Real analytic geometry and model theory

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Miller proposes to continue his research on the model theory ofexpansions of the field of real numbers, concentrating on furtherdeveloping the model theory and analytic geometry associated witho-minimal, and certain other well-behaved, expansions of the field ofreal numbers. He intends to do this by applying techniques fromdescriptive set theory and geometric measure theory in addition to themodel-theoretic and analytic-geometric techniques usually associatedwith o-minimality. In turn, Miller hopes to apply model-theoretictechniques to questions in descriptive set theory and geometric measuretheory. Miller has also begun to collaborate with applied mathematiciansand control engineers on applications of model theory to hybrid controlsystems, and intends to continue.Many results of so-called classical mathematics are very general; theyapply to a wide variety of input, so to speak, so we must expect to haveto deal with a correspondingly wide variety of output. However, wecould hope that if the input is, in some respect, particularly wellbehaved, then the output would be similarly well behaved. This turns outto be true in many important cases, but to see this usually requiresnew, more constructive proofs of classical results, as well as a deeperunderstanding of the good properties of the input. Before we can evenbegin such projects, though, we need some way of deciding whichmathematical objects (inputs) should be considered as well behaved, andwhich should be considered as troublesome. This can be a difficultmatter. The theory of o-minimal structures on the real field, asub-discipline of mathematical logic, has been developed in large partto deal with this issue. This has been a rapidly-developing area for thelast decade, with many contributions from---and cooperationbetween---several branches of mathematics and logic. Applications, andpotential applications, of these developments have been found in areasas diverse as theoretical economics, neural-net learning theory, andhybrid control systems.