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.
This kind of finite/infinite connection can sometimes be used to prove qualitative properties of large finite structures using the powerful known methods and results coming from infinite model theory, and in the other direction, quantitative properties in the finite structures often induce desirable model-theoretic properties in their ultraproducts. These ideas were used by Hrushovski to apply ideas from geometric model theory to additive combinatorics, locally compact groups and linear approximate subgroups.
More examples of this fruitful interaction were given by Goldbring and Towsner who provided proofs of the Szemerédi’s regularity lemma and Szemerédi’s theorem via ultraproducts of finite structures.
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.