The PI will research in several complex variables. Lempert and the PI (initially working independently) have recently identified a class of analytic sheaves over complex Banach manifolds that is analogous to the class of coherent analytic sheaves over finite dimensional complex manifolds. They proved very general analogs of Theorems A and B of Cartan-Oka-Serre over pseudoconvex open subsets of, say, Hilbert space, and their closed complex Hilbert submanifolds. This proposal aims to find a class of complex Banach manifolds that is analogous to the class of Stein manifolds in finite dimensions in the sense that, say, the above mentioned analogs of Theorems A and B hold, and they holomorphically embed in a complex Banach space as complex Banach submanifolds the complex tangent spaces to which have direct complements at every point. The identification of the correct class of complex Banach manifolds that forms an analog of Stein manifolds is important and seems close to indispensable for further work on complex Banach manifolds. The creation of classical Stein theory was one of the early triumphs of several complex variables that in part defined it as a field of study in its own right. It may be expected that a strong analog of Stein theory for Banach manifolds will prove beneficial for complex analysis on Banach spaces in general.
Complex analysis on Banach spaces is a natural extension of complex analysis on Euclidean spaces. Complex manifolds appear naturally and unavoidably when one studies the solution set of holomorphic (e.g., polynomial) equations. As there are theories in physics that model the universe as finite or infinite dimensional complex manifolds, their investigation is natural and unavoidable. A thorough understanding of complex Banach manifolds is thus potentially very useful in the human quest for knowledge. Complex analysis is a beautiful chapter of analysis. Analysis is a major field of mathematics that gave us a large part of the understanding necessary for the technological revolution, e.g., through the Maxwell laws of electromagnetism, and quantum mechanics. In part it also gave us the laser light and methods of data compression that enable us to watch a movie on a DVD.
