Interactions between topology, model theory, and operator algebras

我的研究计划在于数学的两个部分，即模型理论和拓扑结构之间的接口，并寻求产生应用到其他领域。模型论是数理逻辑的一部分，涉及数学结构和它们所满足的形式陈述;它提供了一种数学观点，在应用于数学的其他部分方面有着成功的历史。拓扑学是对抽象空间的研究，它本身就是一门学科，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论中与数学分析产生的结构有关的部分。虽然模型理论家从学科一开始就使用拓扑语言和方法，但相互作用很少使用拓扑学家开发的强大工具。最近的结果，包括我自己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。 我的程序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.