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

该方案由几个关于算子代数和函数方程的项目组成。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。它们最初来自量子物理学。许多复杂的非线性现象可以用线性算子成功地建模。有直接应用于量子计算，量子信息，信号处理，表示理论和微分几何。在这个领域，我的研究关注算子代数的结构（特别是非自伴随）和这些思想的一些应用，例如与有向图的高维推广的高秩图相关的代数，一次研究多个算子的多元算子理论，以及（一元）对偶算子代数，这是冯·诺伊曼代数的非自伴随模拟。泛函方程是未知量为函数的方程。它们在信息论、社会和行为科学、概率论和统计学以及经济学中有许多重要的应用。我在这一领域的研究尽可能多地关注函数方程与其他领域的密切联系，特别是谐波分析、表示理论和算子代数。在我的研究中使用的现代方法不仅可以解决一些经典方法无法解决的方程，而且还为函数方程与其他现代领域之间的联系架起了一座桥梁。这将丰富泛函方程的研究领域，使泛函方程的研究在未来取得更大的成果。

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.