Abstract: In these series of lectures I shall cover some of the main ideas and results in cyclic cohomology and its application to problems in index theory.

Abstract: Classical Hardy's inequalities have been recently considered in a more general context than Lebesgue spaces (e.g., monotone functions, weighted inequalities, Hardy operator minus the identity, etc.) I will present some of the main results obtained in these new settings, as well as an extension to discrete averaging isometries. (Joint work with Santiago Boza.)