Topics in Complex and Harmonic Analysis

贝伦斯坦教授打算继续他对复分析、交换代数和积分几何的几个问题的研究。例如，成员资格问题出现在机器人学、控制理论和计算几何中，它包括确定一个多项式是否属于给定的理想。因此，估计其复杂性并获得关于其可解性的先验知识是非常有意义的。利用复数分析的方法(多维剩余、积分表示公式)，Berenstein与Yger合作，已经在这一领域取得了重要的结果。例如，关于n元多项式系数的次数和大小的精确估计，解决了任意特征场的希尔伯特零点定理。贝伦斯坦计划继续研究这些问题，并解决超验数论中的相关问题。贝伦斯坦教授还将研究断层成像中的几个问题。例如，Berenstein和Walut等人获得了一种算法，该算法允许显著减少CAT扫描的辐射剂量。(这项发明将被授予专利)。展望了这项工作的继续，并可能应用于肺癌检测。电阻抗断层成像是另一种非侵入性医学成像，也用于检测材料中的裂纹。双曲平面上的测地线Radon变换在算法中起着重要作用。另一个成像问题位于庞贝变换的核心，该变换对任意几何形状的均匀探测器进行建模。与盖伊共同完成的关于这个问题的早期工作，导致了对海森堡小组复杂分析问题的考虑，以及对材料的非破坏性评估的其他几个问题的考虑。

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.