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; 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.

该项目广泛关注开发在重要类别的奇异空间上研究几何和分析问题的新技术。几何和数学物理学传统上专注于称为歧管的平滑几何对象，但是一类称为分层空间的几何对象一类稍微普遍存在。尽管如此，直到最近才开发出适当的特定技术来研究这些空间上的详细分析问题。该项目的一个目标是延续PI的长期目标，即建立某些强大的分析技术来研究在这种分层空间上设定的几何分析中的各种问题。该项目的其他目标包括将这些技术应用于仪表理论和数学相对论中发生的问题。这在部分微分方程，几何分析和数学物理学之间具有重大的跨学科吸引力，并在开发通用技术的并行目标与解决各种学科中感兴趣的特定问题之间进行了调节。几何微局部分析已成为研究差异和PseudoDodididifential fircifential operators contrications corking corne corne corne consection。此类操作员在几何分析和数学物理学中出现了许多应用。 PI目前的研究兴趣集中在迭代边缘伪差算子的演算中，可以用作研究椭圆，抛物线和双曲线操作员对分层假魔法的研究工具。在此的最终目标是将Boutet de Monvel微积分平行和推广的理论，该理论提供了在具有边界的歧管上研究椭圆边界问题的最一般和自然的环境。 PI打算利用该积分来研究特定问题，包括沿正常的交叉部分和Kapustin-witten方程的奇异溶液的Kaehler-Einstein指标。紧密相关的研究项目包括一种研究某些规格模量空间的渐近几何形状的新方法，尤其是在黎曼表面上Hitchin方程的解决方案的模量空间，以及一个程序的延续，以了解数学关系中Vacuum Einstein约束方程的解决方案的爆炸型。