Mathematical Sciences: Complex and Harmonic Analysis

工作将集中在拟共形群理论的最新发展，普遍的Teichmuller空间和在近似理论中产生的域称为约翰域。关于拟共形映射的一些经典问题也将根据一些研究人员的新结果进行重新研究。在这些问题中，有关于准共形映射在高维中由相同膨胀的可微映射逼近的问题。描述准共形群（首先是离散群）的努力激励了包括首席研究员在内的几位顶级研究人员。最基本的问题是确定一个群体何时是一个有效的莫比乌斯群体。由于情况并非总是如此，因此应该对填充中间的问题进行一些令人兴奋的研究。其他面向复分析中的功能分析结构的工作将集中在分析函数空间中的循环向量的研究上。这些是空间的元素，它们的力量跨越空间。循环性的一般条件只在特殊空间中已知。然而，对于布洛赫空间，只有部分结果可用，工作将继续完成这里的图片。此外，还研究了局部同胚、拟对称群和马尔可夫过程的注入性准则。在多复变量领域的分析主要集中在多盘中的外函数和球中的两个复变量，特别是在零集很小的情况下。本文将继续讨论B. Levin关于四分之一平面上可积函数的一个老问题。这项工作涉及许多重要的数学领域，包括三流形理论、偏微分算子和解析函数的Toeplitz变换空间。它也用于涉及系统理论的工程应用。

Work will focus on recent developments in the theory of quasiconformal groups, the universal Teichmuller space and domains arising in approximation theory termed John domains. Some classical questions regarding quasiconformal mappings will also be reexamined in light of new results obtained by several researchers. Among these are questions about the approximation of quasiconformal maps in higher dimensions by differentiable maps of the same dilatation. Efforts to characterize the quasiconformal groups (discrete ones, first) have energized several top investigators including the principal investigator. The fundamental question is to determine when a group is effectively a Mobius group. Since this is not always the case, the problem of filling in the middle should generate some exciting research. Other work oriented more toward functional-analytical structures in complex analysis will focus on the study of cyclic vectors in spaces of analytic functions. These are elements of the spaces whose powers span the space. General conditions for cyclicity are only known for special spaces. However, for Bloch spaces only partial results are available and work will continue to complete the picture here. In addition, investigations will be carried out on injectivity criteria for local homeomorphisms, quasisymmetric groups and Markov processes. Analysis in the field of several complex variables will focus on outer functions in the polydisk and ball in two complex variables, especially when the zero set is small. An old problem of B. Levin on integrable functions on the quarter plane will be pursued. This work relates to a number of important areas of mathematics including three-manifold theory, partial differential operators and Toeplitz transformation spaces of analytic functions. It is also used in engineering applications involving systems theory.