Pluripotential Theory and Applications to Complex Geometry and Number Theory

该研究项目是在复分析，复几何和潜在的理论领域。复分析研究的是依赖于复变量的函数，很多时候，具体的问题都是在复数的背景下考虑的。复分析和势理论是现代数学的核心，它们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。这个数学研究项目涉及自然产生的多能理论问题，并对复几何或超越数论有重要应用。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作为一些工具。第一个研究方向是关于复空间上的量子化问题。这些理论在统计物理学（量子混沌）和数论（模形式的量子唯一遍历性）中都有应用。科曼将考虑复空间上的奇异埃尔米特全纯线丛序列，并将研究使用此度量数据定义的平方可积全纯截面的伯格曼空间。特别是，他将研究伯格曼核函数的渐近性和与这些空间相关的Fubini-Study流的收敛性，以及全纯部分的m元组随机序列的公共零点的渐近分布。在特殊情况下的一个单一的线丛的权力序列，科曼将研究渐近的部分伯格曼核对应的空间的全纯部分消失到高阶沿着一个复杂的超曲面。第二个方向的研究解决问题的多能理论紧凑Kaehler流形。这里有一些有趣的新现象，不同于当地的设置。目标是描述复Monge-Ampere算子的定义域，并研究相应的绿色函数及其奇异性。科曼还将考虑问题的延伸和正规化的（准）pluisubharmonic函数解析子品种的环境流形。第三个方向的研究涉及的问题，从pluripotential理论在复杂的欧几里德空间，并认为有关几何性质的积极封闭的电流和他们的近似解析品种的问题，以及有关的行为多项式沿着超越解析品种。预计后者将继续有应用超越数论，如研究的代数独立性的价值观的整个职能。

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.