I will report on current work in progress on the construction of anticyclotomic p-adic L-functions for Rankin--Selberg products. I will explain how by p-adically interpolating the branching law for the spherical pair (U(n)xU(n+1), U(n)) we can construct a p-adic L-function attached to cohomological automorphic representations of U(n) x U(n+1), including anticyclotomic variation. Due to the recent proof of the unitary Gan--Gross--Prasad conjecture, this p-adic L-function interpolates the square root of the central L-value. Time allowing, I will explain how we can extend this result to the Coleman family of an automorphic representation.

This talk will cover the greatest hits of pointwise ergodic theory, beginning with Birkhoff's theorem, then Bourgain's work, and finishing with more modern directions.