Model theory of analytic functions

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Model theory, a branch of mathematical logic, provides powerful tools to classify mathematical structures according to the complexity of their definable sets. This has generated striking applications in various fields, from real analytic geometry to number theory. This project will study the model theoretic properties of expansions of fields related to several analytic functions of interest in mathematics, but whose behaviour is not yet captured by current model theory, with the aim of obtaining model theoretic control that has potential for new applications. Possible directions include: explore connections between surreal numbers, transseries and model theoretic classes of non-oscillating functions explore applications of Pila-Wilkie type results in tame expansions of the reals investigate o-minimality and model completeness for analytic functions built on definable reals in models of arithmetic