Pluripotential Theory and Applications to Geometry, Number Theory, and Dynamics

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Abstract for Proposal DMS-0900934, PI: Dan ComanThis project addresses problems from pluripotential theory, some of which have important applications to transcendental number theory, complex geometry, and algebraic geometry. A unifying theme is that at its core lie plurisubharmonic functions and positive closed currents, either as main objects of investigation or as main tools to be employed. One direction of research deals with problems in pluripotential theory on compact complex manifolds, where there are new interesting phenomena, different from the local setting. The main goals are the study of the complex Monge-Ampere operator and of the corresponding Green functions. Another direction is concerned with problems from pluripotential theory in the complex Euclidean space. The questions to be considered involve geometric properties of positive closed currents and their approximation by analytic varieties, the study of pluricomplex Green functions and their connection to problems in algebraic geometry. A third direction of research is to analyze the behavior of polynomials along transcendental analytic varieties and to study the algebraic independence of entire functions. It is expected this will continue to have applications to transcendental number theory. The project also contains problems from complex dynamics, concerning polynomial automorphisms of complex Euclidean spaces, where pluripotential theory provides important tools.Complex analysis and potential theory are central areas of Mathematics. Over the years, they have provided methods and powerful tools that helped solve many important problems from other fields of pure and applied Mathematics, as well as from Physics, Biology, etc. Thanks to the powerful methods of complex analysis, it has been often the case that progress is made in the study of concrete problems by formulating them first in the context of complex numbers. This project deals with the developing and further applications of new techniques from complex analysis and potential theory to problems in several important areas of modern Mathematics, such as number theory, complex and algebraic geometry, dynamical systems, as well as possible applications to Mathematical Physics. The problems to be studied belong to the main stream of current research in several complex variables. Making progress on these problems will contribute to the advancement of knowledge and understanding in the field. The proposed research will impact human resources development through summer funding of two graduate students. They will work for their dissertation under the investigator?s supervision on topics related to this project. In this way the project integrates research and education.