Nonlinear optics of collective excitations and coherent control of matter with light Eugene Demler Department of Physics, Harvard University, This talk will review several examples of coherent control of quantum many-body systems with light. An emphasis will be on understanding pump and probe experiments, including terahertz light amplification by Higgs oscillations in buckyball superconductors, and generation of Josephson plasmons in high Tc cuprates using parametric resonance with phonons. An example of equilibrium phase transition enhanced by coupling to a cavity will also be discussed.