Mathematical Sciences: Complex Analysis and Harmonic Analysis

摘要 贝伦斯坦9622249 贝伦斯坦将研究复分析中的几个问题 和积分几何，它们可以应用于 计算代数和医学成像。 最初的联系 这些问题在于使用来自复分析的方法， 尤其是多维残基。第一项具体问题 主题包括以下字段上的Nullstellenestertion的有效界限： 积极的特征。第二，利用小波和 双曲几何，分别研究，本地化版本的 欧氏Radon变换和Neumann-to-Dirichlet映射 欧氏平面上的拉普拉斯算子。 该项目涉及纯数学（复杂）之间的相互作用 和谐波分析）和应用到几个成像和检测 问题最值得注意的是Radon变换的本地化， 小波我们预计，这种算法将有助于显着减少 患者在CAT扫描期间接受的辐射剂量，以及 减少处理时间。最后一个考虑因素是， 重要的是，如果它可以应用于MRI扫描，我们计划考虑如何 为了使我们的方法适应这种广泛使用的医学成像过程， （最终帮助降低成本）。我们研究的另一个问题是 由不同的成像方法，电阻抗断层扫描激发。 这种低分辨率但成本非常低的方法可应用于 医学和材料科学（裂纹检测）。最后 建议中提到的交换代数问题也来自于 工程应用，例如，机器人运动规划 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.