Pluripotential Theory and Applications to Complex Geometry and Number Theory

这个数学研究项目由丹科曼地址问题从多能理论，其中一些自然出现或有重要的应用在复杂的几何或超越数论。一个统一的主题是，建议的问题集中在pluisubharmonic函数和积极的封闭电流作为调查对象或一些工具。研究的一个方向涉及复流形上的量子化问题。给定流形上的奇异Hermitian全纯线丛，存在由度量数据定义的平方可积全纯截面的自然Hilbert空间。科曼，将研究收敛的序列Fubini研究电流的这些希尔伯特空间和渐近分布的随机全纯部分的高张量权力的线束。科曼也将在复空间上的线丛的更一般情况下考虑这些问题。这在统计物理学（量子混沌）和数论（模形式的量子唯一遍历性）中有应用。第二个研究方向将研究紧致复流形上的多能理论问题，其中有新的有趣现象，不同于局部设置。主要目标是研究复Monge-Ampere算子、相应的绿色函数及其奇异性，以及（拟）多重次调和函数在环境流形的解析子簇上的延拓和正则化问题。第三个方向的研究涉及的问题，从多能理论在复杂的欧几里德空间。所要考虑的问题涉及几何性质的正封闭电流和他们的近似解析品种，和多项式的行为沿着超越解析品种。预计后者将继续应用于超越数论，例如研究整函数值的代数独立性。该数学研究项目属于复分析和势理论领域;这些学科是数学的中心领域，为解决纯数学和应用数学其他领域的重要问题提供了强大的工具（例如，图像和信号处理）和物理学（例如，量子力学;统计物理学）。在这个项目的研究问题上取得进展将有助于在这些领域的知识和理解的进步。该项目研究新技术的发展和应用，从复分析和势理论到数学物理，复几何和代数几何，数论等重要领域的问题。由于强大的复分析方法，通常情况下，首先在复数的背景下制定具体问题，从而取得进展。该项目将通过夏季资助两名研究生来影响人力资源的开发，他们将在调查员的监督下就与该项目有关的主题撰写论文。通过这种方式，该项目有效地结合了研究和教育。

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.