This mathematics research project by Dan Coman addresses problems from pluripotential theory, some of which arise naturally or have important applications in complex geometry or in transcendental number theory. A unifying theme is that the proposed problems focus on plurisubharmonic functions and positive closed currents as objects of investigation or some of the tools to be employed. One direction of research deals with quantization problems on complex manifolds. Given a singular Hermitian holomorphic line bundle on the manifold, there are natural Hilbert spaces of square-integrable holomorphic sections defined using the metric data. Coman, will study the convergence of the sequence of Fubini-Study currents of these Hilbert spaces and the asymptotic distribution of simultaneous zeros of random holomorphic sections of the high tensor powers of the line bundle. Coman will also consider these problems in the more general case of line bundles over complex spaces. This has applications in statistical physics (quantum chaos), as well as in number theory (quantum unique ergodicity for modular forms). A second direction of research will study 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, the corresponding Green functions and their singularities, and the problem of extension and regularization of (quasi) plurisubharmonic functions on analytic subvarieties of the ambient manifold. A third direction of research deals 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, and 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.
This mathematics research project is in the areas of complex analysis and potential theory; these subjects are central areas of mathematics, providing 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. The project investigates the development and applications of new techniques from complex analysis and potential theory to problems in important areas such as mathematical physics, complex and algebraic geometry, number theory. Thanks to the powerful methods of complex analysis, it has been often the case that progress is made by formulating concrete problems at first in the context of complex numbers. The project will impact the development of human resources through summer funding of two graduate students who will work for their dissertation under the investigator's supervision on topics related to this project. In this way the project effectively integrates research and education.
