Mathematical Sciences: Complex and Harmonic Analysis

工作将集中在理论的最新发展， 拟共形群、泛Teichmuller空间和 在近似理论中出现的域称为约翰域。 关于拟共形映射的一些经典问题， 也可以根据几个人获得的新结果进行重新检查 研究人员 这些问题中有关于近似的问题 高维拟共形映射的可微性 同样的扩张。 努力描述 准共形群（首先是离散群）激发了 包括首席研究员在内的几位顶级研究员。 最基本的问题是确定一个群体何时 实际上是一个莫比乌斯群 由于情况并非总是如此， 中间填充的问题应该会产生一些 令人兴奋的研究 其他工作更倾向于功能分析 在复杂的结构分析将集中在循环的研究 解析函数空间中的向量 这些都是 这些空间的力量遍布整个空间。 的一般条件 循环性只在特殊空间中才有。 然而，对于布洛赫来说， 只有部分结果可用，工作将继续 来完成这幅图。 此外，还将对以下方面进行调查： 拟对称局部同胚内射性准则 群和马尔可夫过程 在几个领域的分析 复变量将集中在polydisk中的外部函数上 和球在两个复杂的变量，特别是当零集 很小。 B的一个老问题。上的可积函数 四分之一平面将被追踪。 这项工作涉及到一些重要领域， 数学，包括三流形理论，偏微分 算子和Toeplitz变换空间 功能协调发展的它还用于工程应用， 系统理论

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.