Simple Theories II

Massachusetts Institute Of Technology, Cambridge MA

Investigators

Abstract

One of the major quests of Kim's research project is to investigate stable relations occurred in simple structures. Major problem is to know when the simple structure has generic stable reduct. A generic stable reduct of a simple structure is the definitional stable reduct preserving forking independence. If a stable reduct of a simple structure is found, then every problem is reduced to the stable context from the simple context. Hence then stationarity can be used, which will resolve many open problems such as definable infinite group existence in 1-based structures, and group configuration problems. Unlike the well-known opposite process of constructing simple structures from stable structures, questions on finding stable fragments in simple structures seem to be rather difficult. Over the last 3 to 4 decades, model theorists have worked on stable structures. Kim's thesis in 1996 opened new research areas in model theory enlarging the scope of study of stable structures. Namely he proved the fundamental notion of independence in stable structures also exists in the wider class of structures, called simple. After then the study of simple structures takes place one of the major research area in model theory, and grows rapidly. Kim's continued work on simple structures will deepen and strengthen the model theoretic understanding of mathematical structures more.