Mathematics 146
Model theory (232985)
Michal Szachniewicz2026 Fall (4 Credits)
Schedule: MW 0900 AM - 1015 AM
Instructor Permissions: None
Enrollment Cap: n/a
Model theory divides mathematical theories into regions where the geometry of definable sets shares common features. For example, algebraic equations involving an order have different geometry than differential equations. In the model theoretic “map of the universe” (see https://forkinganddividing.com/) different properties of definable sets have different names like NSOP, NIP, stable, etc. In particular, the theory of real closed fields is NIP but has SOP, and the theory of differentially closed fields is stable. Many of the properties considered in the map originated in works of Shelah who understood when models of a theory can be classified, in some precise mathematical sense. During the course we will explore the dictionary of model theory and study possible behaviors of definable sets. The goal is to give participants tools necessary to understand modern applications of the area (o-minimality, pseudo-finite fields, Ax-Schanuel type theorems, continuous logic, …). The following will be talked about during the course: 1. Compactness theorem; 2. Types and basic properties of models, stability; 3. Saturated models, back-and-forth, Ehrenfeucht–Fraïssé games; 4. Quantifier elimination, Skolemization, Morleyization, imaginaries; 5. Ultraproducts, interpretations, categorical logic; 6. Examples; 7.Prime models, ranks, indiscernibles; 8. Categoricity, Morley’s theorem; 9. Local stability, forking; 10. Properties from the map of the universe, geometric stability.
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