Quantitative weak mixing for substitution dynamical systems

Weak mixing is one of the most basic properties in ergodic theory, which has several equivalent, seemingly rather different, formulations. For instance, weak mixing is characterized by convergence of absolute Ces\`aro averages or, alternatively, by the absence of non-trivial discrete spectrum. A natural question is how to make these properties quantitative. I will start by introducing the notion of quantitative weak mixing in a general setting. Next I will briefly survey some recent work of several authors on quantitative weak mixing and spectral properties for parabolic systems, such as interval exchanges and translation flows. I will then focus on substitutions, which often serve as "test cases" for the more complicated systems. This part is based on collaboration with A.I. Bufetov during the last 10 years and on recent work with A.I. Bufetov and J. Marshall-Maldonado.