This project proposes the further study of the geometric function theory in complex analysis and its applications. The geometric function theory, whose goal is to obtain qualitative characteristics of investigated objects when quantitative approach fails, is represented by several topics. (1) Studies of quotients of holomorphic loop spaces on complex manifolds, which are responsible for several intrinsic properties of the manifolds. Introducing different equivalence relations it is possible to construct objects ranging from holomorphic fundamental groups similar to classical fundamental groups to expansions of complex manifolds, where all holomorphic and plurisubharmonic functions can be extended. The initial work in this direction has shown an interesting mixture of analysis, geometry and topology. (2) Pluripotential theory on compact sets. In this setting such major tool as the Monge-Ampere equation does not work, while many classical questions have a natural reduction to the compact case. In particular, it is a problem of the existence of pluriharmonic fibrations of maximal plurisubharmonic functions. Since it is known that, in general, fibrations by complex curves do not exist, the PI will look for fibrations by compact sets such that the restrictions of a maximal plurisubharmonic function to these sets are pluriharmonic. (3) The theory of Hardy and Bergman spaces on hyperconvex domains. This is another topic that can be approached by geometric methods. Here the primary goal is the connection of the Poisson kernels with the geometry of the domains and applications to composition operators. As applications of the geometric function theory the PI is going to study such related subject as: uniform algebras, algebraic dependence of entire functions and approximation theory.
If the major advances in mathematics achieved in 19th and 20th centuries were due to our better quantitative skills, the modern mathematics became too complicated to be described by formulas. For example, such important objects as fractals cannot be described by any equations. This project intends to study complex mathematical problems using geometry. Geometry, as a part of mathematics, aims to describe qualitative, rather than quantitative, links between different objects. However, the geometric links can provide quantitative information about the subjects of study. The main goal of the project is to develop tools which can allow us to find geometric links between properties of functions and properties of their domains when complex numbers are involved. In its turn the knowledge of these links will lead us to better understanding of other areas of mathematics such as: the geometry of complex spaces, the transcendental numbers and the theory of approximation of functions by polynomials. As a broader impact, funding for this project will support the infrastructure of the Syracuse University's analysis group, which has historically been very strong. Some of the questions raised by this research will lead to PhD dissertation topics for graduate students, and others will provide an opportunity to involve undergraduate students in mathematical research. The funds will be used to allow PhD students to attend conferences for learning and disseminating their own achievements. A textbook which will be written as a part of the proposal will allow students in mid-size universities to access this important subject.
The Intellectual Merit of this proposal is the development of a geometric approach to different subjects complex analysis and its applications. One of the most interesting questions of mathematics is how far can we extend a mathematical object? For example, how far can I draw a line? And if above this line there is a graph can I extend this graph with the line preserving its properties? The first part of PI's work on the project last year was devoted exactly to these questions, where the objects were complex manifolds and plurisubharmonic functions. It was shown that under some conditions we can close holes in complex manifolds and extend there some useful functions. The second part of his work was devoted to introduction of algebraic methods to geometric problems. It is known that combinations of algebra and geometry bring new results and better understanding. PI's work led to discovery of algebraic structures where they were not known and hopefully these structures will be beneficial for the subject. The third part was dealing with adding flexibility to existing spaces of functions. Their properties play an important role in solving different problems however are dificult to establish. PI's work suggested tweaked versions of these spaces in expectation that they will be easier to use. As a broader impact, funding for this project supported the infrastructure of the Syracuse University's analysis group, which has historically been very strong. The project made an impact on the graduate program at SU: some of the questions raised by this research led to the PhD dissertation topics for graduate students. The book which the PI is writing as a part of the project will allow students in mid-size universities to access this important subject.