CAREER: Model Theory and Operator Algebras

模型论是数理逻辑的一个分支，它通过理解一阶逻辑中关于结构可以表达的东西来研究结构类。除了本身就是一门有趣的学科外，模型理论几乎对数学的其他所有分支都产生了重大影响。在这个项目中，我们主要关注模型理论在算子代数中的应用，即Hilbert空间上有界算子代数的各种子代数，它们在伴随下是闭的，并且在不同的拓扑中是闭的。模型理论和算子代数的结合已经被证明是卓有成效的，我们计划继续算子代数中模型理论方法的新发展。我们还计划继续使用非标准分析来解决数学中不同领域的问题，包括无限维李论、拓扑图理论和组合数论。非标准分析利用理想化的元素来代替极限过程，并提供了对困难问题的新见解。算子代数的研究最初是为了研究量子物理中的各种现象而建立的严格的数学公式。希尔伯特空间是由向量组成的空间，这些向量可以被标量相加和相乘，角度的概念对其来说是有意义的。Hilbert空间上的算子是Hilbert空间的连续变换，它尊重加法和标量乘法；算子本身可以被加法和乘法，还有一个算子的伴随的概念，在某种意义上，这类似于取一个矩阵并取它的转置。算子代数是Hilbert空间上的算子的集合，它在加法、标量乘法和伴随下是闭的，在适当意义下取极限是闭的。半个多世纪以来，了解各种算子代数的性质并试图对它们进行分类一直是泛函分析中的一项重要工作。在这个项目中，我们建议继续使用逻辑学的方法来研究算子代数及其模型论性质，即它们拥有的可以用逻辑术语表示的性质。

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.