Overview

Summary

Note: All bibliographic references are to articles in this volume. See the individual articles for more references.

Even the most reserved among model theorists would no doubt agree that the subject has grown dramatically over the last thirty years. This period has produced a substantial and beautiful abstract theory as well as a range of remarkable applications that extend into several areas of mathematics and incorporate the most sophisticated theoretical developments in the field. During the last decade in particular, results obtained by model theorists have attracted the attention of researchers outside logic and have opened up broad avenues for interaction. The Model Theory of Fields program held at the Mathematical Sciences Research Institute from January to June 1998 sought to capitalize on the intense activity that has taken place in the discipline by bringing together for an extended period of time model theorists and mathematicians working in the areas of some of the most exciting applications.

Model theory's stock-in-trade is the analysis of the so-called definable subsets of a mathematical structure. The definable subsets of classical mathematical structures have long occupied a central position in algebraic and geometric investigations; the constructible sets in algebraic geometry and the semialgebraic sets in real geometry provide two notable examples. Although the model-theoretic viewpoint may have supplied these areas with some basic results, it did not offer enough until recently for practitioners to notice that the objects that they study with their own sophisticated methods could be illuminated by model theory.