Complex Analysis and CR Geometry

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The principal investigator will study ten related problems in several complex variables. The problems divide roughly into two areas, positivity conditions in complex analysis and CR geometry. The ideas on positivity conditions have already had an impact on the principal investigator's work on complex variables analogues of Hilbert's seventeenth problem, his work with Catlin on isometric imbedding, and his work with Varolin on stability criteria for Hermitian metrics. One of the main concerns here is a nonlinear Cauchy-Schwarz inequality, both for metrics on holomorphic line bundles and in a general setting. The second area primarily concerns CR mappings between spheres and hyperquadrics. Two main issues arise: group-invariant CR mappings and the complexity of smooth CR mappings between spheres in different dimensions. The group-invariant mappings exhibit surprising connections with number theory and combinatorics. The principal investigator intends to explore the connection with number theory. For instance, he will investigate certain cubic and higher order Diophantine equations that arise when considering CR mappings from Lens spaces to hyperquadrics. These equations involve an injectivity result that is elementary in the simplest case, where binomial coefficients arise, but quite subtle in general. He will continue to develop notions of complexity for CR mappings, to study proper holomorphic mappings, and to seek connections with sub-Riemannian geometry.Mapping theorems in one complex dimension have played a crucial role in mathematics, physics, and engineering for at least a century. The situation in higher dimensions is much more subtle and new phenomena arise. Eventually the applications of higher dimensional complex analysis will permeate all of science, as one-dimensional complex analysis does now. The crucial point of departure in this research is to pass from the unit circle to the unit sphere. The role of CR mappings between spheres in different dimensions has become more prominent in recent years. CR mappings are boundary analogues of complex analytic mappings. Their study leads to an unusual combination of analysis, geometry, and algebra. Progress has led to complex variables analogues of Hilbert's seventeenth problem, to isometric imbedding theorems, to a new kind of complexity theory, and to helical CR structures. Recent investigations have led to number-theoretic questions that form the foundation for the proposed work. This work will continue to impact complex analysis, especially via the portion concerning positivity conditions, but it will also broaden the scope of CR Geometry to include applications to complexity theory and to number theory. The proposer has organized several meetings around these topics, a program in 2005 at MSRI for graduate students and a workshop in 2006 at AIM for researchers. He will be giving a course at PCMI to graduate students on related material. Finally, he has introduced both his postdoctoral fellow Jiri Lebl and his current graduate student Dusty Grundmeier to the possible new applications of CR geometry.