Model Theory of Unstable Structures

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

Model theory is a branch of mathematical logic that studies mathematical structures at a very general level. Whereas algebra studies various operations (addition, multiplication etc.) on various sets by classifying them according to what rules, or axioms, they satisfy, model theory takes a broader perspective and looks at all possible operations at once and attempts to classify them according to their combinatorial complexity. Some tame classes of operations are singled out for which then general theorems can be proved. Those in turn will apply to specific situations in different areas of mathematics (algebra, geometry, number theory...). This project lies at the border between the abstract and the applied side. The investigator plans to study two rather broad notions of tameness and apply this study to certain algebraic structures. More precisely, the aim of this project is the study of definable groups in NIP and NTP_2 structures and more specifically in valued fields, and PRC fields and the general development of NTP_2. Our guiding intuition is that in NTP_2, ideals should work as a common generalization of non-forking extensions in simple theories and measures in NIP. Also, there should be an analogue of distality in this setting. On the more applied side, we plan to develop tools for the study of non-definably amenable groups both in NIP and NTP_2; test questions being the classification of all groups definable in PRC fields and a decomposition theorem for non-abelian metastable groups.