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结构的表达方式来研究结构类别。 除了本身是一个有趣的学科之外，模型理论还对数学的几乎所有其他分支产生了重大影响。 在这个项目中，我们重点关注模型理论在算子代数中的应用，即希尔伯特空间上有界算子代数的各种子代数，这些子代数在伴随下是闭的，并且在各种拓扑中也是闭的。 模型理论和算子代数的结合已经被证明是富有成效的，我们计划继续算子代数中模型理论方法的新兴发展。 我们还计划继续使用非标准分析来解决数学不同领域的问题，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元素来代替限制过程，并为难题提供了新的见解。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。 希尔伯特空间是一个由向量​​组成的空间，这些向量可以与标量相加和相乘，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；运算符本身可以相加和相乘，并且还有一个运算符伴随的概念，在某种意义上类似于采用矩阵并对其转置。 算子代数是希尔伯特空间上的算子集合，该空间在加法、标量乘法和伴随下是封闭的，并且在适当意义上取极限时也是封闭的。 半个多世纪以来，了解各种算子代数的性质并尝试对它们进行分类一直是泛函分析中的一项重要事业。 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理论属性，即它们拥有的可以用逻辑术语表达的属性。