Higher Rank Graph Algebras, Multivariate Operator Theory, Free semigroup Algebras, and Functional Equations

高阶图代数、多元算子理论、自由半群代数和函数方程

• 批准号: 358793-2013
• 负责人: Yang, Dilian
• 金额: $ 0.8万
• 依托单位: University of Windsor
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635207
• 关键词: Higher; Rank; Graph; Algebras; Multivariate

The proposal consists of several projects on operator algebras and functional equations.Operator algebras are algebras generated by continuous linear transformations on Hilbert spaces. They are originally from quantum physics. Many complicated nonlinear phenomena can be moddelled successfully using linear operators. There are direct applications to quantum computing, quantum information, signal processing, representation theory and differential geometry. In this area, my research is concerned about some issues in the structure of (particularly nonself-adjoint) operator algebras and some applications of these ideas, such as algebras associated to higher rank graphs which are higher dimensional generalization of directed graphs, multivariate operator theory which studies more than one operator at a time, and (unital) dual operator algebras which are a nonself-adjoint analogue of von Neumann algebras.Functional equations are equations in which the unknowns are functions. They have many significant applications to information theory, social and behavioral sciences, probability and statistics, and economics. My research in this area concerns close connections between functional equations and other areas as many as possible, particularly, harmonic analysis, representation theory and operator algebras. Modern approaches used in my research not only can solve some equations which classical approaches cannot, but also bring a bridge connecting functional equations with other modern areas. This would enrich the area of functional equations and make the study of functional equations much more fruitful in the future.

该提案由几个有关操作员代数和功能方程式的项目组成。操作器代数是由希尔伯特空间上的连续线性变换产生的代数。它们最初来自量子物理学。许多复杂的非线性现象可以使用线性操作员成功地改造。有直接应用于量子计算，量子信息，信号处理，表示理论和差异几何形状。 In this area, my research is concerned about some issues in the structure of (particularly nonself-adjoint) operator algebras and some applications of these ideas, such as algebras associated to higher rank graphs which are higher dimensional generalization of directed graphs, multivariate operator theory which studies more than one operator at a time, and (unital) dual operator algebras which are a nonself-adjoint analogue of von Neumann代数功能方程是未知数是函数的方程式。他们在信息理论，社会和行为科学，概率和统计以及经济学方面有许多重要的应用。我在该领域的研究涉及功能方程与其他尽可能多的领域之间的联系，尤其是谐波分析，表示理论和操作员代数。我的研究中使用的现代方法不仅可以求解经典方法不能不能的一些方程式，还可以带来将功能方程与其他现代领域连接起来的桥梁。这将丰富功能方程的领域，并使对功能方程的研究将来更加富有成果。