CAREER: Model Theory and Operator Algebras

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Model theory is a branch of mathematical logic which studies classes of structures by understanding what can be expressed about the structures in first-order logic. Besides being an interesting subject in its own right, model theory has had major impacts on almost every other branch of mathematics. In this project, we focus on applications of model theory to operator algebras, that is, various subalgebras of the algebra of bounded operators on a Hilbert space that are closed under adjoint and are closed in various topologies. The union of model theory and operator algebras has already proven to be fruitful and we plan on continuing the emerging evolution of model-theoretic methods in operator algebras. We also plan to continue our work in using nonstandard analysis to solve questions in diverse areas of mathematics, including infinite-dimensional Lie theory, topological graph theory, and combinatorial number theory. Nonstandard analysis takes advantage of idealized elements to replace limiting processes and offers new insights into difficult problems.The study of operator algebras originally began as a rigorous mathematical formulation for studying various phenomena in quantum physics. A Hilbert space is a space consisting of vectors that can be added and multiplied by scalars and for which a notion of angle makes sense. An operator on a Hilbert space is a continuous transformation of the Hilbert space that respects the addition and scalar multiplication; operators can themselves be added and multiplied and there is also a notion of an adjoint of an operator, which in some sense is akin to taking a matrix and taking its transpose. An operator algebra is a collection of operators on a Hilbert space that is closed under addition, scalar multiplication and adjoint and is closed under taking limits in a suitable sense. Understanding the properties of various kinds of operator algebras and attempting to classify them has been an important venture in functional analysis for over half a century. In this project, we propose to continue the use of techniques from logic to study operator algebras and their model-theoretic properties, that is, the properties they possess that can be expressed in logical terms.