Problems in Complex Analysis and CR Geometry

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D'Angelo's proposal concerns research in three related parts of complex analysis: positivity conditions in complex geometry, CR mappings, and algebraic aspects of subelliptic multiplier theory. One of D'Angelo's primary contributions to complex analysis in recent years has been a systematic study of positivity conditions for Hermitian symmetric functions on complex manifolds. This work has merged diverse issues such as a complex variables analogue of Hilbert's 17th problem, proper holomorphic mappings between balls, isometric imbedding for holomorphic bundles, and globalizable metrics into a coherent subject. The work on proper mappings led to a surprising result about primality; invariant CR mappings provide large classes of polynomials with integer coefficients exhibiting the same remarkable congruence properties satisfied by the p-th power of x plus y. Proper mappings also led to results in the complexity theory of mappings between balls. The work on subelliptic multiplier theory is related to the complexity and effectiveness of Kohn's algorithm, and thus fits into the same general area.Complex variable theory in one and several variables is a striking part of mathematics; it provides a beautiful example of pure mathematics, which has provided applications throughout engineering and the physical sciences. D'Angelo's work in CR geometry (a geometric part of complex analysis) has clarified one of the most basic issues, the roles of Hermitian symmetry and positivity conditions. His work on proper mappings between balls has led to new sorts of complexity questions, and even to a new primality test. D'Angelo has published two research level books in several complex variables, one undergraduate textbook (with D. West) with exciting problem sets, and has played an active role in mathematics education. He will be guiding a research experience for graduate students at MSRI in summer 2005 that will initiate students from many departments into CR geometry and the workdescribed in this proposal. Progress on these problems should impact complex variable theory, CR geometry, and possibly number theory.