Random Holomorphic Sections and Complex Geometry

伯纳德Shiffman将继续他的研究应用pluripotential理论和伯格曼Szego内核的统计随机函数的几个复杂的变量和更普遍的随机部分的积极线丛紧凑复杂的流形。 该项目的主要重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，充分线丛的全纯截面和整函数的理解。 在随机部分的研究的一个基本组成部分是伯格曼-Szego内核，这个项目涉及检查这个内核的大功率的线丛的渐近性。 Shiffman将调查渐近统计多项式的程度增加，全纯部分增加权力的一个线丛，或在整个功能的情况下，域的大小增加。 一个问题是估计“空洞概率”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还将研究真实的和复杂的“fewnomial”系统的零点分布--多项式的高次少项。他还将研究球谐函数随维数增加的临界点的渐近分布。随机函数的临界点和零点的分布与物理学、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，经常需要处理无序，即在系统中插入一定量的随机性。 随机多项式为许多系统提供了基本模型，例如原子和分子及其组成粒子-质子，中子和电子的系统。 量子力学通过波函数描述这些粒子，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线。 多变量多项式对应于具有多个自由度的系统，并且那些高次多项式对应于高激发态的波函数。 虽然Shiffman最近的研究主要是为了证明这种大型系统的平均状态是典型的，就像大数定律一样，但这个项目还将包括对“罕见事件”的研究，这些事件可以应用于当前感兴趣的各个领域，例如经济学和极端气候的研究。

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.