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: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=572778
• 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.

该提案包括几个项目的运营商代数和功能方程。 算子代数是由希尔伯特空间上的连续线性变换生成的代数。 它们起源于量子物理学。许多复杂的非线性现象可以用线性算子成功地模拟。它们直接应用于量子计算、量子信息、信号处理、表示论和微分几何。在这方面，我的研究关注的是结构中的一些问题，（特别是非自伴）算子代数和一些应用这些想法，如代数相关联的高秩图是高维广义的有向图，多元算子理论研究一个以上的运营商在同一时间，和（unital）对偶算子代数这是一个非自伴模拟冯诺依曼代数。 函数方程是未知数是函数的方程。它们在信息论、社会和行为科学、概率和统计学以及经济学中有许多重要的应用。我在这方面的研究关注密切联系之间的功能方程和其他领域尽可能多的，特别是调和分析，表示论和算子代数。本文所采用的现代方法不仅可以解决一些经典方法无法解决的问题，而且还为函数方程与其他现代领域的联系架起了一座桥梁。这将丰富函数方程的研究领域，使函数方程的研究在未来取得更大的成果。