Mathematical Sciences: Complex Analysis and Harmonic Analysis

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Abstract Berenstein 9622249 Berenstein will investigate several problems in complex analysis and integral geometry which have applications to subjects like computational algebra and medical imaging. The original link between these problems lies in the use of methods from complex analysis, especially multidimensional residues. Specific questions on the first subject include effective bounds for the Nullstellensatz over fields of positive characteristic. On the second, the use of wavelets and hyperbolic geometry to study, respectively, localized versions of the Euclidean Radon transform and the Neumann-to-Dirichlet map for the Laplace operator in the Euclidean plane. The project involves an interplay between Pure Mathematics (complex and harmonic analysis) and applications to several imaging and detection problems. Most notable is the localization of the Radon transform using wavelets. We expect that this algorithm will help to significantly decrease the radiation dosage received by patients during CAT scans, as well as reduce processing time. The last consideration would be especially important if it could be applied to MRI scans and we plan to consider how to adapt our method to this widely used medical imaging procedure also (eventually helping reduce its cost). Another question we study is motivated by a different imaging method, Electrical Impedance Tomography. This low resolution but very low cost procedure has applications to Medicine as well as Material Sciences (detection of cracks). Finally, the problems of Commutative Algebra mentioned in the proposal also arise from engineering applications, for instance, robot motion planning and control of distributed parameter systems.