The project considers several exciting developments in complex analysis and geometry, focusing in particular on problems unified by the use of Hilbert-space methods as tools. The underlying theme behind many of the problems under investigation is the issue of interpolating functions, or more generally cross sections, on subspaces. In this way, one can study functions on the spaces by applying the principle of mathematical induction with respect to the dimension of the space. One of the main problems encountered in regard to the theory of interpolation and sampling (the same theory that gives us digital music and movie technology) is how to interpolate or sample data when the missing information has more than one degree of freedom. This same problem represents a fundamental gap in many problems in algebraic geometry. While some partial results are known, this project aims to uncover the key idea underlying the possibility of such interpolation and sampling.
Many of the problems to be addressed by this project, though fundamental to the fields of complex analysis and geometry in higher dimensions, require the creation and development of new ideas and techniques. These new techniques will help to forge links between different areas of mathematics and will establish pathways to other areas of science, advancing knowledge not only in complex analysis and geometry, but also in algebraic and differential geometry, partial differential equations, topology, mathematical physics, and engineering systems and signal analysis. The collection of ideas that will be pursued lie close to the foundations of complex analysis and geometry. Consequently, there are many problems related to those met in the project that can serve as good thesis problems and problems for young Ph.D.'s. The principal investigator has three graduate students at the moment, and the research undertaken in the project will directly affect them. Because certain components of the research are especially amenable to elementary considerations, one finds in them problems that can teach basic research ideas to undergraduates and even to talented high-school students. The principal investigator has previously supervised REU students and enjoys the process. Stony Brook also has a high-school outreach program with which he will be involved. The principal investigator will continue to disseminate his work through publication, teaching of graduate courses, lecturing at conferences and seminars, and visiting colleagues in departments across the globe. He is also working on a book that presents a lot of background material in complex analytic geometry in a way suited to graduate students and complex analysts less familiar with geometric methods. Finally, he is certain that some of the results of this project will eventually find their way into the toolboxes of engineers.
