Title: Zilber's Trichotomy and it's Applications
Abstract: Zilber's trichotomy is a general phenomenon in model theory, originating with influential work (and an ultimately false conjecture) of Zilber in the 1970s and 1980s. The idea is that under strong enough model theoretic assumptions, only three basic types of mathematical structures can emerge: graphs, modules, and fields. The trichotomy has manifested in a plethora of results and conjectures over the last 40 years, which have led to surprising applications in diophantine geometry, anabelian geometry, and combinatorics. In this talk, I will give a broad overview of Zilber's trichotomy and its applications, starting without any assumed knowledge of model theory. I will then discuss some new work, including the recent solution of one of the oldest open problems on the trichotomy: Zilber's original conjecture restricted to any mathematical structure which can be defined using varieties over algebraically closed fields.
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