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)提供资金的。主要研究人员建议对具有边型奇点的紧致流形上的椭圆型偏微分方程的理论进行系统的发展。该项目的核心是精确地描述奇异轨迹上的边值问题，并研究在适当的椭圆性条件下的适定性。作为理论的直接应用，该项目将研究自然椭圆复形，如奇点光滑的紧凑解析变种的Dolbeault复形。主要研究人员预计，在更一般奇点的情况下，这里将发展的技术也将增加我们对多尔博特复合体的理解。该项目将为光谱分析(解析器、Zeta函数等)铺平道路。与不完整的楔形几何相关联的椭圆运算符。在复杂分析的背景下，该项目有可能为更一般的紧凑分析品种的上同调研究提供一些启示。即将发展的理论将为工程、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共同努力，促进在研究和与教育有关的问题上进行更广泛的互动。这项工作的某些方面为毕业生和高级本科生提供了研究经验的机会。

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.