TOWARD SOLUTION OF THE MULTIVARIATE CORONA PROBLEM

本文的研究目的是在n重上有界复解析函数代数H（Dn）的拓扑信息和电晕问题附近的分析之间找到一个更紧密的联系 开单位圆的直积Dn是现代复变函数分析中的一个重要问题。这个问题是问Dn在H（Dn）的极大理想空间M（H（Dn））中是否稠密（即，在非零复形空间中 H（Dn）的同态具有一定的拓扑）。电晕问题的H（D）是由角谷在1941年和解决了卡尔森在一个著名的文件于1962年。自那时以来，多元日冕问题吸引了许多 复杂的分析师的注意力，但在50年或更长的时间里几乎没有进展。(In部分原因是Carleson的证明方法对n 2不起作用。） 我最近在这一领域的工作提出了一种新的方法，电晕问题的基础上产生的一个新的方法有界的解决方案的具体微分方程的光盘和仔细分析的拓扑 M（H（D））的结构。因此，我在极大理想空间上发展了复变函数理论，并在这个框架下证明了复分析的许多经典结果的类似物（Cartan定理，Runge逼近定理，Grauert和Ramspott定理），并解决了这一领域中的几个重要问题，如切片代数的极大理想空间的描述，H（Dn）的一个重要子代数，算子值复解析函数在具有相对紧凑图像的圆盘上的完备化问题（著名的Sz. 1978年提出的Nagy Corona问题）。 我的近期目标是发展一个类似的理论上的最大理想空间的切片代数有界复解析函数的直积某些黎曼曲面。 与冠问题密切相关的是极大理想空间M（H（Dn））的拓扑刻画问题。在我最近的工作中，我证明了这一领域的一些基本结果， M（H（D））。我的目标是

The objective of the proposed research is to find a closer link between the information encoded in the topology and analysis in the vicinity of the corona problem for the algebra H(Dn) of bounded complex analytic functions on the n-fold direct product Dn of open unit discs, one of the major open problems of modern complex analysis. The problem asks whether Dn is dense in the maximal ideal space M(H(Dn)) of H(Dn) (i.e., in the space of nonzero complex homomorphisms of H(Dn) equipped with a certain topology). The corona problem for H(D) was posed by Kakutani in 1941 and solved by Carleson in a famous paper of 1962. Since then the multivariate corona problem attracted a lot of attention of complex analysts but there was little headway on it for 50 years or more. (In part because the Carleson method of the proof does not work for n 2.) My recent work in this area proposes a new approach to the corona problem based on a new method producing bounded solutions of specific differential equations on the disc and careful analysis of the topological structure of M(H(D)). As a result, I developed complex function theory on the maximal ideal space and in this framework proved analogs of many classical results of complex analysis (Cartan theorems, Runge approximation theorems, Grauert and Ramspott theorems) and solved several significant problems in this area such as the description of the maximal ideal space of the slice algebra, an important subalgebra of H(Dn), the completion problem for operator-valued complex analytic functions on the disc with relatively compact images (a modification of the famous Sz.-Nagy corona problem posed in 1978). My nearer-term objective is to develop an analogous theory on the maximal ideal space of the slice algebra of bounded complex analytic functions on the direct product of certain Riemann surfaces. Closely related to the corona problem is the problem on the topological characterization of the maximal ideal space M(H(Dn)). In my recent work I proved some fundamental results in this area for M(H(D)). My goal is to exte