Pseudofinite structures, simplicity and uni- modularity. (Exposé dans le cadre des rencontres Franco-Colombienne)
by Dario Garcia (University of Leeds)
at Ens Lyon Site Monod ( 4ème étage salle 435 )
The fundamental theorem of ultraproducts (Los’ Theorem) provides a transference principle between the finite structures and their limits. It states that a formula is true in the ultraproduct M of an infinite class of structures if and only if it is true for "almost every" structure in the class, which presents an interesting duality between finite structures and their infinite ultraproducts.
In this talk I will review the main concepts of pseudofinite structures, and present joint work with D. Macpherson and C. Steinhorn where we explored conditions on the (fine) pseudofinite dimension that guarantee good model-theoretic properties (simplicity or supersimplicity) of the underlying theory of an ultraproduct of finite structures, as well as a characterization of forking in terms of decrease of the pseudofinite dimension. I will also present the concept of *unimodularity* (for definable sets) - which is satisfied by both pseudofinite structures and omega-categorical structures -and, if time permits, a result (joint with F. Wagner) about the equivalence between difference notions of unimodularity.