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

算子代数是Hilbert空间上由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）算子代数和这些思想的一些应用，例如与高秩图相关的代数，它是有向图的高维推广，多元算子理论，它一次研究一个以上的算子，vonNeumann代数的一个非自伴的类似物（酉）对偶算子代数。函数方程是未知数为函数的方程。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。

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.