CAREER: Model Theory and Operator Algebras

职业：模型理论和算子代数

• 批准号: 1708802
• 负责人: Isaac Goldbring
• 金额: $ 33.6万
• 依托单位: University of California-Irvine
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-09-01 至 2021-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1708802&HistoricalAwards=false
• 关键词: CAREER; Model; Theory; Operator; Algebras

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.

模型论是数理逻辑的一个分支，它通过理解一阶逻辑中的结构可以表达什么来研究结构的类别。 除了本身是一门有趣的学科外，模型论对数学的几乎所有其他分支都产生了重大影响。 在这个项目中，我们专注于应用模型论的算子代数，也就是说，各种子代数的有界算子代数上的希尔伯特空间，是封闭的伴随下，并在各种拓扑封闭。 模型论和算子代数的结合已经被证明是富有成效的，我们计划继续在算子代数中的模型论方法的新兴发展。 我们还计划继续使用非标准分析来解决数学各个领域的问题，包括无限维李理论，拓扑图论和组合数论。 非标准分析利用理想化的元素来代替极限过程，并为困难的问题提供新的见解。算子代数的研究最初是作为研究量子物理中各种现象的严格数学公式开始的。 希尔伯特空间是由向量组成的空间，这些向量可以与标量相加或相乘，并且角度的概念是有意义的。 希尔伯特空间上的算子是希尔伯特空间的连续变换，它遵循加法和标量乘法;算子本身可以相加和相乘，也有算子的伴随的概念，在某种意义上类似于取矩阵并取其转置。 算子代数是希尔伯特空间上的算子的集合，该空间在加法、标量乘和伴随下是封闭的，并且在适当的意义下在取极限下是封闭的。 理解各种算子代数的性质并试图对它们进行分类是半个多世纪来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技术来研究算子代数及其模型论性质，即它们所具有的可以用逻辑术语表示的性质。