Mathematical Sciences: Group Proposal in Complex Analysis

泰勒9307987该奖项支持的大部分工作集中在与函数论有关的数学问题和几个复变量理论中的几何问题。这项工作包括寻找一个代数簇必须满足的几何条件，以使该簇上的多重亚调和函数在经典的Phragman-Lindelof定理的精神下满足估计。研究这条线的原因是观察到常系数偏微分算子的许多性质都是通过对簇上多重亚调和函数的估计来刻画的。第二条研究路线涉及将复杂分析和多服务器队列相结合的持续工作。这种联系源于这样一个事实，即排队过程背后的马尔可夫链是由正直角上的整点上的齐次向量值随机游动给出的。得到的函数方程的系数是矩阵值解析函数，未知数是矢量值解析函数。其他工作集中在三维Cauchy-Riemann流形的可嵌入性上。这里的目的是测试3-球面上的所有可嵌入的Cauchy-Riemann结构是否一定是严格伪凸区域微分同胚于复2-球的边界。最后一个主题涉及到2-shpere上的调和映射到射影的Hilbert空间的分类，以及它们是否必须通过有限维子射影空间的问题。本世纪初出现了几个复变量，这是一个复变量函数研究的自然结果。很早就很明显，这一理论与其前身有很大不同。基本的几何学要难得多，而函数论与一阶偏微算符的亲和力要强得多。因此，它成长为一门混合学科，结合了微分几何和微分方程式的深刻特征。许多基本结构都是在过去30年里定义的。目前的研究仍然集中在理解这些基本的数学形式上。***

Taylor 9307987 Much of the work supported by this award focuses on mathematical problems related to function theory and geometric questions in the theory of several complex variables. The work includes a search for geometric conditions an algebraic variety must satisfy in order that the plurisubharmonic functions on the variety satisfy estimates in the spirit of the classical Phragmen-Lindelof Theorem. The reason for this line of investigation stems from the observation that many properties of constant coefficient partial differential operators are characterized by estimates for plurisubharmonic functions on a variety. A second line of study concerns ongoing work combining complex analysis and multiserver queues. The connection derives from the fact that the Markov chains underlying the queuing processes are given by homogeneous, vector-valued random walks on the integer points in the positive orthant. The resulting functional equation has coefficients that are matrix-valued analytic functions and the unknowns are vector-valued analytic functions. Other work centers on the embeddability for three-dimensional Cauchy-Riemann manifolds. The goal here is to test whether all embeddable Cauchy-Riemann structure on the 3-sphere must be the boundary for a strictly pseudoconvex domain diffeomorphic to the complex 2-ball. A final topic concerns the classification of harmonic maps on the 2-shpere into the projectivized Hilbert space and the question of whether they must factor through a finite-dimensional sub-projective space. Several complex variables arose at the beginning of the century as a natural outgrowth of studies of functions of one complex variable. It became clear early on that the theory differed widely from it predecessor. The underlying geometry was far more difficult to grasp and the function theory had far more affinity with partial differential operators of first order. It thus grew as a hybrid subject combining deep characteristics of differential geometry and differential equations. Many of the fundamental structures were defined in the last three decades. Current studies still concentrate on understanding these basic mathematical forms. ***