Microlocal Methods in Geometric Analysis

几何分析中的微局部方法

• 批准号: 1608223
• 负责人: Rafe Mazzeo
• 金额: $ 23.5万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2021-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1608223&HistoricalAwards=false
• 关键词: Microlocal; Methods; Geometric; Analysis

This project is broadly concerned with the development of new techniques to study geometric and analytic problems on an important class of singular spaces. Geometry and mathematical physics have traditionally focused on smooth geometric objects called manifolds, but a slightly more general class of geometric objects called stratified spaces is equally ubiquitous. Despite this, only recently have specific techniques been developed appropriate to study detailed analytic problems on these spaces. One goal of this project is a continuation of the PI's long-term goal of establishing certain strong analytic techniques to study a wide variety of problems in geometric analysis which are set on such stratified spaces. Other goals of this project include applications of these techniques to problems occurring in gauge theory and in mathematical relativity. This has significant cross-disciplinary appeal between partial differential equations, geometric analysis and mathematical physics, and modulates between the parallel goals of developing general techniques and solving specific problems of interest in various disciplines.Geometric microlocal analysis has emerged as a powerful tool to study a range of problems concerning differential and pseudodifferential operators with degeneracies. Such operators arise in many applications in geometric analysis and mathematical physics. The PI's current research interest centers on the development of a calculus of iterated edge pseudodifferential operators which can be used as an investigative tool to study elliptic, parabolic and hyperbolic operators on stratified pseudomanifolds. The ultimate goal in this is a theory paralleling and generalizing the Boutet de Monvel calculus, which provides the most general and natural setting to study elliptic boundary problems on manifolds with boundary. The PI intends to use this calculus to study specific problems, including Kaehler-Einstein metrics with edge singularities along normal crossing divisors and singular solutions of the Kapustin-Witten equations. Closely related research projects include a new approach to study the asymptotic geometry of certain gauge-theoretic moduli spaces, in particular the moduli space of solutions to the Hitchin equations on a Riemann surface, and the continuation of a program to understand the blowup profiles of families of solutions of the vacuum Einstein constraint equations in mathematical relativity.

该项目主要关注研究一类重要的奇异空间上的几何和分析问题的新技术的发展。几何学和数学物理学传统上关注称为流形的光滑几何对象，但一个稍微更一般的称为分层空间的几何对象也同样普遍存在。尽管如此，直到最近才开发出适合于研究这些空间的详细分析问题的特定技术。这个项目的一个目标是继续PI的长期目标，建立一定的强大的分析技术，研究各种各样的问题，在几何分析，这是设置在这样的分层空间。这个项目的其他目标包括应用这些技术的问题发生在规范理论和数学相对论。这有显着的偏微分方程，几何分析和数学物理之间的跨学科的呼吁，并调制之间的并行目标的发展一般技术和解决具体问题的兴趣在各个学科。几何微局部分析已成为一个强大的工具来研究一系列的问题，关于微分和pseudodiomatic算子退化。这种算子在几何分析和数学物理中有许多应用。PI目前的研究兴趣集中在迭代边伪微分算子的微积分的发展上，该算子可以用作研究椭圆，抛物和双曲算子的分层伪流形的研究工具。在这方面的最终目标是一个理论平行和推广的Boutet de Monvel演算，它提供了最普遍和自然的设置来研究椭圆边界问题的流形上的边界。PI打算使用这种微积分来研究特定的问题，包括具有沿着法向交叉因子的边缘奇点的凯勒-爱因斯坦度量和Kapustin-Witten方程的奇异解。密切相关的研究项目包括一种新的方法来研究某些规范理论模空间的渐近几何，特别是黎曼曲面上希钦方程解的模空间，以及继续一个程序来理解数学相对论中真空爱因斯坦约束方程解族的爆破剖面。