Variations on conservation laws for the wave equation

The first part of this paper, presented as an Emmy Noether lecture in connection with the ICM in Berlin in August 1998, gives some examples of using Noether's theorem for conservation laws for Tricomi-like equations and for the wave equation. It is also shown that equations which are semilinear variations of the wave equation can very often be handled similarly. The type of estimate obtained can even be used to get otherwise unobtainable local estimates for regularity. The fourth part is an introduction to the relation of black holes to the wave equation mainly showing the results of D. Christodoulou. His results use much more difficult estimates not corresponding at all to those in the first part of the paper.