Random Holomorphic Sections and Complex Geometry

该项目的主要关注点是复杂几何和概率之间的相互作用。特别是，Bernard Shiffman将继续他的研究，将多势理论和Bergman-Szego核应用于几个复变量的随机函数的统计，以及更一般的紧致复流形上正线丛的随机截面的统计。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。研究随机截面的一个基本成分是Bergman-Szego核，这个项目涉及到使用曲率不变量来描述该核对于线丛的幂的非对角渐近。Shiffman将应用非对角线Bergman核的渐近性，得到全纯全纯截面空间的最优超范数。他还将继续研究多项式系统或更一般的随机截面的随机零点的分布，以便获得光滑域中零点个数随着次数的增加的中心极限定理。他将研究包含固定数量的单项式的递增次数随机多项式的零点。希夫曼还将研究由随机全纯截面的临界点给出的点过程。在物理学中，通常需要处理无序，即在系统中插入一定数量的随机性。随机函数可以用来模拟许多系统，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。多元多项式可用于研究具有多个自由度的系统，高次多项式对应于高能态的波函数。点过程的数学--随机事件的空间和/或时间分布--已被用于许多不同的领域，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。这个数学研究项目包括发展几何方法来研究由具有一些随机输入的数学方程产生的点过程的统计。

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.