Mathematical Sciences: Analytic Discs and Integral Formulas in Several Complex Variables

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Tumanov 9401652 This award supports mathematical research on problems arising in the theory of functions of several complex variables. The work concentrates on boundary values of holomorphic functions; more precisely that of identifying functions on the boundary of a domain which can be boundary values of holomorphic functions. Such functions must satisfy partial differential equations known as the boundary Cauchy-Riemann equations. Solutions are known as CR-functions. Techniques for extending CR-functions from manifolds include a newly developing process of attaching discs to manifolds and looking at analytic extensions through the discs to obtain holomorphic extension. Then the discs and extensions must be sewn together appropriately. Another method involves integral formulas which give the extension by means of an integration process using a kernel which depends on the manifold (Szego kernel). A new line of research on a long-standing problem of embedding CR-structures in domains of several complex variables will also be undertaken. Several complex variables arose at the beginning of the century as a natural outgrowth of studies of functions of one complex variable. It became clear early on that the theory differed widely from it predecessor. The underlying geometry was far more difficult to grasp and the function theory had far more affinity with partial differential operators of first order. It thus grew as a hybrid subject combining deep characteristics of differential geometry and differential equations. Many of the fundamental structures were defined in the last three decades. Current studies still concentrate on understanding these basic mathematical forms. ***