Model theory and its applications: dependent class.. (DEPENDENTCLASSES)
Model theory and its applications: dependent classes
Start date: Mar 1, 2014,
End date: Feb 28, 2019
Model theory deals with general classes of structures (called models).Specific examples of such classes are: the class of rings or the class ofalgebraically closed fields.It turns out that counting the so-called complete types over models in theclass has an important role in the development of model theory in general andstability theory in particular.Stable classes are those with relatively few complete types (over structuresfrom the class); understanding stable classes has been central in model theoryand its applications.Recently, I have proved a new dichotomy among the unstable classes:Instead of counting all the complete types, they are counted up to conjugacy.Classes which have few types up to conjugacy are proved to be so-called``dependent'' classes (which have also been called NIP classes).I have developed (under reasonable restrictions) a ``recounting theorem'',parallel to the basic theorems of stability theory.I have started to develop some of the basic properties of this new approach.The goal of the current project is to develop systematically the theory ofdependent classes. The above mentioned results give strong indication that thisnew theory can be eventually as useful as the (by now the classical) stabilitytheory. In particular, it covers many well known classes which stability theorycannot treat.
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