TOWARD SOLUTION OF THE MULTIVARIATE CORONA PROBLEM

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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.