Tameness in expansions of the real field

点击翻译按钮获取中文摘要

Miller will continue his research on first-order structures on the field of real numbers, concentrating on further developing the model theory and analytic geometry associated with o-minimal and certain other classes of well-behaved structures on 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 obtain results in control theory (specifically, classifying expansions of structures on the real field by trajectories of definable vector fields), descriptive set theory, and geometric measure theory.Many results of classical mathematics are very general: They apply to a wide range of input, so to speak, and thus tend to produce a wide range of output. But one could hope that if the input is particularly well behaved in some respect, then the output would be similarly well behaved. This turns out to be true in many important cases, but usually requires new, more constructive, proofs of classical results, as well as a deeper understanding of which inputs should be regarded as well behaved. 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 twenty-five years, with many contributions from, and cooperation between, researchers from several branches of mathematics and logic. Applications have been found in areas as diverse as theoretical economics, neural-net learning theory, and hybrid control systems, as well as in pure mathematics. However, o-minimality has a drawback: It allows only for the modelling of locally finitely connected behavior, and thus has rather limited use in understanding noisy or oscillatory settings. Miller proposes to develop extensions of o-minimality that can deal with at least some of these non-o-minimal phenomena.