Geometric Function Theory in Several Complex Variables

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Proposal: DMS-9877194Principal Investigator: Jean-Pierre RosayAbstract: The first goal of Professor Rosay is to continue his study, in collaboration with E.L. Stout, of the theory of boundary values (in the sense of hyperfunctions). Indeed, the two have started developing an entirely new theory of boundary values. This is research in function theory, with strong connections to several complex variables and partial differential equations. Rosay's project also entails the continuation of his study of various questions on the dynamics of biholomorphisms of complex n-space (collaborations with P. Ahern and F. Forstneric). Several other topics (e.g., uniqueness in the Cauchy problem, the nonlinear d-bar equation) will also be explored.Rosay's main area of activity is the theory of functions of several complex variables. This field is a central one in mathematics, with direct ties to such areas as approximation theory and partial differential equations that are very close to immediate applications (e.g., image processing, modeling, scientific computation). It should be stressed that mathematical results that may at first glance look quite abstract can in fact lead to very practical applications, by creating the possibility of new computational tools. (Even if, as usual in fundamental science, benefits may take many years to surface.) In order to illustrate how a shift from what seems to be purely theoretical to something more applied or at least applicable is possible, one need look no further than the present project for an example. Stout and Rosay recently gave a totally new proof of a very classical and fundamental theorem known as the Paley-Wiener theorem. This new proof was unlikely to be found in the classical setting, because the approach only became natural in the generalized and more abstract setting of "analytic functionals." However, despite the fact that it arose in a generalized setting, it turns out that the new proof is constructive in nature, and therefore much more closely linked to the possibility of practical computations. From one perspective, this development is not so surprising. It has been known for centuries that the full explanation of many basic mathematical phenomena (as fundamental as the convergence of power series expansion) is to be found only in the realm of complex numbers. It has likewise been known for centuries that many computations are best done with complex variables, even if one is interested in a final answer expressed in terms of real numbers. Of course, no more so than in any other science can one claim that all the research in the field of complex analysis will lead to applications, and often the applications that are found will come as complete surprises. But complex analysis clearly furnishes an essential tool to both pure and applied mathematicians.