coupling of phonons to electronic degrees of freedom. Examine the quantum geometry and quantum metric of Weyl systems, comparing low-energy theories with full band structures for various systems. Discover how second-order optical responses can distinguish topological phases of matter in superconductors. Investigate the theory behind "giant" g-factors in chiral phonons of insulating transition metal oxides. Gain insights into relevant experimental results and their theoretical implications in this cutting-edge exploration of material physics and optics.