Topics in Complex and Harmonic Analysis

贝伦斯坦教授打算继续他的研究在几个问题的复杂分析，交换代数和积分几何。 例如，隶属度问题，它包括决定一个多项式是否属于一个给定的理想，出现在机器人学，控制理论和计算几何。 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。 使用方法从复杂的分析（多维残基，积分表示公式），贝伦斯坦，在合作与伊格尔，已经取得了重要成果，在这一领域。 例如，尖锐的估计的次数和大小的系数的n-变量多项式解决希尔伯特的零点场的任意特征。 Berenstein教授计划继续研究这些问题，并解决超越数论中的相关问题。Berenstein教授还将研究层析成像中的几个问题。 例如，Berenstein与Walnut等人获得了一种算法，该算法允许显著降低CAT扫描的辐射剂量（本发明将获得专利）。 继续这项工作，与可能的应用程序，肺癌检测的设想。 电阻抗断层成像是另一种非侵入式医学成像，也用于检测材料中的裂纹。 双曲平面上的测地Radon变换在算法中起着重要的作用。 另一个成像问题是庞培变换的核心，它模拟了一个任意几何形状的均匀探测器。 与盖伊共同完成的关于这个问题的早期工作，导致了对海森堡群中复分析问题的考虑，以及对材料的非破坏性评估的其他几个问题。

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.