Mathematical Sciences: Complex and Harmonic Analysis

工作将集中在拟共形群理论、普适TeichMuller空间和被称为John域的逼近理论中出现的域的最新发展。一些关于拟共形映射的经典问题也将根据几个研究人员的新结果被重新考察。其中包括关于高维拟共形映射用相同伸缩的可微映射的逼近的问题。描述准共形群(首先是离散群)的努力激励了包括首席调查员在内的几位顶级调查人员。基本问题是确定一个群什么时候是有效的Mobius群。由于情况并不总是如此，填补中间位置的问题应该会产生一些令人兴奋的研究。在复分析中，其他更倾向于泛函分析结构的工作将集中于研究解析函数空间中的循环向量。这些都是空间的元素，它们的力量跨越了整个空间。循环性的一般条件只对特殊空间是已知的。然而，对于Bloch空间，只有部分结果可用，工作将继续在这里完成。此外，还将研究局部同胚、拟对称群和马尔可夫过程的内射性准则。在多个复变量领域的分析将重点放在外函数中的多面体和球中的两个复变量，特别是当零点集很小时。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。这项工作涉及到许多重要的数学领域，包括三维流形理论、偏微分算子和解析函数的Toeplitz变换空间。它还被用于涉及系统论的工程应用中。

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.