DEPENDENTCLASSES (338821)

  https://cordis.europa.eu/project/id/338821

  FP7 (2007-2013)

  Model theory and its applications: dependent classes

  ERC Advanced Grant - Mathematical foundations (ERC-AG-PE1)

  mathematical logic

  2014-03-01 Start Date (YY-MM-DD)

  2019-02-28 End Date (YY-MM-DD)

  € 1,748,000 Total Cost

  Description

Model theory deals with general classes of structures (called models). Specific examples of such classes are: the class of rings or the class of algebraically closed fields. It turns out that counting the so-called complete types over models in the class has an important role in the development of model theory in general and stability theory in particular. Stable classes are those with relatively few complete types (over structures from the class); understanding stable classes has been central in model theory and 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 of dependent classes. The above mentioned results give strong indication that this new theory can be eventually as useful as the (by now the classical) stability theory. In particular, it covers many well known classes which stability theory cannot treat.

  Complicit Organisations

1 Israeli organisation participates in DEPENDENTCLASSES.