Title: Toward a Topological Zilber Trichotomy
Abstract: Many notions of linearity have been proposed over the last 40 years of model theory, in attempt to extend the Zilber trichotomy beyond the strongly minimal setting. For example, Peterzil and Loveys identified the `linear' and `non-linear' o-minimal structures, and this eventually led to the celebrated o-minimal trichotomy theorem of Peterzil and Starchenko. In this talk, I will discuss the basic idea behind linearity dividing lines, and argue that the most successful such dividing lines occur in so-called `topologically tame structures' (as opposed to tame theories in the Shelah sense). Then I will introduce a linearity dividing line for a large class of topologically tame structures, and present a theorem showing that the `linear' half of Zilber's trichotomy holds in the best possible way for these structures. This recovers many existing linearity notions, while (for example) extending the literature on linear o-minimal theories to the weakly o-minimal case.