Jan Fornal

Pointwise Ergodic Theorem along the primes of the form x² + ny²

In the last 40 years analysts tried to establish Birkhoff Ergodic Theorem for nonconventional ergodic averages. For instance, Bourgain proved convergence of ergodic averages along values of polynomial, Wierdl did similar thing for ergodic averages along primes. Later on, Mirek, Trojan, Zorin-Kranich and Krause proved also strong variational inequalities for these averages, that is, one has very rapid convergence to appropriate limit for these averages. The goal of the talk is to survey on those methods and justify why they lead the speaker to establish convergence and variational inequality for polynomial images of the primes of the form x² + ny².