RUI: Collaborative Research: Elliptic Partial Differential Equations on Singular Manifolds and Applications in Complex Geometry

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。主要研究人员建议进行系统的发展理论的椭圆型偏微分方程的一个紧凑的流形与奇点的边缘型。该项目的核心是精确描述奇异轨迹上的边值问题，并在适当的椭圆度条件下研究适定性。作为该理论的直接应用，该项目将研究自然椭圆复形，例如其奇异轨迹光滑的紧致解析簇的Dolbeault复形。主要研究人员希望，在这里开发的技术也将增加我们的理解的Dolbeault复杂的情况下，更一般的奇点。该项目将为光谱分析铺平道路（解析式，zeta函数等）。与不完全楔形几何相关的椭圆算子。在复分析的背景下，该项目有可能在更一般的紧致解析簇类中的上同调研究上有所启发。该理论将为工程、数学物理和量子化学中的应用问题的分析提供理论基础。该项目依赖并鼓励具有不同数学背景的研究人员的共同努力，促进研究和教育相关问题的更广泛互动。这项工作的某些方面代表了研究生和高年级本科生的研究经验的机会。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The principal investigators propose to undertake a systematic development of a theory for elliptic partial differential equations on a compact manifold with singularities of edge type. The core of the project is a precise description of boundary value problems at the singular locus and the investigation of well-posedness under suitable ellipticity conditions. As a direct application of the theory, the project will study natural elliptic complexes such as the Dolbeault complex of a compact analytic variety whose singular locus is smooth. The principal investigators expect that the techniques to be developed here will also increase our understanding of the Dolbeault complex in the case of more general singularities. The project will pave the way for a spectral analysis (resolvents, zeta functions, etc.) of elliptic operators associated with incomplete wedge geometries. In the context of complex analysis, the project has the potential to shed some light on the study of the cohomology in more general classes of compact analytic varieties. The theory to be developed will provide a theoretical underpinning for the analysis of applied problems in engineering, mathematical physics, and quantum chemistry. The project relies on and encourages a joint effort from researchers with different mathematical backgrounds, promoting a broader interaction in research and in issues pertaining to education. Some aspects of the work represent opportunities of research experiences for graduates and advanced undergraduate students.