Degenerate Microlocal Methods and Geometric Analysis

简并微局部方法和几何分析

• 批准号: 0204730
• 负责人: Rafe Mazzeo
• 金额: $ 23.85万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0204730&HistoricalAwards=false
• 关键词: Degenerate; Microlocal; Methods; Geometric; Analysis

NSF Proposal DMS - 0204730: Rafe MazzeoThe proposed work in this project involves the continuingdevelopment of analytic tools to study a variety of problemsin geometric analysis. These include the global theory ofthe moduli space of constant mean curvature surfaces in Euclideanspace, the application of new gluing techniques to constructnew types of Einstein metrics, a more detailed study of theanalytic and geometric behaviour of conformally compact Einsteinmetrics and the deformation theory of such metrics, with specialattention to self-dual conformally compact Einstein metrics infour dimensions. The proposed techniques here include furtherextensions of Cauchy data matching, as developed by the PI andPacard, as well as refinements of the PI's `edge calculus'.Abstract for Other parts of the proposal involve use of the pseudodifferentialcalculus of fibred boundary operators, as developed by the PI andMelrose, to the study of gravitational instantons, particularlytheir L2 cohomology. Finally, the PI and Vasy propose toinvestigate the connections between geometric scattering theoryon symmetric spaces of rank greater than one, as well as aclass of spaces asymptotically modelled on these, and themicrolocal theory of quantum N-body scattering. For the humanresources component, the PI proposes to continue his directorshipof the Stanford University Math Camp, a residential summerprogram for talented high school students, and also to continuehis other outreach efforts to disseminate mathematics appreciationto the general public.From a more general point of view, the PI's research concernsproblems arising in geometry and analysis involving what areknown as curvature equations (the theory of Einstein metrics ingeneral relativity being the best-known case) as well as scatteringtheory on spaces which possess high degrees of symmetry `at infinity'.A central concern throughout is the application of somewhat noveltechniques from harmonic and microlocal analysis to these problems. Thetheme is that one should develop analytic techniques which arespecifically adapted to each geometric problem, and these geometricsettings in turn should suggest new developments in the analytictechnology. This approach has proved very successful in the PI'sprevious research. The problems considered here are inspired by maintrends in various aspects of mathematical physics, most specificallythe two somewhat separate fields of quantum scattering and some partsof string theory. Some of the current and proposed work has alreadystimulated interest on the part of some communities of physicists,and their intuitions provide an interesting guide for further mathematicaldirections in this work. Beyond these motivations, the PI regards thisparticular interplay between geometry and analysis as an important one,particularly because the types of geometric objects studied here arebecoming increasingly important in many other places in mathematics.The PI has also undertaken extensive human resources development,including the above-mentioned summer program, and is active inmentoring a number of young researchers.

美国国家科学基金会提案DMS-0204730：Rafe Mazzeo这个项目中提议的工作包括继续开发分析工具来研究几何分析中的各种问题。这些内容包括欧氏空间中常平均曲率曲面的模空间的整体理论，新的粘合技术在构造新型爱因斯坦度量中的应用，更详细地研究共形紧致爱因斯坦度量的解析和几何行为以及这种度量的形变理论，特别是四维自对偶共形紧致爱因斯坦度量。这里提出的技术包括由PI和Pacard发展的柯西数据匹配的进一步扩展，以及PI的“边缘演算”的改进。对于建议的其他部分，涉及使用由PI和Melrose发展的纤维边界算符的伪微积分来研究引力瞬子，特别是它们的L2上同调。最后，Pi和Vasy建议研究秩数大于1的对称空间上的几何散射理论以及一类渐近模拟的空间与量子N体散射的微域理论之间的联系。在人力资源方面，PI建议继续担任斯坦福大学数学夏令营的负责人，这是一个面向有才华的高中生的暑期住宿计划，并继续他的其他外展努力，向公众传播数学欣赏。从更广泛的角度来看，PI的研究涉及几何和分析中出现的问题，涉及到众所周知的曲率方程(广义相对论中的爱因斯坦度量学理论)以及具有高度对称性的空间的散布理论。贯穿始终的一个中心问题是应用调和和微局部分析中的一些新技术来解决这些问题。主题是，一个人应该开发专门适用于每个几何问题的分析技术，而这些几何背景反过来应该建议分析技术的新发展。这种方法在PI的前人研究中被证明是非常成功的。这里考虑的问题的灵感来自于数学物理各个方面的主流趋势，最具体的是量子散射的两个不同领域和弦理论的一些部分。目前和拟议中的一些工作已经激发了一些物理学家群体的兴趣，他们的直觉为这项工作中进一步的数学方向提供了有趣的指导。除了这些动机之外，国际数学协会还认为几何和分析之间的这种特殊的相互作用是一个重要的因素，特别是因为这里研究的几何对象的类型在数学的许多其他地方变得越来越重要。国际数学协会还承担了广泛的人力资源开发，包括上述暑期项目，并积极指导一些年轻的研究人员。