Mathematical Sciences: Holomorphic Mappings and Projections

这个项目的工作结合了几何和函数理论 在几个复杂的变量，研究如何几何的一个 域影响定义的全纯函数的性质 在域上。 虽然最近取得了进展， 理解那些复杂的几何结构并不过分 退化，所谓的有限型域，一般状态 知识仍然有限。 在全纯函数的最易处理的类中， 定义在域上，那些平方可积的是 可能是最有用的。 可以将任何正方形 区域上可积函数到其最近全纯函数 通过对核函数进行积分的过程， 伯格曼内核 每一个都有一个独特的内核 域和集成将每个功能投影到最佳 全纯逼近 也许最重要的问题 与整环上的全纯函数有关的是 确定投影是否保留功能 它们在域的边界上是可微的。 是 它们的图像在边界上也是可微的 甚至 有经验的数学家会考虑这种可能性， 一个事件一个事故。 但事实并非如此。 如果边界 具有平滑的弯曲形状（伪凸）， 总是可微的。 将继续努力将目前的成果推广到 不是伪凸的域。 在那些更容易驾驭的 要分析的域是Hartogs域。 这些是 域具有某种圆对称性。 几 函数论中的反例在Hartogs 域. 最后，本文将把这一工作应用于一般问题 分析域之间的双全纯映射， 复杂变量 这些地图被认为是平滑的 在许多情况下，延伸到各自的边界。 这些光滑的域的完整描述 扩展是否存在可能是最重要的问题， 在这个时候，多个复杂的变量。

Work on this project combines geometry and function theory in several complex variables to study how the geometry of a domain influences the properties of holomorhpic functions defined on the domain. Although progress has been made recently in understanding domains whose complex geometry is not excessively degenerate, so-called domains of finite type, the general state of knowledge is still limited. Among the most manageable classes of holomorphic functions defined on a domain, those which are square integrable are probably the most useful. It is possible to convert any square integrable function on a domain into its nearest holomorphic neighbor by a process of integration against a kernel function - the Bergman kernel. There is a unique such kernel for each domain and integration projects each function onto the best holomorphic approximant. Perhaps the most important question related to holomorphic functions on domains is that of determining whether or not the projection preserves functions which are differentiable on the boundary of the domain. Are their images also differentiable on the boundary? Even experienced mathematicians would consider the likelihood of such an event an accident. The facts are otherwise. If the boundary has a smooth, curved shape (pseudoconvex) then the projections are always differentiable. Work will continue in an effort to extend present results to domains which are not pseudoconvex. Among the more tractable domains to be analyzed are the Hartogs domains. These are domains possessing some circular symmetry. Several counterexamples in function theory have been found in Hartogs domains. Ultimately, this work will be applied to the general problem of analyzing biholomorphic mappings between domains in several complex variables. These maps are known to have smooth extensions to the respective boundaries in many instances. A complete description of the domains for which these smooth extensions exist is probably the most important question under investigation in several complex variables at this time.