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Dive into the fascinating world of gravitational waves with this comprehensive lecture by Sebastiano Bernuzzi. Explore Einstein's groundbreaking theory, modern observational techniques, and the mathematical foundations of gravitational wave physics. Learn about the harmonic gauge, Einstein equations, and the quadratic age of the universe. Gain insights into formal solutions, interpretations, and approximations, including slow motion and far-from-source scenarios. Examine the expanding hypothesis and its implications for our understanding of the cosmos. This in-depth presentation offers a thorough introduction to the cutting-edge field of gravitational wave research, suitable for physics enthusiasts and aspiring astrophysicists alike.