Pluripotential Theory and Applications to Complex Geometry and Number Theory

本研究项目涉及复数分析、复数几何和位势理论等领域。复数分析研究依赖于复变量的函数，很多时候通过在复数的背景下考虑具体问题来回答这些问题。复数分析和位势理论是现代数学的核心，它们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(例如图像和信号处理)和物理(例如量子力学；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。这个数学研究项目涉及自然产生的、对复杂几何或超越数论有重要应用的多势论的问题。一个统一的主题是，所提出的问题集中在多重亚谐函数和正闭合电流作为研究对象或作为一些可使用的工具。第一个研究方向是关于复空间上的量子化问题。它们在统计物理(量子混沌)以及数论(模形式的量子唯一遍历性)中都有应用。科曼将考虑复空间上奇异厄米全纯线丛的序列，并将研究使用这些度量数据定义的平方可积全纯截面的Bergman空间。特别是，他将研究Bergman核函数的渐近性和与这些空间相关的Fubini-Study电流的收敛，以及全纯截面的m元组随机序列的公共零点的渐近分布。在单个线丛的幂序列的特殊情况下，Coman将研究部分Bergman核的渐近性，该部分Bergman核对应于沿复超曲面消失到高阶的全纯截面空间。第二个研究方向是紧致Kaehler流形上的多势理论问题。这里有一些与当地环境不同的有趣的新现象。其目的是描述复Monge-Ampere算子的定义域，并研究相应的格林函数及其奇性。Coman还将考虑环境流形的解析子簇上的(拟)多重次调和函数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，并考虑正闭流的几何性质及其解析簇的逼近问题，以及多项式沿超越解析簇的行为问题。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。

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.