Random Holomorphic Sections and Complex Geometry

该项目的主要重点是复杂几何和概率之间的相互作用。 特别是，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。 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。 随机截面研究中的一个基本组成部分是Bergman-Szego核，这个项目涉及使用曲率不变量来描述这个核的非对角渐近性。 Shiffman将应用非对角Bergman核渐近性来获得充足线丛的幂的全纯截面的空间的正交基的最优次范数。 他还将继续他的调查随机零点的系统的多项式，或更普遍的随机部分，以获得中央极限定理的数量零光滑域的程度增加。 他将调查零随机多项式的增加程度包含一个固定数目的单项条款。 Shiffman还将研究点过程的临界点随机全纯部分。在物理科学中，它往往是必要的处理混乱，其中一定数量的随机性插入到一个系统。 随机函数可以用来模拟许多系统，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。 多元多项式可用于研究多自由度系统，而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几何方法的发展，以研究从数学方程产生的点过程的统计与一些随机输入。

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.