Mathematical Sciences: Complex and Harmonic Analysis

工作将集中于拟共形群理论、通用 Teichmuller 空间和近似理论中出现的域（称为 John 域）的最新发展。 关于拟共形映射的一些经典问题也将根据几位研究人员获得的新结果重新审视。 其中包括关于通过相同膨胀的可微图来近似更高维度的拟共形图的问题。 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。 基本问题是确定一个群何时实际上是莫比乌斯群。 由于情况并非总是如此，因此填充中间的问题应该会产生一些令人兴奋的研究。 其他更多面向复分析中的泛函分析结构的工作将集中于解析函数空间中循环向量的研究。 这些是空间的元素，其力量跨越空间。 循环性的一般条件仅在特殊空间中已知。 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，还将研究局部同胚、拟对称群和马尔可夫过程的内射性准则。 多复变量领域的分析将重点关注两个复变量中的多圆盘和球中的外函数，特别是当零集较小时。 将继续研究 B. Levin 关于四分之一平面上可积函数的老问题。 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托普利茨变换空间。它还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.