TOWARD SOLUTION OF THE MULTIVARIATE CORONA PROBLEM

所提出的研究的目的是在n重有界复解析函数的代数H(Dn)的冠层问题附近找到拓扑中所编码的信息与分析之间的更紧密的联系 开单位圆盘的直积Dn，是现代复分析的主要公开问题之一。问题是Dn在H(Dn)的极大理想空间M(H(Dn))中(即在非零复空间中)是否稠密 具有某种拓扑的H(Dn)的同态)。H(D)的日冕问题是Kakutani在1941年提出的，Carleson在1962年的一篇著名论文中解决了这一问题。从那时起，多元日冕问题引起了人们的极大关注。 引起了复杂的分析师的注意，但在50年或更长的时间里，这方面的研究几乎没有取得什么进展。(部分原因是证明的Carleson方法对n2不起作用。) 我最近在这方面的工作提出了一种新的方法来解决日冕问题，该方法基于产生圆盘上特定微分方程组的有界解的新方法和对拓扑的仔细分析 M(H(D))结构。因此，我发展了极大理想空间上的复函数理论，并在这个框架下证明了许多经典的复分析结果(Cartan定理、Runge逼近定理、Grauert和Rampott定理)的类似结果，并解决了这一领域的几个重要问题，如切片代数的极大理想空间的刻画、H(Dn)的一个重要子代数、具有相对紧像的圆盘上算子值复解析函数的完成问题(对1978年提出的著名的Sz.-Nagy冠状问题的修正)。 我的近期目标是在某些黎曼曲面的直积上的有界复解析函数的切片代数的极大理想空间上发展一个类似的理论。 与日冕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))。我的目标是退出

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