Mathematical Sciences: Interpolation by Meromorphic Matrix Functions

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Work to be done on this project combines the mathematical theories of closed multiple-valued, matrix valued functions. The underlying motivation for the studies lies in the expected significant applications to systems theory and functional analysis. The research can be viewed from the point of view of interpolation theory and meromorphic matrix functions, or as explicit work on the correspondence between the Weil divisor approach and the factor of automorphy approach to complex vector bundles over closed Riemann surfaces. In the former one wants to construct meromorphic matrix valued functions interpolating given points on a closed Riemann surface. In the latter, one takes local solutions to form a holomorphic vector bundle and asks for conditions which guarantee that the bundle be trivial. This is where Weil's work enters. The work described in this project arises in systems analysis where one is interested in selecting optimal functions from classes all of whose members satisfy the same conditions. The conditions may be given as prescribed pointwise values of the functions. It then becomes essential that one first solve the interpolation problem to classify all such functions before establishing the optimal ones.