Degenerate Microlocal Methods and Geometric Analysis

NSF提案DMS - 0204730: Rafe mazzeo本项目提出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。这些包括欧几里得空间中常平均曲率曲面模空间的整体理论，新胶合技术在构造新类型爱因斯坦度量中的应用，对共形紧致爱因斯坦度量的解析和几何行为的更详细的研究以及这些度量的变形理论，特别关注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开发的柯西数据匹配的进一步扩展，以及PI的“边缘演算”的改进。该提案的其他部分涉及使用由PI和melrose开发的纤维边界算子的伪微分学来研究引力瞬子，特别是它们的L2上同调。最后，PI和Vasy提出研究秩大于1的对称空间的几何散射理论，以及在这些空间上渐近建模的一类空间，与量子n体散射的微局部理论之间的联系。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主任，这是一个针对有才华的高中生的住宿暑期项目，同时也继续他的其他外展工作，向公众传播数学欣赏。从更普遍的角度来看，PI的研究涉及几何和分析中出现的问题，涉及所谓的曲率方程（爱因斯坦度量理论广义相对论是最著名的例子），以及在“无限”处具有高度对称性的空间上的散射理论。贯穿始终的一个中心问题是从谐波和微局部分析到这些问题的一些新技术的应用。主题是，人们应该开发专门适用于每个几何问题的分析技术，而这些几何设置反过来应该建议分析技术的新发展。这种方法在PI之前的研究中被证明是非常成功的。这里考虑的问题受到数学物理各个方面的主流趋势的启发，最具体的是量子散射和弦理论的某些部分这两个有些独立的领域。一些当前的和提议的工作已经激发了一些物理学家团体的兴趣，他们的直觉为这项工作的进一步数学方向提供了一个有趣的指导。除了这些动机，PI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领域变得越来越重要。PI还开展了广泛的人力资源开发，包括上述夏季项目，并积极指导一些年轻研究人员。

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.