Title: Making nonelementary classes more elementary
Speaker: William Boney, Harvard University
Abstract : Classification theory seeks to organize classes of structures (such as algebraically closed fields, random graphs, dense linear orders) along dividing lines that separate classes into well-behaved on one side and chaotic on the other. Since its beginning, classification theory has discovered a plethora of dividing lines for classes axiomatizable in first-order logic and has been applied to solve problems in algebraic geometry, topological dynamics, and more.
However, when looking at examining nonelementary classes (such as rank 1 valued fields or pseudoexponentiation), the lack of compactness is a serious impediment to developing this theory. In the past decade, Grossberg and VanDieren have isolated the notion of tameness. Tameness can be seen as a fragment of compactness that is strong enough to allow the construction of classification theory, but weak enough to be enjoyed by many natural examples. We will discuss the motivation for classification theory in nonelementary classes and some recent results using tameness, focusing on illuminating examples. No logic background will be assumed.