Random Holomorphic Sections and Complex Geometry

Bernard Shiffman将继续他的关于几个复变量的随机函数的统计研究，更广泛地说，关于紧复流形上正线束的随机幂部分的统计研究。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、充足线束的全纯截面和整个函数的理解。本文主要研究了次递增多项式的渐近结果、线束幂递增全纯截面的渐近结果，以及整个函数在大小递增区域上的渐近结果。一个问题是在一个定义域内找出随机多项式系统或整个函数的同时零的数量的方差。另一个问题是估计“空穴概率”，即随机函数在固定域内无解（或临界点）的概率。随机截面研究的一个基本要素是Bergman-Szego核，本项目涉及对该核的渐近性的检验。Shiffman还将研究实数和复杂的“少项”系统的零点分布——高次多项式的少项——他还将研究球面谐波的类似问题。关于临界点的研究计划有两个方面：一是研究紧复流形上正线束上的度规如何影响全纯截面上临界点的分布；二是如何计算弦理论中真空的分布。随机函数的临界点和零点分布与物理、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，通常有必要处理无序，其中一定数量的随机性被插入系统中。然后寻找一种统计描述随机“景观”的方法——几个变量函数的图形的山丘和山谷。该项目研究的景观几何的一个方面是局部最大值（峰值）、局部最小值和鞍点（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——质子、中子和电子的系统。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给出了关于原子和分子状态的重要信息；在量子化学和物理学中，零点被称为节点线。几个变量的多项式对应于几个自由度的系统，而那些高阶的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”。弦理论是统一广义相对论和量子力学的框架，它假设空间有6个隐藏的维度，卷成一个小而紧凑的形状，称为卡拉比-丘流形。这个流形由许多参数来描述，这些参数的可能值作为函数的临界点给出。这个项目调查了与我们宇宙的可能描述相对应的临界点的枚举和分布。

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.