December 5
Mostafa Mirabi,
Wesleyan University
Asymptotic Classes of Trees and $\aleph_0$-Categoricity
In this talk we introduce a framework for analyzing $\aleph_0$-categorical theories of trees using a combinatorial notion called a tree plan. We show that every $\aleph_0$-categorical theory of trees arises from a Fraïssé class $K$ determined by a tree plan, and that each such $K$ forms an asymptotic class in the sense of Macpherson--Steinhorn. We then discuss the model-theoretic consequences of this perspective, including the behavior of ultraproducts, the emergence of $\aleph_0$-categoricity from asymptotic classes of finite trees, and a characterization of supersimple finite-rank theories of trees in terms of their associated tree plans.