Geometric and topological properties of marginally outer trapped surfaces

The modern theory of gravity was introduced by Albert Einstein in 1915. In General Relativity there is a one-to-one relationship between geometry and gravity. Black holes are one of the most interesting predictions of general relativity. The Schwarzschild solution or Schwarzschild black hole is named in honor of Karl Schwarzschild, who found this exact solution in 1915 and published it in January 1916. It was the first exact solution of the Einstein field equations other than the trivial flat space solution. Since then black holes have become an important as well as interesting part of GR. In the early days of general relativity, nobody believed that black holes actually exist. However observational evidence of their existence is now overwhelming [13, 1]. One definition of black holes which is very common is that a black hole is a region of spacetime from which even light cannot escape. But this global definition is not very useful for understanding the dynamics of black holes. In this thesis, we want to answer these questions: how can we define a black hole locally? And how can that definition be used to better understand things like black hole mergers? We begin with the definition of a marginal outer trapped surface (MOTS) and then we will discuss what we know about them and what is our goal for the future.