Real analytic geometry and model theory

Ohio State University Research Foundation -Do Not Use, Columbus OH

Investigators

Abstract

Miller proposes to continue his research on the model theory of expansions of the field of real numbers, concentrating on further developing the model theory and analytic geometry associated with o-minimal, and certain other well-behaved, expansions of the field of real numbers. He intends to do this by applying techniques from descriptive set theory and geometric measure theory in addition to the model-theoretic and analytic-geometric techniques usually associated with o-minimality. In turn, Miller hopes to apply model-theoretic techniques to questions in descriptive set theory and geometric measure theory. Miller has also begun to collaborate with applied mathematicians and control engineers on applications of model theory to hybrid control systems, and intends to continue. Many results of so-called classical mathematics are very general; they apply to a wide variety of input, so to speak, so we must expect to have to deal with a correspondingly wide variety of output. However, we could hope that if the input is, in some respect, particularly well behaved, then the output would be similarly well behaved. This turns out to be true in many important cases, but to see this usually requires new, more constructive proofs of classical results, as well as a deeper understanding of the good properties of the input. Before we can even begin such projects, though, we need some way of deciding which mathematical objects (inputs) should be considered as well behaved, and which should be considered as troublesome. This can be a difficult matter. The theory of o-minimal structures on the real field, a sub-discipline of mathematical logic, has been developed in large part to deal with this issue. This has been a rapidly-developing area for the last decade, with many contributions from---and cooperation between---several branches of mathematics and logic. Applications, and potential applications, of these developments have been found in areas as diverse as theoretical economics, neural-net learning theory, and hybrid control systems.