Topics in analysis related to resolution of singularities

这个项目是在接口的分析和解决的奇异性，后者领域往往被认为是一个部分的代数几何。工作将在几个科目，其中大部分都与数学的各个部分。这些科目包括限制定理的傅里叶变换和相关问题的偏微分方程，最大平均超过表面，和振荡积分算子与真实的阶段。研究人员的早期研究发现，在标量振荡积分与真实的阶段和积分的稳定性问题的应用，本研究的另一个目标是提供在这些领域的进一步发展。调查人员也有兴趣在追求类似的振荡积分的结果，以积分在其他领域，如p-adics，这反过来又与各种主题的数论。大部分的研究建议是谐波分析（广义解释）的一部分，一个领域与应用领域，如小波，物理学和电气工程有实质性的联系。从长远来看，提高对谐波分析理论方面的理解有可能导致更多应用学科的应用，这反过来又对整个社会产生广泛的影响。研究人员希望与学生进行多次互动，帮助发展这个国家的科学人力资源。此外，调查者所在的大学是一所城市州立大学，学生群体是全国最多样化的大学之一，学生来自许多族裔和民族。这种情况发生在本科生和研究生身上。因此，与学生合作可能有助于增加数学和相关领域的多样性。

This project is at the interface of analysis and resolution of singularities, the latter area often being considered to be a part of algebraic geometry. Work will be done in several subjects, most of which have connections to various parts of mathematics. These subjects include restriction theorems for the Fourier transform and associated problems in partial differential equations, maximal averages over surfaces, and oscillatory integral operators with real phase. The investigator's earlier research has found applications in scalar oscillatory integrals with real phase and in stability problems for integrals, and another goal of this research is to provide further developments in these fields. The investigator is also interested in pursuing analogues of these oscillatory integral results to integrals over other fields such as the p-adics, which in turn are connected with various topics in number theory. Much of the research proposed is part of harmonic analysis (broadly construed), a field with substantial connections to applied fields such as wavelets, physics, and electrical engineering. In the long term, improved understanding of theoretical aspects of harmonic analysis has the potential to lead to applications in more applied subjects, which in turn have a broad impact on society as a whole. The investigator expects to have numerous interactions with students, helping in the development of the scientific human resources of this country. Furthermore, the investigator's university is an urban state university with one of the most diverse student populations in the country, having students of many ethnic and national origins. This occurs in both the undergraduate and graduate student body. Hence it is likely that working with students will help increase diversity in mathematics and related areas.