Abstract: This will be an introductory talk on the study of wave
equation on black hole spacetimes.
I will describe some of the fundamental geometric properties of black
holes and how they are reflected in the PDE estimates. I will discuss
the geometric origin of vectorfield estimates for the linear wave
equation on Schwarzschild (and time permitting also the Kerr spacetimes)
and explain in what ways they are robust to perturbations. The
connection to the stability of black holes and the analysis of the
Teukolsky equation will also be made.