Cooperative International Research Project with Russia: Blaschke Inductive Limits of Disc Algebras

摘要：本课题主要研究复泛函理论、交换Banach代数、空间的归纳极限和投影极限之间相互作用所引起的一系列问题。复分析的经典和现代技术将应用于由Blaschke积和内函数生成的盘代数的归纳极限，其中的一个特殊情况是广义解析函数的代数，即在特定意义上具有有序对偶的紧阿贝尔群上的函数的解析。由一个内函数生成的归纳极限代数与复杂动力系统之间的关系将被研究。结果表明，所涉及的思想在有序对偶紧群的分析和一系列主题的进一步发展中具有很大的潜力：Wiener-Hopf算子，某些函数空间的不变子空间，动力系统，交换Banach代数理论，解析函数理论，corona型问题，Bourgain代数和Hankel型算子，小扰动下的稳定性。该国际合作研究项目由美国国家科学基金会数学科学部和国际项目部共同资助。其目标将通过美国蒙大拿大学的Thomas Tonev和俄罗斯鞑靼斯坦喀山州立大学的Suren Grigoryan博士的直接合作研究来实现。该项目的工作将通过将其与俄罗斯的一个类似项目和其他有关研究活动联系起来，有助于使目前在抽象谐波分析领域进行的国内研究工作国际化。研究将利用双方的智力资源，直接接触将使项目以最佳速度进行。这是动力系统理论，通过它与抽象谐波分析的联系，这是提出的研究的根源。这一领域目前发展非常迅速，在许多物理问题上有重要的应用。例如，众所周知，物理世界中的大多数过程具有几乎周期性而不是严格周期性的行为。动力系统理论的胜利之一是创造了研究近周期现象的数学工具。这就解释了人们对近周期函数及其近亲——广义解析函数的极大兴趣，而广义解析函数是本次合作研究的主要对象之一。这个项目的成功完成将导致新的发现，最初主要是数学上的兴趣，但具有巨大的未来应用潜力。

Proposal: DMS-9820775 Principal Investigator: Thomas TonevAbstract: This project focuses on a circle of problems arising from the interaction between complex function theory, commutative Banach algebras, and inductive and projective limits of spaces. Classical and modern techniques of complex analysis will be applied to inductive limits of disk algebras generated by Blaschke products and inner functions, a special case of which is the algebra of generalized analytic functions, i.e., of functions on a compact abelian group with ordered dual that are analytic in a particular sense. Relationships between inductive limit algebras generated by one inner function and complex dynamical systems will be investigated. It appears that the ideas involved have great potential for further developments in the analysis on compact groups with ordered duals and for a host of topics: Wiener-Hopf operators, invariant subspaces of some function spaces, dynamical systems, commutative Banach algebra theory, analytic function theory, corona type problems, Bourgain algebras and Hankel type operators, stability under small perturbations.This international cooperative research project is jointly funded by NSF's Division of Mathematical Sciences and Division of International Programs. Its objectives will be achieved through direct cooperative research between Thomas Tonev of the University of Montana in the U.S. and Dr. Suren Grigoryan of Kazan State University in Tatarstan, Russia. Work on the project will serve to internationalize an ongoing domestic research effort in the area of abstract harmonic analysis by linking it with a similar project and other related research activities in Russia. The research will draw on the intellectual resources of both sides, and the direct contacts will allow the project to proceed at an optimal rate. It is the theory of dynamical systems, through its connections to abstract harmonic analysis, that lies at the root of the proposed research. This area, which is developing very rapidly at present, has important applications to many physical problems. It is well-known, for instance, that most processes in the physical world have an almost-periodic rather than strictly periodic behavior. One of the triumphs in the theory of dynamical systems has been the creation of mathematical tools for studying almost-periodic phenomena. This explains the great interest in almost-periodic functions and their close relatives, generalized analytic functions, which are among the principal objects of investigation in this cooperative endeavor. A successful completion of this project would lead to new discoveries of at first primarily mathematical interest but with great future potential for application.