Partial Differential Equations in Several Complex Variables

University Of Notre Dame, Notre Dame IN

Investigators

Abstract

Since the pioneering work of Kohn and Hormander several decades ago, partial differential equations have been among the main tools for studying several complex variables. Morever, advances in one of these fields has frequently triggered advances in the other. Two of the most important examples of such equations are the Cauchy-Riemann equations and the tangential Cauchy-Riemann equations. The problems under consideration in this project include: the Cauchy-Riemann equations and the tangential Cauchy-Riemann equations on complex projective spaces, the geometric aspects of these equations with curvature terms, and their relationship to function theory in complex manifolds. The behavior of these equations under the curvature condition has not been explored systematically earlier, and an understanding of them in this context holds promise for impact on algebraic and complex geometry. New applications of these equations to complex foliation theory also yield important results in topology, geometry, and dynamics. For example, the principal investigator's recent work on the nonexistence of Levi-flat hypersurfaces in complex projective spaces is a holomorphic version of the classical Poincare-Bendixson theorem in dynamical systems. An important aspect of the research that will be done in this project rests in the fact that these problems are central to several different areas, namely, complex variables, complex geometry, and partial differential equations. These classical fields in mathematical analysis have contributed to our understanding of many important phenomena in physics. Since physical problems quite often involve variations in both time and space, the equations that govern them often need to be expressed in the language of complex variables. Many such equations involving several complex variables in a geometric setting have yet to be thoroughly analyzed, and many of the associated physical phenomena, such at the existence of a conic drop in an alternating-current electric field, have yet to be satisfactorily explained. The principal investigator hopes to tackle such remaining and new problems in geometry and topology. She interacts with mathematicians from many different areas in different parts of the world. She has taught courses on the subjects to students and researchers alike in many countries. Such interdisciplinary and international approaches bring new perspective and greater depth to each of the fields involved. In particular, recent progress in the so-called Dirichlet and Neumann problem on nonsmooth domains has found application in other disciplines, for instance, in physics and engineering.