Random Holomorphic Sections and Complex Geometry

随机全纯截面和复杂几何

• 批准号: 1201372
• 负责人: Bernard Shiffman
• 金额: $ 27.4万
• 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: 2012
• 资助国家: 美国
• 起止时间: 2012-06-01 至 2016-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1201372&HistoricalAwards=false
• 关键词: Random; Holomorphic; Sections; Complex; Geometry

The principal focus of the project is the interplay between complex geometry and probability. In particular, 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. 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 using curvature invariants to describe the off-diagonal asymptotics of this kernel for large powers of the line bundle. Shiffman will apply the off-diagonal Bergman kernel asymptotics to obtain optimal sup-norms for orthonormal bases of spaces of holomorphic sections of powers of ample line bundles. He will also continue his investigation of the distribution of random zeros of systems of polynomials, or more generally random sections, in order to obtain central limit theorems for the numbers of zeros in smooth domains as the degree increases. He will investigate the zeros of random polynomials of increasing degree containing a fixed number of monomial terms. Shiffman will also study point processes given by critical points of random holomorphic sections.In the physical sciences it is often necessary to handle disorder, where a certain amount of randomness is inserted into a system. Random functions can be used to model 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 can be used to study systems with several degrees of freedom, and those polynomials of high degree correspond to wave functions for high energy states. The mathematics of point processes--the spatial and/or time distribution of random occurrences--has been used in many diverse fields such as signal and image processing, quantum mechanics, epidemiology, seismology, astronomy, and economics. This mathematics research project includes the development of geometric methods to study the statistics of point processes arising from mathematical equations with some random input.

该项目的主要重点是复杂的几何形状和概率之间的相互作用。 尤其是，伯纳德·希夫曼（Bernard Shiffman）将继续他对多个复杂变量的随机函数的统计数据以及在紧凑型复杂歧管上的正线捆绑包的随机段的随机函数的统计数据继续研究。 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。 随机部分研究中的一种基本要素是伯格曼 - 塞格内核，该项目涉及使用曲率不变性来描述该核的非对角线渐近线，以实现线条束的大幂。 Shiffman将应用非对角线的Bergman内核渐近学，以获得最佳的SUP-NORM，以用于宽敞线条束的全体形态截面的正顺序基础。 他还将继续研究多项式系统系统的随机零或更一般的随机切片的分布，以便获得平滑域中的零零数的中心极限定理，随着程度的增加。 他将研究包含固定数量单项项的随机多项式的零。 什夫曼还将研究由随机霍明型部分的临界点给出的点过程。在物理科学中，经常有必要处理障碍，其中将一定量的随机性插入到系统中。 随机功能可用于建模许多系统，例如原子和分子系统及其成分颗粒 - protons，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可用于研究具有多个自由度的系统，而那些高度的多项式对应于高能状态的波函数。 点过程的数学（随机发生的空间和/或时间分布）用于许多不同的领域，例如信号和图像处理，量子力学，流行病学，地震学，天文学和经济学。该数学研究项目包括开发几何方法，以研究用一些随机输入的数学方程式产生的点过程统计数据。