TOWARD SOLUTION OF THE MULTIVARIATE CORONA PROBLEM

建议的研究的目的是找到一个更紧密的联系之间的信息编码的拓扑结构和分析附近的电晕问题的代数H8（Dn）的有界复解析函数的n重直积Dn的开放单位盘，现代复分析的主要开放问题之一。这个问题是问Dn在H8（Dn）的极大理想空间M（H8（Dn））中是否稠密（即，在H8（Dn）的非零复同态空间中，具有一定的拓扑）。H8（D）的日冕问题是角谷在1941年提出的，并由Carleson在1962年的一篇著名论文中解决。从那时起，多变量电晕问题吸引了很多复杂的分析师的注意，但在50年或更长的时间里，它几乎没有进展。(In部分原因是Carleson的证明方法对n = 2不起作用。）我最近在这方面的工作提出了一种新的方法，电晕问题的基础上，一种新的方法产生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结构的M（H8（D））。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），并解决了这一领域的几个重要问题，如H8（Dn）的一个重要子代数切片代数的极大理想空间的描述，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_8（Dn））的拓扑刻画问题。在我最近的工作中，我证明了M（H8（D））在这方面的一些基本结果。我的目标是将这些结果推广到描述定义在某些黎曼曲面上的这类代数的极大理想空间的拓扑结构。我的研究的长期目标将是对H8（Dn）的电晕问题的解决方案取得实质性进展，并描述这个代数的最大理想空间。为此，我想修改我的一些方法来构造有界解的一些重要类别的微分方程的Dn。此外，我打算证明的极大理想空间M（H8（Dn））的特征类似于霍夫曼在1960年代建立的一维经典的一些结果。总之，我的建议是致力于新的方法来解决多元电晕问题。使用一些分析和几何方法开发在我的早期工作中，我希望取得实质性进展，在这个基本问题的复分析。

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 H8(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(H8(Dn)) of H8(Dn) (i.e., in the space of nonzero complex homomorphisms of H8(Dn) equipped with a certain topology). The corona problem for H8(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(H8(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 H8(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(H8(Dn)). In my recent work I proved some fundamental results in this area for M(H8(D)). My goal is to extend these results to describe the topological structure of the maximal ideal spaces of these type algebras defined on certain Riemann surfaces. The longer term objective of my research would be to make essential progress toward the solution of the corona problem for H8(Dn) and to describe the maximal ideal space of this algebra. To this end, I would like to modify some of my methods to construct bounded solutions for some important classes of differential equations on Dn. Moreover, I intend to prove some results on the characterization of the maximal ideal space M(H8(Dn)) analogous to the classical ones established by Hoffman in dimension one in the 1960s. To summarize, my proposal is devoted to new approaches to the multivariate corona problem. Using some analytic and geometric methods developed in my earlier work, I expect to make essential headway in this fundamental problem of complex analysis.