Pluripotential Theory and Applications to Complex Dynamics and Number Theory

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Abstract for Proposal DMS-0500563, PI: Dan ComanThis project addresses problems from potential theory and dynamics in several complex variables. The problems in complex dynamics use tools from pluripotential theory and some are likely to require developing new such tools. The first direction of research deals with the study of the fundamental solutions of the complex Monge-Ampere operator, which are called pluricomplex Green functions. This will have applications in understanding the singularities of currents and to some questions in algebraic geometry. The second direction of research deals with the dynamics of polynomial automorphisms of complex Euclidean spaces, in the regular case, and for special classes of irregular mappings. The main interest lies in obtaining a detailed understanding of the dynamics, and in the study of the ergodic properties of the dynamical Green currents and measures. The third direction of research of this project is to analyze the behavior of polynomials along transcendental analytic varieties. This will be used to study arithmetic properties of entire functions, and the algebraic independence of sets of values of transcendental functions. Complex analysis and potential theory are central areas of Mathematics. Over the years, they have provided methods and powerful tools which helped to solve many important problems from other fields of pure and applied Mathematics, as well as from Physics, Biology, Economics, etc. This project deals with the developing and further application of new techniques from analysis and potential theory in several complex variables to problems in dynamical systems and number theory. 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.