FRG: Model Theory and its Applications

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

The focus of this project is on the application of model theory (a branch of mathematical logic) within mathematics. Model theory comes into the picture in two ways: (a) it provides a new set of tools; (b) it provides a new language within which to formulate results and problems. Recent successes demonstrate that these can provide powerful new methods for many areas of mathematics and can be the basis for breakthroughs on critical problems. This project focuses, therefore, on a well defined and coherent research direction, made timely by recent advances, not on a single mathematical problem. Model-theoretic methods typically involve identifying and axiomatizing the first order theories of classes of structures, studying their nonstandard models, identifying the class of definable sets in a structure (quantifier-elimination), and making use of the compactness theorem to work within rich nonstandard models. There is a sophisticated machinery available (via Shelah and others) for understanding definable sets in certain general classes of structures. For example, once a concrete structure is identified as being stable, this machinery comes into play. The research problems to be emphasized in this project involve potential application of model theoretic ideas and tools within symbolic dynamics and algebraic geometry, the algebraic theory of differential equations, analysis, and geometry. One of the goals of this project to find new and effective ways to help mathematicians learn how to use model theory. In particular, the researchers selected for support by this project will often be mathematicians who work outside model theory, and who are interested in finding ways to use it in their own work.