Mathematical Sciences: Multivariate Splines

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This project aims at solving the outstanding problem of the dimension and basis for the multivariate spline spaces and applying the results to the most effective surface interpolation and computer representation of three dimensional objects via computer graphics. The structure of multivariate spline spaces is vastly more complex than in the univariate case and fundamental questions are still unanswered. Among these questions are the dimension, the construction of bases, existence of a local basis, and application to interpolation problems. The main tool of this research will be the symbolic manipulation language "Reduce", which will enable the investigator to solve exactly the large linear systems that arise from the smoothness conditions linking polynomials on neighboring triangles. The investigator will blend the Bezier-Bernstein techniques with the symbolic manipulation and will attempt to apply his results to surface representation problems arising in engineering.