Why do circles in the spectrum matter?

I plan to present several results linking the numerical range of a Hilbert space operator to the circle structure of its spectrum. I'll try to explain how the numerical ranges approach helps to unify, extend or supplement several results where the circular structure of the spectrum is crucial, e.g. Arveson's theorem on almost-wandering vectors of unitary actions and Hamdan's recent result on supports of Rajchman measures. Moreover, I'll give several applications of the approach to new operator-theoretical constructions inverse in a sense to classical power dilations. If time permits, I'll also address the same or similar issues in a more general setting of operator tuples. This is joint work with V. M\" uller (Prague).