Short talk: Given a sequence $(f_n)$ consider the moving averages $\frac{1}{L} \sum_{n = v + 1}^{v + L}$ as $v$ and $l$ tend to infinity. Moving averages like these have been a focus of study in real analysis, probability theory, and ergodic theory for some time. We discuss various aspects of the convergence of the moving averages when $f_n = f \circ T^n$ for a measurable function $f$ and a map $T$.
Seminar talk: It turns out that control of moving averages is connected with whether or not f is a coboundary: that is, $f = F - F \circ T$ for some function $f$. In recent work, Adams and Rosenblatt have obtained a number of new results on coboundaries. While much is now understood, there remain some puzzling issues that we have not yet been able to resolve.