Pluripotential Theory and Applications to Complex Geometry and Number Theory

丹·科曼的这项数学研究项目解决了多势论中的问题，其中一些问题是自然产生的，或者在复杂几何或超越数论中有重要应用。一个统一的主题是，所提出的问题集中在多个亚谐函数和正闭合电流作为研究对象或使用的一些工具。研究的一个方向是复流形上的量子化问题。给定流形上的奇异Hermite全纯线丛，利用度量数据定义了具有平方可积全纯截面的自然Hilbert空间。Coman，将研究这些Hilbert空间的Fubini学习电流序列的收敛，以及线丛的高张量幂的随机全纯截面的同时零点的渐近分布。科曼还将在复杂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(量子混沌)以及数论(模形式的量子唯一遍历性)中都有应用。第二个研究方向将研究紧致复流形上的多势理论问题，在紧致复流形上存在着不同于局部环境的新的有趣现象。主要目的是研究复Monge-Ampere算子、相应的Green函数及其奇性，以及(拟)多重亚调和函数在环境流形的解析子簇上的延拓和正则化问题。第三个研究方向涉及复欧氏空间中的多势理论问题。要考虑的问题涉及正闭合电流的几何性质及其解析簇的逼近，以及多项式在超越解析簇上的行为。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无关性。这项数学研究项目是在复分析和位势理论领域；这些学科是数学的中心领域，为解决其他领域的重要问题提供了强大的工具，这些领域包括纯数学和应用数学(如图像和信号处理)和物理(如量子力学；统计物理)。在这一项目的研究问题上取得进展将有助于增进对这些领域的认识和理解。该项目研究新技术的发展和应用，从复数分析和位势理论到数学物理、复数和代数几何、数论等重要领域的问题。由于复杂分析的强大方法，通常情况下，进步是通过首先在复数的背景下阐述具体问题来取得的。该项目将通过夏季资助两名研究生来影响人力资源的发展，这两名研究生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，该项目有效地将研究和教育结合在一起。

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.