Partial Differential Equations and Several Complex Variables

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Two of the most important equations in several complex variables are the Cauchy-Riemann equations and the induced tangential Cauchy-Riemann equations. The understanding of these equations have been the focal point of research in complex analysis in the past few decades. The problems addressed in this project include the Cauchy-Riemann equations and the tangential Cauchy-Riemann complex on complex manifolds, especially on complex projective spaces (which is compact and with positive curvature) and negatively curved manifolds. Understanding the geometric aspects of these equations under the curvature conditions and their relations with function theory in complex manifolds are some of the most challenging and important problems in complex analysis and geometry. The study of several complex variables in a geometric or non-smooth setting has provided interesting new questions with fresh insight to problems in topology, foliation theory, complex dynamics, algebraic and complex geometry. Complex geometric theory has only just begun to develop and Shaw will continue her efforts in this direction. She will also continue her research on applying the geometric measure theory and harmonic analysis to several complex variables for non-smooth domains. Since the pioneering work of Poincare and Hartogs more than a century ago, the field of several complex variables has played a major role in modern mathematics. The use of partial differential equations has been the main tool for studying several complex variables, as well as complex geometry in the past few decades. The broader impacts from the proposed activity are that these problems are at the intersection of analysis, geometry and topology with applications in applied mathematics and physics. Other than the mathematical areas described in the proposal, recent progress in the Dirichlet and Neumann problem on nonsmooth domains has found applications in other disciplines like physics and engineering. The Hodge theorem is an extension of the classical Dirichlet Principle, the canonical solution to the energy minimizing problem arising from the heat transfer problem. Recent applications of the theorem on domains with corners and wedges have been used in electrokinetics and other fields in engineering and physics. The PI will use all of these ideas in her work mentoring students and the writing of a text that makes some of these partial differential equations topics more accessible to a wider range of mathematicians, especially those working in geometry and complex analysis.