Special Year Learning Seminar

The goal of this learning seminar is to explain some of the core model theoretic notions which are behind Tao’s algebraic regularity lemma about definable graphs in finite fields (Tao 2012).

We will assume minimal knowledge of model theory and introduce local stability theory and amalgamation problems from scratch. Stability is a combinatorial notion imposed on formulas which determines rich model theoretic structure. Amalgamation is a model theoretic property which describes how the behaviour of small sets of variables can determine their global behaviour.

We will see that these distinct notions interact in surprising ways in the theory of pseudofinite fields. If time permits, we will give two different proofs of Tao’s algebraic regularity lemma, one relying on the geometry of Hilbert spaces, and the other relying on 3-amalgamation in algebraically closed fields.