Mathematical Sciences: Geometric Function Theory in SeveralComplex Variables

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Three main mathematical themes will be developed in this project, built around the theory of several complex variables. In the first, techniques from partial differential equations, especially real variable microlocal techniques, will be applied to provide insight into the relationship between the smoothness of a function and the type of curve (surface) which contains the curve. In one complex dimension one either concludes that the function must be smooth or else its Fourier coefficients decay slowly. In higher dimensions this phenomenon is believed to be related to the nature of the wave front set. Related work will involve questions of uniqueness for Cauchy -Riemann functions. Emphasis will be placed on the characterization of functions vanishing of infinite order along curves which are not complex tangential and on the relevance of the hypothesis of nonflatness as it relates to the regularity of CR-mappings. Additionally, work will be done on fundamental questions concerning holomorphic maps from complex n-dimensional space to itself. Typical of these are maps whose Jacobian is unity having images of finite volume (they exist) and automorphisms which, for reasons unknown, do not distinguish countable dense sets, but act differently on discrete countable sets. This research will have application to the general understanding of the geometry of complex space and to complex partial differential operators defined on manifolds.