Mathematical Sciences: Complex Analysis and Harmonic Analysis

摘要9622249贝伦斯坦将研究复数分析和积分几何中的几个问题，这些问题在计算代数和医学成像等学科中都有应用。这些问题之间最初的联系在于使用复杂分析的方法，特别是多维残基。关于第一个主题的具体问题包括零点在具有正特征的域上的有效界。其次，利用小波和双曲几何分别研究了欧氏平面上Laplace算子的欧几里德Radon变换和Neumann-to-Dirichlet映射的局部化形式。该项目涉及纯数学(复数和调和分析)与几个成像和检测问题的应用之间的相互作用。最值得注意的是使用小波的Radon变换的局部化。我们预计，该算法将有助于显著减少患者在CAT扫描过程中受到的辐射剂量，并减少处理时间。最后一个考虑将特别重要，如果它可以应用于核磁共振扫描，我们计划考虑如何使我们的方法也适用于这种广泛使用的医学成像程序(最终有助于降低其成本)。我们研究的另一个问题是另一种成像方法--电阻抗断层成像。这种分辨率低但成本非常低的程序在医学和材料科学(裂缝检测)中都有应用。最后，该方案中提到的交换代数问题也源于工程应用，例如机器人运动规划和分布参数系统的控制。

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.