Beijing Institute of Mathematical Sciences and Applications

Last semester, we defined the Berkovich affine (and projective) line and studied its topological properties. However, this construction is not very suitable if we want to study the potential theory and dynamics on the Berkovich projective line. I will start this course by explaining how to define the Berkovich projective line by gluing two copies of the Berkovich affine line (basically, it is the analogue of the "Proj" construction in algebraic geometry). I will then show that this second construction is homeomorphic to the first one. In the second part of this course, we will start studying the potential theory on the Berkovich projective line (e.g. potential and Hsia kernels, Laplacian etc).