Delve into the fourth lecture of the Numerical Relativity series, focusing on the mathematical formulation presented by Thomas Baumgarte from Bowdoin College. Explore advanced concepts in solving Einstein's field equations computationally, essential for modeling high-energy astrophysical phenomena such as black hole mergers, neutron star collisions, and stellar collapses. Gain insights into the mathematical framework underpinning numerical simulations crucial for interpreting gravitational wave signals and multi-messenger observations. Enhance your understanding of this pivotal tool in modern astrophysics, particularly relevant for upcoming gravitational wave, electromagnetic, and neutrino observatories.