Pluripotential Theory and Applications to Complex Geometry and Number Theory

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This research project is in the areas of complex analysis, complex geometry and potential theory. Complex analysis deals with the study of functions that depend on complex variables, and many times concrete questions were answered by considering them in the context of complex numbers. Complex analysis and potential theory are central to modern mathematics and they provide powerful tools for solving important problems from other fields of pure and applied mathematics (e.g., image and signal processing) and physics (e.g., quantum mechanics; statistical physics). Making progress on the research problems in this project will contribute to the advancement of knowledge and understanding in these fields. This mathematics research project deals with problems from pluripotential theory which arise naturally and have important applications to complex geometry or to transcendental number theory. A unifying theme is that the proposed problems focus on plurisubharmonic functions and on positive closed currents as objects of investigation or as some of the tools to be employed. The first direction of research is concerned with quantization problems on complex spaces. These have applications to statistical physics (quantum chaos), as well as to number theory (quantum unique ergodicity for modular forms). Coman will consider sequences of singular Hermitian holomorphic line bundles over complex spaces, and will study the Bergman spaces of square-integrable holomorphic sections defined using this metric data. In particular, he will study the asymptotics of the Bergman kernel functions and the convergence of the Fubini-Study currents associated to these spaces, and the asymptotic distribution of common zeros of random sequences of m-tuples of holomorphic sections. In the special case of the sequence of powers of a single line bundle, Coman will study the asymptotics of partial Bergman kernels corresponding to spaces of holomorphic sections of vanishing to high order along a complex hypersurface. The second direction of research addresses problems in pluripotential theory on compact Kaehler manifolds. Here there are some interesting new phenomena different from the local setting. The goals are to describe the domain of definition of the complex Monge-Ampere operator and to study the corresponding Green functions and their singularities. Coman will also consider the problem of extension and regularization of (quasi) plurisubharmonic functions on analytic subvarieties of the ambient manifold. The third direction of research deals with problems from pluripotential theory in complex Euclidean spaces and considers questions about geometric properties of positive closed currents and their approximation by analytic varieties, and about the behavior of polynomials along transcendental analytic varieties. It is expected that the latter will continue to have applications to transcendental number theory, such as to the study of the algebraic independence of values of entire functions.