Mathematical Sciences: Research in Complex Analysis and Complex Dynamics

9401398贝德福德该奖项支持对几个复杂变量理论中出现的问题进行数学研究。这项工作将从三个方向进行。在第一部分中，工作将完成复分析方法在两个复变量的多项式映射动力学中的应用。假设映射是单值的。本研究的核心是使用多势理论，类似于在一维动力学研究中令人印象深刻的使用势理论。在第二个方向上，将继续研究具有非紧自同构群的域的分类。由于用这样的自同构群构造非光滑域是很简单的，所以将注意力限制在具有光滑边界的域上。首先讨论的两个领域将是凸域和具有真实分析边界的域。在第三个方向，李维平面的使用将与有界解析函数的电晕定理联系起来研究。工作的中心目标是将Wolff的d-bar方法推广到单连通域上的电晕问题到多连通域和黎曼曲面。本世纪初出现了几个复变量，作为一个复变量函数研究的自然产物。很明显，这个理论与它的前身有很大的不同。底层的几何结构更难掌握，而函数理论与一阶偏微分算子的关系要密切得多。因此，它成长为一门混合学科，结合了微分几何和微分方程的深层特征。许多基本结构是在过去三十年中确定的。目前的研究仍然集中在理解这些基本的数学形式。＊＊＊

9401398 Bedford This award supports mathematical research on problems arising in the theory of several complex variables. The work will be carried out in three directions. In the first, work will be done on applications of the methods of complex analysis to the dynamics of polynomial mappings for two complex variables. The mappings are assumed to be univalent. Central to this study will be the use of pluri-potential theory analogous to the impressive use of potential theory in the study of one-dimensional dynamics. In the second direction, work will continue on the classification of domains with noncompact automorphism groups. Since it is trivial to construct nonsmooth domains with such automorphism groups, attention will be restricted to domains with smooth boundaries. The first two domains addressed will be those which are convex and those with real analytic boundaries. In the third direction, the use of Levi flat surfaces will be studied in connection with the corona theorem for bounded analytic functions. A central goal of the work is the generalization of Wolff's d-bar approach to the corona problem on simply connected domains to multiply connected domains and Riemann surfaces. 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. ***