Topics in Complex and Harmonic Analysis

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Professor Berenstein intends to continue his research in several problems of complex analysis, commutative algebra, and integral geometry. For instance, the membership problem, which consists of deciding whether a polynomial belongs to a given ideal, appears in robotics, control theory and computational geometry. For that reason, it is of great interest to estimate its complexity and obtain a priori knowledge about its solvability. Using methods from complex analysis (multidimensional residues, integral representation formulas), Berenstein, in collaboration with Yger, has already obtained important results in this area. For instance, sharp estimates about the degrees and sizes of the coefficients of n-variable polynomials solving Hilbert's Nullstellensatz for fields of arbitrary characteristics. Berenstein plans to continue working on these problems and solve related questions in trancedental number theory.Several problems in tomography will also be investigated by Prof. Berenstein. For instance, an algorithm that allows a significant reduction in the radiation dosage for CAT scans was obtained by Berenstein with Walnut et al. (A patent will be awarded to this invention). Continuation of this work, with possible applications to lung cancer detection is envisioned. Electrical impedance tomography is another kind of non-intrusive medical imaging and also used in the detection of cracks in materials. The geodesic Radon transform in the hyperbolic plane plays a role in the algorithms. Another imaging problem lies at the heart of the Pompeiu transform, which models a homogeneous detector of arbitrary geometrical shape. Earlier work on this problem, done jointly with Gay, has lead both to the consideration of problems of complex analysis in the Heisenberg group, as well as several other questions of non-destructive evaluation of materials.