Forking and dimensions in pseudofinite structures

Darío García, University of Leeds. Part of the logic seminar series.

The notions of dimensions or ranks have is one of the most important concepts in modern model theory and has been used to give a geometrical/combinatorial description of the definable sets in first order structures. One of the recurrent themes in these notions of rank is their relationship with the notion of forking-independence, defined by Shelah. It is often desired that any instance of forking (on types or formulas) can be detected by a decrease of the dimension, as in the well-known cases of algebraically closed fields or vector spaces.

The concept of pseudofinite dimension for ultraproducts of finite structures was introduced by Hrushovski and Wagner as a way to generalize the concept of dimension allowing values different from the integers. The main examples is the so-called "logarithmic pseudofinite dimension" which is defined on ultraproducts of finite structures by taking the logarithm of the cardinality of nonstandard finite sets and factor it out by the convex hull of the non-standar reals.

In this talk, I will present joint work with D. Macpherson and C. Steinhorn in which we explored conditions on the (fine) pseudofinite dimension that guarantee simplicity or supersimplicity of the underlying theory of an ultraproducts of finite structures, as well as a characterization of forking in terms of droping of the pseudofinite dimension. Also, under a suitable assumption, a measure-theoretic condition is showed to be equivalent to local stability.

Darío García, University of Leeds