Introduction

This paper provides an overview of recent work by the authors and others on two topics in the model theory of finite structures. The point of view here differs from that usually associated with the term ‘finite model theory’, as presented for example in [21] or [46], in which the emphasis and motivation come primarily from computer science. Instead, the inspiration for this work has its origins in contemporary (infinite) model theoretic themes such as dimension, independence, and various measures of the complexity of definable sets. Each of the topics deals with classes of finite structures for first-order logic that are isolated by conditions that are drawn from these model-theoretic considerations. Moreover, in both cases, connections exist to areas in infinite model theory such as stability and simplicity theory, and o-minimality. This survey is intended for both mathematical logicians and computer scientists whose work focuses on logical aspects of the subject.

The first theme concerns asymptotic classes of finite structures. This subject has its origins in the model theory of finite fields, via the work of Chatzidakis, van den Dries and Macintyre [13] (see Theorem 4.2.1) and the earlier model theory of finite fields developed by Ax [4], and ultimately rests on the Lang-Weil bounds for the number of points in a finite field of an irreducible variety defined over that field.

These two volumes contain both expository and research papers in the general area of model theory and its applications to algebra and analysis. The volumes grew out of the semester on “Model Theory and Applications to Algebra and Analysis” which took place at the Isaac Newton Institute (INI), Cambridge, from January to July 2005. We, the editors, were also the organizers of the programme. The contributors have been selected from among the participants and their papers reflect many of the achievements and advances obtained during the programme. Also some of the expository papers are based on tutorials given at the March – April 2005 training workshop. We take this opportunity, both as editors of these volumes and organizers of the MAA programme, to thank the Isaac Newton Institute and its staff for supporting our programme and providing a perfect environment for mathematical research and collaboration.

The INI semester saw activity and progress in essentially all areas on the “applied” side of model theory: o'minimality, motivic integration, groups of finite Morley rank, and connections with number theory and geometry. With the exception of motivic integration and valued fields, these topics are well represented in the two volumes.

The collection of papers is more or less divided into (overlapping) themes, together with a few singularities. Aspects of the interaction between stability theory, differential and difference equations, and number theory, appear in the first six papers of volume I.

These two volumes contain both expository and research papers in the general area of model theory and its applications to algebra and analysis. The volumes grew out of the semester on “Model Theory and Applications to Algebra and Analysis” which took place at the Isaac Newton Institute (INI), Cambridge, from January to July 2005. We, the editors, were also the organizers of the programme. The contributors have been selected from among the participants and their papers reflect many of the achievements and advances obtained during the programme. Also some of the expository papers are based on tutorials given at the March-April 2005 training workshop. We take this opportunity, both as editors of these volumes and organizers of the MAA programme, to thank the Isaac Newton Institute and its staff for supporting our programme and providing a perfect environment for mathematical research and collaboration.

The INI semester saw activity and progress in essentially all areas on the “applied” side of model theory: o-minimality, motivic integration, groups of finite Morley rank, and connections with number theory and geometry. With the exception of motivic integration and valued fields, these topics are well represented in the two volumes.

The collection of papers is more or less divided into (overlapping) themes, together with a few singularities. Aspects of the interaction between stability theory, differential and difference equations, and number theory, appear in the first six papers of volume I.

Valuations are among the fundamental structures of number theory and of algebraic geometry. This was recognized early by model theorists, with gratifying results: Robinson's description of algebraically closed valued fields as the model completion of the theory of valued fields; the Ax-Kochen, Ershov study of Henselian fields of large residue characteristic with the application to Artin's conjecture work of Denef and others on integration; and work of Macintyre, Delon, Prestel, Roquette, Kuhlmann, and others on p-adic fields and positive characteristic. The model theory of valued fields is thus one of the most established and deepest areas of the subject.

However, precisely because of the complexity of valued fields, much of the work centers on quantifier elimination and basic properties of formulas. Few tools are available for a more structural model-theoretic analysis. This contrasts with the situation for the classical model complete theories, of algebraically closed and real closed fields, where stability theory and o-minimality make possible a study of the category of definable sets. Consider for instance the statement that fields interpretable over ℂ are finite or algebraically closed. Quantifier elimination by itself is of little use in proving this statement. One uses instead the notion of ω-stability; it is preserved under interpretation, implies a chain condition on definable subgroups, and, by a theorem of Macintyre, ω-stable fields are algebraically closed. With more analysis, using notions such as generic types, one can show that indeed every interpretable field is finite or definably isomorphic to ℂ itself.