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

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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.