Taylor 9307987 Much of the work supported by this award focuses on mathematical problems related to function theory and geometric questions in the theory of several complex variables. The work includes a search for geometric conditions an algebraic variety must satisfy in order that the plurisubharmonic functions on the variety satisfy estimates in the spirit of the classical Phragmen-Lindelof Theorem. The reason for this line of investigation stems from the observation that many properties of constant coefficient partial differential operators are characterized by estimates for plurisubharmonic functions on a variety. A second line of study concerns ongoing work combining complex analysis and multiserver queues. The connection derives from the fact that the Markov chains underlying the queuing processes are given by homogeneous, vector-valued random walks on the integer points in the positive orthant. The resulting functional equation has coefficients that are matrix-valued analytic functions and the unknowns are vector-valued analytic functions. Other work centers on the embeddability for three-dimensional Cauchy-Riemann manifolds. The goal here is to test whether all embeddable Cauchy-Riemann structure on the 3-sphere must be the boundary for a strictly pseudoconvex domain diffeomorphic to the complex 2-ball. A final topic concerns the classification of harmonic maps on the 2-shpere into the projectivized Hilbert space and the question of whether they must factor through a finite-dimensional sub-projective space. 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. ***
