Resolution of Singularities in Analysis

在过去的几年里，Greenblatt一直致力于奇异点的解析，他最近证明了实解析函数的n维局部奇异点解析算法。这个方法是显式的、基本的，并且在坐标中完成。在他随后的研究中，他将运用他的方法，在适当的时候使用额外的想法，来证明涉及振荡积分、拉东变换和其他他已经做过研究的主题的定理。除了这些领域之外，他还将扩展到与奇点解析相关的其他几个不同领域。例如，他打算研究振荡积分算子的多线性推广，与积分稳定性有关的问题，以及代数几何中的相关问题，如涉及乘法器束的问题。此外，在[G1]的发展过程中出现了有趣的算法和计算问题，他计划在计算代数几何中研究这些问题。振荡积分算子是傅里叶分析的一部分，傅里叶分析在信号处理、密码学和统计学等科学和工程领域有着广泛的应用。因此，本研究提高了对振荡积分算子的理解，有可能有助于使用傅里叶分析方法的科学应用的发展。此外，氡变换是核磁共振成像和其他医学成像应用的基础，也可用于从石油勘探到国土安全等各个领域。因此，从这项研究中得到的关于氡变换的理论知识的改进可能会导致这些领域的进步。

Over the last several years, Greenblatt has been working on resolution of singularities, and he recently has proved an n-dimensional local resolution of singularities algorithm for real-analyticfunctions. This method is explicit, elementary, and done in coordinates. In his subsequent research,he will apply his methods, using additional ideas when appropriate, to prove theorems involving oscillatory integrals, Radon transforms, and other subjects in which he has done research. In addition to these areas, he will branch out into several other of the diverse areas that relate to resolution of singularities. For example, he intends to work on multilinear generalizations of oscillatory integral operators, problems related to the stability of integrals, and associated problems in algebraic geometry such as those involving multiplier sheaves. In addition, intriguing algorithmic and computational questions arose during the development of [G1], and he plans to investigate such issues in computational algebraic geometry.Oscillatory integral operators are a part of Fourier analysis, a field with wide application inscience and engineering, such as in signal processing, cryptography, and statistics. As a result,improved understanding of oscillatory integral operators resulting from this research has the potential to help in the development of scientific applications that use Fourier analytic methods. In addition, Radon transforms are fundamental to MRI and other medical imaging applications, and also find uses in diverse fields ranging from oil exploration to homeland security. As a result, improved theoretical knowledge of Radon transforms resulting from this research may lead to advances in such fields.