Problems in Complex Analysis

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The principal investigator plans to work on various problems in complex analysis. There are two main parts to the project. The first focuses on several complex variables, the second on complex dynamics. However, the line between these two areas is not clear-cut. Within the several complex variables portion, the central concept is the Cauchy-Riemann equation. The problems are concerned with either deriving estimates for this equation, or improving understanding of key concepts basic to it, or applying estimates for this equation to function theory. The second part of the project deals with complex dynamics. Its objective is to obtain an understanding of foliations by Riemann surfaces. These arise naturally from flows of holomorphic vector fields and lead to basic questions about currents. (The latter could just as well be thought of as belonging to several complex variables.) The principal investigator has a long-standing collaboration with Sibony in which the two develop the theory of complex dynamics. Most recently they have carried out some investigations of Riemann surface laminations. These occur in the iteration of a holomorphic map, in that its associated Julia set might have a lamination along which the map is more regular than it is in other directions. (Of course, laminations are also an interesting topic in their own right.) Until now it has been necessary for the study to impose extra conditions, such as hyperbolicity, on the singular points. The plan is to continue the study while allowing for more general singularities and to search for ergodic properties of the laminations. In two dimensions, positive closed currents can be approximated by currents of integration of Riemann surfaces. It would be useful for the study of dynamics in higher dimensions to have a similar geometric interpretation of currents. The PI and Sibony propose to work on this topic with Coman. The case of immediate interest is that of currents in three dimensions of bidimension (1,1).Mathematical analysis is an important tool for studying the world from a quantitative perspective. But already when one tries to find roots of polynomial equations, it becomes clear that complex numbers and complex analysis are needed. Hence it is important to develop complex analysis, which can then be used to build up other areas of mathematics (e.g., algebraic geometry, number theory). This project explores various aspects of complex analysis. The plan is for the principal investigator to work with two senior mathematicians, Diederich and Sibony. Together the three have a good overview of the fields of several complex variables and complex dynamics, which benefits not only their joint projects but also their work with younger mathematicians. They will work with two midcareer mathematicians (Coman and Lanzani) and with a group of postdocs (Heier, Herbig, Lee, Sahutoglu, Siano, and Wold). The project will also involve a number of graduate students, including five women, from both the University of Michigan and other institutions.