I will explain some of our new results on spectral eta invariant of certain families of Dirac operators on NC tori.

Mixed norm Lebesque spaces are Banach spaces of multi-variable functions for which a different Lp norm is used in each variable. In many cases, there are simple, easy-to-compute conditions that ensure one given mixed norm space is contained in another. In other cases, only complicated, hard-to-compute conditions will do.

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.

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.)