Nick Ramsey, "The interpretability order and simple theories"
112
ICJ
The interpretability order--sometimes called the triangle-star order--was introduced by Shelah (and studied in depth by Džamonja and Shelah and later Malliaris and Shelah) as a tool for comparing the complexity of first-order theories. It is closely connected to Keisler's order and, assuming some set theory, its structure on the stable theories and among theories with TP_1 is completely understood. Nonetheless, the behavior of simple unstable theories from the point of view of this order is quite complicated. We will talk about new work in progress towards understanding the interpretability order on concrete examples and constructions of simple theories: ACFA, pseudo-finite fields, vector spaces over finite fields with non-degenerate bilinear forms, etc. We will describe how, somewhat unexpectedly, Ramsey theory turns out to be the main mathematical ingredient towards concrete classification. This is joint work with Danielle Ulrich