Random Holomorphic Sections and Complex Geometry

随机全纯截面和复杂几何

• 批准号: 0100474
• 负责人: Bernard Shiffman
• 金额: $ 34.94万
• 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: 2001
• 资助国家: 美国
• 起止时间: 2001-07-01 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0100474&HistoricalAwards=false
• 关键词: Random; Holomorphic; Sections; Complex; Geometry

Abstract:Bernard Shiffman will continue his research on the statistics of randompolynomials in several complex variables and more generally of randomsections of powers of positive line bundles on compact complex manifolds andon almost complex symplectic manifolds. He will investigate the "scalinglimit" statistics as the degree of the polynomial or the power of the linebundle goes to infinity when distances are rescaled so that densities arenormalized. He will study spacing of zeros, hole probabilities, paircorrelations for local maxima, and other topics. He will also studycorrelations of critical points of spherical harmonics. In anotherdirection, he will study the compact singularities of equidimensionalmeromorphic mappings into compact complex manifolds. He will also look fornew examples of Kobayashi hyperbolic hypersurfaces in complex projective3-space. Kobayashi hyperbolic spaces do not carry any entire holomorphiccurves; simple examples are the Cartesian squares of curves of genus greaterthan 1 and symmetric squares of generic curves of genus greater than 2. Hewill look for low-degree hyperbolic birational images of these surfaces incomplex projective 3-space.This research project is motivated by a need to understand complex quantummechanical systems. Quantum mechanics is the fundamental theory thatdescribes the behavior of atoms and molecules and their componentparticles--protons, neutrons, and electrons. These particles are describedby wave functions, which are solutions of Schrodinger's equation. The zerosand local maxima of wave functions give important information on states ofatoms and molecules; the zeros are known in quantum chemistry and physics asnodal lines. The behavior of random polynomials provide an elementary modelsimilar to complex quantum systems. Polynomials in several variablescorrespond to systems with several degrees of freedom, and those polynomialsof high degree correspond to wave functions for highly excited states. Theproject includes statistics on symplectic manifolds, which serve as themathematical models for the states of quantum systems. Another component ofthe research involves understanding the geometry of complex algebraicmanifolds, which play an important role in quantum field theory and providemodels for diverse physical phenomena.

摘要：伯纳德·希夫曼 (Bernard Shiffman) 将继续研究多个复变量中的随机多项式的统计，更一般地研究紧复流形和几乎复辛流形上正线丛幂的随机部分的统计。 他将研究“缩放极限”统计数据，因为当重新缩放距离以使密度归一化时，多项式的次数或线束的幂会趋于无穷大。 他将研究零点间距、空洞概率、局部最大值的配对相关性以及其他主题。他还将研究球谐函数临界点的相关性。 在另一个方向上，他将研究等维亚纯映射到紧复流形的紧奇点。 他还将寻找复杂射影空间中小林双曲超曲面的新例子。 小林双曲空间不包含任何完整的全纯曲线；简单的例子是大于 1 的亏格曲线的笛卡尔平方和大于 2 的一般曲线的对称平方。他将在复射影 3 空间中寻找这些曲面的低次双曲双有理图像。该研究项目的动机是需要理解复杂的量子力学系统。 量子力学是描述原子和分子及其组成粒子（质子、中子和电子）行为的基础理论。 这些粒子由波函数描述，波函数是薛定谔方程的解。 波函数的零值和局部极大值提供了有关原子和分子状态的重要信息；零点在量子化学和物理学中被称为节点线。 随机多项式的行为提供了类似于复杂量子系统的基本模型。 多个变量的多项式对应于具有多个自由度的系统，而那些高次多项式对应于高激发态的波函数。 该项目包括辛流形的统计数据，作为量子系统状态的数学模型。 该研究的另一个组成部分涉及理解复杂代数流形的几何形状，它在量子场论中发挥着重要作用，并为各种物理现象提供了模型。