I will revisit the (super classical) notion of recurrence and expected return time
for an example of a classical walk and then define these notions for (unitary)
quantum walks. We will see how very classical pieces of analysis developed for
different purposes around 1900-1920 give a good way to study these properties
in the quantum case. Among the main characters of the story are Polya, Riesz
and Schur. There are still lots of open questions, some surprises and even some physics experiments trying to test some of the predictions of the theory.