allmath@iastate.edu
The logic seminar will meet on Tuesday 12/2 at 10am in Carver 401. Our speaker is Ben Castle from UIUC. The presentation will be virtual, and all are welcome to participate in person or via the zoom link below Here are the title and abstract.
Title: Toward a Zilber trichotomy for topological structures
Abstract: The Zilber Trichotomy is a highly influential program in modern model theory: roughly, it refers to the general expectation that sufficiently model-theoretically tame structures, if non-trivial, should always arise from modules and fields. Put another way: any sufficiently tame model-theoretic setting should admit a linear/non-linear dividing line, so that non-trivial structures in that setting only arise from (a) modules, if on the linear side, and (b) fields, if on the non-linear side. The trichotomy was classically stated in the 1970s-1980s for combinatorially tame (e.g. stable) structures, but turned out to be false in essentially every possible formulation; the only truly abstract stable setting where it worked was extremely restrictive and required the structure be controlled by a well-behaved system of Noetherian topological spaces. Later (1990s), a true (and simpler) trichotomy theorem was found for ordered (o-minimal) structures: these are unstable structures, but they similarly carry a natural topology, and this topology leads to `stability-like’ phenomena in the class of definable sets. In this talk, I propose that the Zilber trichotomy is really a topological phenomenon rather than a stability-theoretic one, and thus the abstract study of the Zilber trichotomy should proceed in a general setting of topological structures. As a first step, I will discuss a topological linear/non-linear dividing line, paired with a theorem giving the strongest possible classification for structures on the linear side. This is joint work with Assaf Hasson.
Zoom link:
https://iastate.zoom.us/j/93127761644