Random Holomorphic Sections and Complex Geometry

随机全纯截面和复杂几何

• 批准号: 0600982
• 负责人: Bernard Shiffman
• 金额: $ 20.68万
• 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: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0600982&HistoricalAwards=false
• 关键词: Random; Holomorphic; Sections; Complex; Geometry

Bernard Shiffman will continue his research on the statistics of random functions of several complex variables and more generally of random sections of powers of positive line bundles on compact complex manifolds. 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. The research focuses on asymptotic results 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 find the variance of the number of simultaneous zeros of systems of random polynomials or entire functions in a domain. Another problem is to estimate "hole probabilities;" i.e., the probabilities that a random function has no solutions (or critical points) in fixed domains. 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. Shiffman will also study the distribution of zeros of real and complex "fewnomial" systems---polynomials of high degree with few terms---and he will investigate similar problems for spherical harmonics. The research program on critical points has two aspects: studying how the metric on a positive line bundle on a compact complex manifold affects the distribution of critical points of holomorphic sections, and secondly, how to count the distribution of vacua in string theory.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. One then looks for a way to statistically describe a random "landscape"---the hills and valleys of the graph of a function of several variables. One aspect of the geometry of the landscape that this project studies is the distribution of the local maxima (peaks), local minima, and saddle points (passes), the totality of which are called "critical points." 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. Another application of the research is to the "string theory landscape." String theory, which is a framework that unifies general relativity and quantum mechanics, postulates that space has 6 hidden dimensions rolled up into a small compact shape, called a Calabi-Yau manifold. This manifold is described by many parameters, whose possible values are given as critical points of a function. This project investigates the enumeration and distribution of the critical points corresponding to possible descriptions of our universe.

伯纳德·希夫曼（Bernard Shiffman）将继续他对几个复杂变量的随机函数的统计数据，更普遍地是紧凑型复杂流形的正线束的随机段。 研究的目标之一是完善我们对零分布和多项式临界点的分布的理解，充分的线条束的全态部分以及整个功能。 该研究的重点是增加程度的多项式，线条束的增加的尸体形态部分，或在整个功能的情况下，在大小增加的域上。 一个问题是找到随机多项式系统或域中整个功能的同时零的数量的差异。 另一个问题是估计“孔概率”；即，随机函数在固定域中没有解决方案（或关键点）的概率。 随机部分研究中的一种基本要素是伯格曼 - 塞格内核，该项目涉及对该核的渐近学检查。 Shiffman还将研究真实和复杂的“少数”系统的零 - 高度多项式的零，几乎没有术语 - 他将研究球形谐波的类似问题。 有关关键点的研究计划有两个方面：研究紧凑型复合歧管上的正线束上的指标如何影响全体形态部分的临界点的分布，其次，如何计算弦乐理论中真空的分布。随机函数的关键点和零在物理，信号和图像和图像处理中，随机功能的分布与许多领域相关。 在物理科学中，通常有必要处理障碍，其中将一定数量的随机性插入到系统中。 然后，人们正在寻找一种方法来统计描述随机的“景观” ---几个变量函数的丘陵和山谷。 该项目研究的景观几何形状的一个方面是局部最大值（峰值），局部最小值和鞍点（通过）的分布，其总数称为“关键点”。 随机多项式为许多系统提供了基本模型，例如原子和分子系统及其组分颗粒---质子，中子和电子。 量子力学通过波函数描述了这些粒子，这些粒子是施罗丁方程的解决方案。 波功能的零和局部最大值提供了有关原子和分子状态的重要信息。零在量子化学和物理学中已知为淋巴结线。 几个变量中的多项式对应于具有几个自由度的系统，而高度的多项式对应于高度激发态的波函数。 该研究的另一个应用是“弦理论景观”。 字符串理论是一个统一一般相对论和量子力学的框架，它假定空间具有6个隐藏的维度，使其滚成一个小的紧凑形状，称为calabi-yau歧管。 该歧管由许多参数描述，其可能的值将其作为函数的临界点。 该项目调查了与我们宇宙可能描述相对应的关键点的枚举和分布。