Interactions between topology, model theory, and operator algebras

我的研究项目位于数学的两个部分，即模型论和拓扑学之间的接口，并寻求在其他领域产生应用。模型论是数理逻辑中有关数学结构及其满足的形式陈述的部分；它提供了一种关于数学的观点，这种观点在应用于其他数学领域方面取得了成功。拓扑学是对抽象空间的研究，它既是一门独立的学科，也是许多其他数学学科的统一框架。我计划使用拓扑学的技术来发展模型理论的一部分，这部分与数学分析产生的结构有关。虽然模型理论家从一开始就使用了拓扑语言和方法，但相互作用很少使用拓扑学家开发的强大工具。最近的一些结果，包括我和合著者的一些结果，表明使用更复杂的拓扑方法可以产生新的模型理论工具。我的计划是专门针对生产工具，将在动力系统和算子代数有用。

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.