Random Holomorphic Sections and Complex Geometry

文摘：Bernard Shiffman将继续研究多个复变量的随机多项式的统计量，以及更一般的紧复流形和几乎复辛流形上正线丛幂的随机截面的研究。他将研究“标度极限”统计，当距离被重新标度，使得密度归一化时，多项式的次数或线丛的幂变成无穷大。他将研究零点的间距、空穴概率、局部极大值的配对相关性，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，他将研究等维亚纯映射到紧复流形的紧奇性。他还将寻找复杂射影三维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全纯曲线；简单的例子是亏格大于1的笛卡尔正方形曲线和亏格大于2的一般曲线的对称正方形。他将在复射影空间中寻找这些曲面的低次双曲双曲映象。这项研究项目的动机是需要了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--行为的基本理论。这些粒子用波函数来描述，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。随机多项式的行为提供了一个类似于复杂量子系统的基本模型。多元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。该项目包括关于辛流形的统计，作为量子系统状态的数学模型。这项研究的另一个组成部分涉及理解复代数流形的几何，它在量子场论中扮演着重要的角色，并为各种物理现象提供演示。

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.