$p$-adic model theory uniformly in $p$, and applications to algebra and number theory
In 1949, Julia Robinson proved that the field of rational numbers is undecidable. Later, in 1965, motivated by a conjecture of Artin on zeroes of p-adic forms, Ax and Kochen proved that the field of p-adic numbers is decidable. This enabled development of model theory for $p$-adic and related fields. These theories have turned out to have a surprising feature, namely, that most of the properties that hold do not depend on the prime $p$, and are in turn controlled by other universal theories. Later, Ax developed a model theory of finite fields for almost all $p$, and this theory beautifully relates to the theory of $p$-adics for almost all $p$. The uniform logical behaviour was then showed by Pas-Denef-Loeser to govern many properties of $p$-adic integrals uniformly in $p$, which enabled a theory of motivic integration, and it is is believed that this is a feature of many of the naturally occurring concepts and structures in number theory.
Recently, in continuation of this line of developments, new results have been obtained in the following topics:
1. On counting conjugacy classes and representations (and other counting problems) in algebraic groups over local fields (this is joint work with Mark Berman, Uri Onn, and Pirita Paajanen) where one can translate asymptotic questions in group theory formulated in terms of a generating Poincare series to questions on p-adic and motivic integrals approachable by model theory.
2. A model theory has been developed for the adeles of a number field (this is joint work with Angus Macintyre). The ring of adeles of the rational numbers is a locally compact ring made of all the $p$-adic fields for all $p$ and the real field and enables using results on the local fields to derive results for the global rational field. It is intimately related to questions on various kinds of zeta functions in arithmetic and geometry.
Going from the local ($p$-adic and real) fields back to the rationals has long been a fundamental local-global transition both for the logic and the algebra. The above results give new tools and results on this. I will give a survey of some of the results and challenging open problems.