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.