Resolution of Singularities in Analysis

在过去的几年里，Greenblatt一直致力于奇异性的解决方案，他最近证明了一个n维的局部解决方案的奇异性算法的真正解析功能。这种方法是显式的，基本的，并在坐标系中完成。在他随后的研究中，他将应用他的方法，在适当的时候使用额外的想法，以证明涉及振荡积分，Radon变换和其他主题的定理，他已经做了研究。除了这些领域，他将分支到其他几个不同的领域，涉及到解决奇点。例如，他打算工作的振荡积分算子的多线性推广，有关问题的稳定性积分，并在代数几何相关的问题，如那些涉及乘子层。此外，有趣的算法和计算问题的发展过程中出现的[G1]，他计划调查这些问题在计算代数geometrics.Oscillatory积分算子的一部分傅立叶分析，一个领域与广泛的应用inscience和engineering，如在信号处理，密码学和统计。因此，从这项研究中产生的振荡积分算子的理解有可能帮助发展的科学应用，使用傅立叶分析方法。此外，Radon变换是MRI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的各种领域。因此，从这项研究中得到的氡变换的理论知识的改进可能会导致在这些领域的进展。

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.