Partial Differential Equations in Several Complex Variables

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Complex analysis in one and several variables occupies a central place in mathematics and mathematical physics. The use of complex numbers is essential in the development of mathematics. The concept of phase in complex analysis, for example, is commonly used in control theory, image analysis and dynamical systems. Certain important structures in physics arise from an integration of complex analysis with partial differential equations. Partial differential equations and several complex variables are also used in String and Twister mathematical physical theories that try to unify different physical force fields. For all these applications, foundational questions concerning the solution of such partial differential equations under certain geometric constraints are still not fully understood and they form some of the most challenging problems in mathematics. The current study is not only important for the development in mathematics, but it may lead to new understanding of physical phenomena with potential applications in other sciences and technology. This research will focus on some of the most important equations in several complex variables, including the Cauchy-Riemann equations and the induced tangential Cauchy- Riemann equations. The problems discussed in this proposal include function theory on complex manifolds, Hausdorff property of Dolbeault cohomology groups, Levi-flat hypersur- faces and complex foliation, the Cauchy-Riemann operators on complex pro jective spaces and negatively curved manifolds. Understanding the geometric aspects of these equations under the curvature conditions and their relations with function theory in complex manifolds is one of the most challenging and important problems in complex analysis and geometry. New approaches have been introduced to study these problems which connect the topology of domains in complex manifolds with the topology of Dolbeault cohomology groups. Regularity of the solutions to the Cauchy-Riemann equations are related to the foliation theory on the boundary. The study of several complex variables in a geometric setting has provided interesting new questions with fresh insight to problems in topology, foliation theory, complex dynamics, algebraic and complex geometry. The solution of these questions will advance our knowledge in all the aforementioned areas. Complex geometric theory has only just begun to develop and more efforts will be directed in this direction. Geometric measure theory and harmonic analysis will also be applied to study several com- plex variables on non-smooth domains. These problems are at the intersection of analysis, geometry and topology with applications in applied mathematics and physics.