Interactions between topology, model theory, and operator algebras

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My research program lies at the interface between two parts of mathematics, namely model theory and topology, and seeks to generate applications to other areas. Model theory is the part of mathematical logic concerned with mathematical structures and the formal statements that they satisfy; it provides a viewpoint on mathematics which has a history of success in applications to other parts of mathematics. Topology is the study of abstract spaces, and is both a subject in its own right as well as a unifying framework for many other mathematical subjects. I plan to use techniques from topology to develop the part of model theory that is related to structures arising from mathematical analysis. Though model theorists have used topological language and methods since the beginning of the subject, the interactions have only infrequently used the powerful tools topologists have developed. Recent results, including some by myself and co-authors, have shown that using more sophisticated topological methods can produce new model-theoretic tools.  My program is specifically directed at producing tools that will be useful in dynamical systems and operator algebras.