The dependence of the rotation number on parameters in families of circle diffeomorphisms–often visualized as a “Devil’s Staircase”–is a fundamental object of study in one-dimensional dynamics. While the monotonicity and continuity of the rotation number is well-established, its regularity properties are more subtle. In this talk, we present Jacek Graczyk’s proof (1991) that for any C^2 one-parameter family of circle diffeomorphisms that also satisfy some other natural assumptions, the rotation number is Hölder continuous with exponent equal to ½. We will derive this result by first organizing frequency locking intervals according to the Farey tree, then establishing the universality of “harmonic scaling” in the parameter space with the help of this organization. 

Metric spaces and metric structures viewed from a model-theoretic perspective have attracted considerable attention in recent years. When the analogue of Scott's analysis is developed in the setting of continuous model theory, the rank of complete separable metric spaces (and structures) in continuous logic is always countable; this was done by Ben Yaacov, Doucha, Nies and Tsankov. An interesting problem arises if we equip a metric space with a natural, but classical, model-theoretic structure instead of a continuous logic structure. This situation was investigated by Fokina, Friedman, Koerwien and Nies (FFKN), and these authors asked if the Scott rank of complete, separable metric space in this way is always countable. In this talk I will give an example of a complete separable metric space which has Scott rank omega_1 when it is viewed as a classical model-theoretic structure as FFKN did. I will also say something about the proof, which is somewhat unusual because of it uses a fair amount of "serious" set theory.