Model theory of analytic functions

模型论是数理逻辑的一个分支，它提供了一个强有力的工具，可以根据可定义集合的复杂性对数学结构进行分类。这在从真实的解析几何到数论的各个领域都产生了引人注目的应用。本项目将研究与数学中感兴趣的几个解析函数相关的领域扩展的模型论性质，但其行为尚未被当前的模型论捕获，目的是获得具有新应用潜力的模型论控制。可能的方向包括：探索超实数，transseries和非振荡函数的模型论类之间的联系探索Pila-Wilkie型结果在实数的驯服展开中的应用研究算术模型中建立在可定义实数上的解析函数的o-极小性和模型完备性

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