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

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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.