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.

Bernard Shiffman将继续研究几个复变量的随机函数的统计量，以及更一般的紧致复流形上正线丛幂的随机截面的统计量。这项研究的目的之一是加深我们对多项式的零点和临界点的分布、充分线丛的全纯截面和整函数的理解。研究集中于递增次多项式、线丛的递增幂全纯截面或整函数在递增区域上的渐近结果。一个问题是求一个域中的随机多项式系统或整函数系统的同时零点个数的方差。另一个问题是估计“空洞概率”，即随机函数在固定区域内没有解(或临界点)的概率。研究随机截面的一个基本成分是Bergman-Szego核，这个项目涉及到对这个核的渐近性的检查。希夫曼还将研究实数和复数“几项式”系统的零点分布-少项高次多项式-他还将研究球谐函数的类似问题。研究紧致复流形上正线丛上的度量对全纯截面的临界点分布的影响，以及弦理论中真空分布的计算两个方面是关于临界点研究的两个方面。随机函数的临界点和零点的分布涉及到物理学、信号和图像处理等工程领域的许多领域。在物理科学中，通常需要处理无序，即在系统中插入一定数量的随机性。然后，人们寻找一种统计方法来描述一幅随机的“风景”-多变量函数图的起伏。这个项目研究的景观几何的一个方面是局部最大值(峰)、局部极小点和鞍点(通道)的分布，它们的总和被称为“临界点”。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的系统-质子、中子和电子。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提供了有关原子和分子状态的重要信息；在量子化学和物理学中，零点被称为节线。多变量多项式对应于具有多个自由度的系统，而高次多项式对应于高激发态的波函数。这项研究的另一个应用是“弦理论景观”。弦理论是广义相对论和量子力学的统一框架，它假定空间有6个隐藏的维度卷成一个小的紧凑形状，称为卡拉比-尤流形。这种流形由许多参数描述，这些参数的可能值作为函数的临界点给出。这个项目调查与我们宇宙的可能描述相对应的临界点的列举和分布。