Random Holomorphic Sections and Complex Geometry

随机全纯截面和复杂几何

• 批准号: 0901333
• 负责人: Bernard Shiffman
• 金额: $ 30.04万
• 依托单位: Johns Hopkins University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-06-01 至 2013-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901333&HistoricalAwards=false
• 关键词: Random; Holomorphic; Sections; Complex; Geometry

Bernard Shiffman will continue his research on applications of pluripotential theory and the Bergman-Szego kernel to the statistics of random functions of several complex variables and more generally of random sections of positive line bundles on compact complex manifolds. The principal focus of the project is the interplay between geometry and probability. One of the goals of the research is to refine our understanding of the distributions of zeros and critical points of polynomials, holomorphic sections of ample line bundles, and entire functions. A fundamental ingredient in the study of random sections is the Bergman-Szego kernel, and this project involves an examination of the asymptotics of this kernel for large powers of the line bundle. Shiffman will investigate asymptotic statistics for polynomials of increasing degree, holomorphic sections of increasing powers of a line bundle, or in the case of entire functions, on domains of increasing size. One problem is to estimate "hole probabilities" and "overcrowding probabilities" (i.e., the probabilities that random systems of equations have no solutions or too many solutions in fixed domains). Shiffman will also study the distribution of zeros of real and complex "fewnomial" systems--polynomials of high degree with few terms. He will also investigate the asymptotic distribution of critical points for spherical harmonics as the dimension increases.The distribution of critical points and zeros of random functions is relevant to many areas in physics, signal and image processing, and other areas of engineering. In the physical sciences it is often necessary to handle disorder, where a certain amount of randomness is inserted into a system. Random polynomials provide an elementary model for many systems, such as systems of atoms and molecules and their component particles--protons, neutrons, and electrons. Quantum mechanics describes these particles by wave functions, which are solutions of Schrodinger's equation. The zeros and local maxima of wave functions give important information on states of atoms and molecules; the zeros are known in quantum chemistry and physics as nodal lines. Polynomials in several variables correspond to systems with several degrees of freedom, and those polynomials of high degree correspond to wave functions for highly excited states. While Shiffman's recent research was concerned primarily with demonstrating that average states of such large systems are typical, as in the law of large numbers, this project will also include the study of "rare events," which has applications to various areas of current interest, such as economics and the study of climatic extremes.

Bernard Shiffman将继续研究多元势理论和Bergman-Szego核在复数变量随机函数和紧复流形上正线束随机截面统计中的应用。该项目的主要焦点是几何和概率之间的相互作用。研究的目标之一是完善我们对多项式的零点和临界点的分布、充足线束的全纯截面和整个函数的理解。随机截面研究中的一个基本成分是Bergman-Szego核，本项目涉及对线束的大幂次核的渐近性的检验。Shiffman将研究级数递增的多项式的渐近统计量，线束幂递增的全纯截面，或在整个函数的情况下，在尺寸递增的域上。一个问题是估计“空穴概率”和“过度拥挤概率”（即随机方程组在固定域中没有解或有太多解的概率）。Shiffman还将研究实的和复杂的“少项”系统的零点分布——高次多项式的少项。他还将研究球面谐波的临界点随维数增加的渐近分布。随机函数的临界点和零点分布与物理、信号和图像处理以及其他工程领域的许多领域有关。在物理科学中，通常有必要处理无序，其中一定数量的随机性被插入系统中。随机多项式为许多系统提供了一个基本模型，例如原子和分子及其组成粒子——质子、中子和电子的系统。量子力学用波函数来描述这些粒子，波函数是薛定谔方程的解。波函数的零点和局部极大值给出了关于原子和分子状态的重要信息；在量子化学和物理学中，零点被称为节点线。几个变量的多项式对应于几个自由度的系统，而那些高阶的多项式对应于高激发态的波函数。虽然Shiffman最近的研究主要关注于证明这种大型系统的平均状态是典型的，就像在大数定律中一样，这个项目也将包括对“罕见事件”的研究，这将应用于当前感兴趣的各个领域，比如经济学和极端气候的研究。