Microlocal Methods in Geometric Analysis

这个项目广泛地涉及研究一类重要的奇异空间上的几何和解析问题的新技术的发展。几何和数学物理传统上专注于光滑的几何对象，称为流形，但同样普遍存在的是一类稍微更一般的几何对象，称为分层空间。尽管如此，直到最近才发展出适合于研究这些空间上的详细分析问题的具体技术。这个项目的一个目标是继续国际和平协会的长期目标，即建立某些强大的分析技术来研究几何分析中的各种问题，这些问题设置在这样的分层空间上。该项目的其他目标包括将这些技术应用于规范理论和数学相对论中出现的问题。这在偏微分方程组、几何分析和数学物理之间具有显著的跨学科吸引力，并在发展一般技术和解决不同学科中感兴趣的特定问题的并行目标之间进行调节。几何微局部分析已经成为研究一系列与具有退化的微分和拟微分算子有关的问题的有力工具。这种算符出现在几何分析和数学物理中的许多应用中。PI目前的研究兴趣集中在发展迭代边缘伪微分算子演算，它可以作为研究分层伪流形上的椭圆、抛物线和双曲算子的工具。本文的最终目的是建立一个与Boutet de Monvel演算平行和推广的理论，它为研究带边界流形上的椭圆边值问题提供了最一般和最自然的环境。PI打算用这个演算来研究特定的问题，包括沿法向交叉因子具有边缘奇异性的Kaehler-Einstein度规和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Hitchin方程的解的模空间，以及在数学相对论中理解真空爱因斯坦约束方程解族的爆破轮廓的程序的继续。

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.