Mass and momentum in general relativity. Lecture 1

In this lecture, we shall explain that the classical notion of mass and momentum encounter difficulty when looking into gravitation energy. And we also explain how these important concepts can be defined using intuitions from geometry and physics. The discussion is based on the works with Schoen, Melissa Liu and Mu tao Wang and Poning Chen. For a closed surface with genus zero, we define the concept of mass and momentum of the region where the surface encloses. We derive properties that they should satisfy. The mathematics behind it is a nontrivial combination of geometry and partial differential equations which we shall attempt to explain to the audience in a reasonable manner.