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.
Miller continued his research on expansions of the field of real numbers, concentrating on further developing the model theory (a branch of mathematical logic), analytic geometry and geometric measure theory associated with o-minimal and certain other well-behaved expansions of the field of real numbers. Techniques were applied 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 applied model-theoretic techniques to questions in descriptive set theory and geometric measure theory. Intellectual Merit. Many results of so-called 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 two and and half decades, 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. Much of Miller's research consists of developing extensions of o-minimality that can deal with at least some of these non-o-minimal phenomena. Broader Impact. Like most research in theoretical mathematics, activities consisted primarily of analyzing the work of others, synthesizing new results, and communicating these results, in both preliminary and published final forms, to colleagues; as a concrete example, this grant provided substantial funding for Miller to spend three months at the Mathematical Sciences Research Institute (Berkeley, CA), where he interacted with other researchers from around the world. In addition, Miller worked, in collaboration with a colleague, toward the production of a substantial research monograph that would collect and organize much of the presently-fragmented information on the construction and analytic geometry of o-minimal expansions of the real field, thus making the material more accessible to graduate students and researchers from other areas.
