Random Gaussian Curvatures, Image Centroids and Caustic Surfaces in Gravitational Lensing

摘要：NSF提案，DMS-0302812（Petters）该提案提出了一个两部分的研究计划，创造了数学和天体物理学之间的协同作用。计划1旨在发展一个数学理论，以确定随机时间延迟函数临界点处高斯曲率的概率分布。 这项研究揭示了新的数学方向，将奇点理论与概率论联系起来（例如，随机多项式的零点和Rayleigh-Levy统计中的极限定理）。此外，计划1通过星系透镜特征中的通量比异常，直接应用于理解银河系中暗物质的性质。 程序2试图将薄屏弱场微透镜中焦散线附近光源的光变曲线和像心的分类扩展到克尔时空的强引力场。 这涉及到研究的全球拓扑结构，几何，和奇异的理论结构的轻焦散面在时空。 程序2指出了新的物理问题，这些问题涉及到光焦散面的几何形状和引力透镜的物质奇异性的拓扑不变量。 这个程序也可以检验爱因斯坦的广义相对论，特别是关于我们星系核中的大质量黑洞。星系晕中的暗物质和我们星系中心的黑洞是天体物理学中两个中心和紧迫的课题。这些问题直接受到几何分析的强大数学方法的影响，因为引力透镜同时是一种独特的工具，探索暗物质分布和黑洞的本质，以及建立在几何、分析和概率概念上的理论。暗物质研究需要发展一种数学理论，使人们能够区分一般的暗物质透镜特征和暗物质所用的特定选择简单透镜模型的特征。 建议的第一部分涉及制定这样一个理论。 这项工作将开辟新的数学方向，桥梁几何方面的奇点理论与几个领域的概率论（瑞利-列维统计，零的随机多项式等）。 第二部分的建议集中在爱因斯坦的广义相对论的基本预测之一，即黑洞的存在。虽然黑洞不能被直接看到，但它极其强大的引力场会扭曲黑洞周围的时空，导致设法穿过该区域的光以令人印象深刻的大角度弯曲（例如，光线在到达观测者之前可以绕黑洞无数次）。 对这种极端引力的探测将是对我们理解空间和时间本质的一次关键考验。 这个提议从数学上探索了由大质量黑洞引起的强引力场的透镜特征，人们通常认为黑洞位于我们银河系的中心。 这应该利用微分几何和奇点理论的技术，并在这些物理主题和黑洞物理之间创造协同效应。

ABSTRACT: NSF Proposal, DMS-0302812 (Petters)This proposal presents a two part research program thatcreates a synergistic interaction between mathematics and astrophysics.Program 1 aims at developing a mathematical theory to determine theprobability distributions of Gaussian curvaturesat critical points of random time-delay functions. This research programopens up new mathematical directions that bridge singularity theory withprobability theory (e.g., zeros of random polynomials andlimit theorems in Rayleigh-Levy statistics). In addition, Program 1 hasdirect applications to understanding the nature of dark matter in galactichalos via the flux ratio anomalies in the lensing signatures of galaxies. Program 2 seeks to extend the classification of light curves and imagecentroids of sources near caustics in thin-screen, weak-field microlensingto the strong gravitational field of a Kerr spacetime. This involves studying the global topology, geometry, and singular theoretic structure oflight caustic surfaces in a spacetime. Program 2 points to new mathematicalissues relating the geometry of light caustic surfaces to topologicalinvariants of the matter singularities of gravitational lenses. This programmay also yield tests of Einstein's General Theory of Relativity, especially asit pertains to the massive black hole believed to lie in the nucleus of ourgalaxy.The dark matter in galactic halos and the black hole inthe center of our galaxy are two central and pressing topics in astrophysics.These issues are directly impacted by the powerful mathematical methods ofgeometric analysis because gravitational lensing is simultaneously a uniquetool for probing the nature of dark matter distributions and black holes, and a theory that is built on geometric, analytical, and probabilistic concepts.The dark matter study requires developing a mathematical theory that allowsone to differentiate generic dark matter lensing signatures from features thatare specific to the particular choice simple lens model used for dark matter. The first part of the proposal deals with formulating such a mathematicaltheory. This work would open up new mathematical directions that bridge thegeometric aspects of singularity theory with several areas in probabilitytheory (Rayleigh-Levy statistics, zeros of random polynomials, etc.). Thesecond part of the proposal focuses on one of the fundamental predictions ofEinstein's General Theory of Relativity, namely,the existence of black holes. Though a black hole cannot be seen directly, its extremely stronggravitational field warps the spacetime about the black hole causing lightrays that managed to get through the region to be bent with impressively largeangles (e.g., rays can loop around a black hole numerous times before arrivingat the observer). The probing of such extremes of gravity would be a criticaltest of our understanding of the nature of space and time. The proposalexplores mathematically the lensing signatures of the strong gravitationalfield due to the massive black hole generally believed to be at the center ofour galaxy. This should draw upon techniques from differential geometry andsingularity theory, and create synergies between these mathematicaltopics and the physics of black holes.