Geometric Analysis, Wave Optics, and Geometric Gravity Models

摘要奖：DMS-0707003主要研究者：Arlie O.这个研究项目的目标是发展一个天体物理学理论的阴影模式投在宇宙中的作用引力对光。 该项目的目的是表征这些阴影模式的逐点总放大率和放大截面函数，无论是在几何和波动光学slimits的某些关键重力模型。 这项研究需要几何分析焦散奇点，diffractionintegrals，和渐近在一个弯曲的时空设置。 该研究为将阴影模式的研究推广到任意洛伦兹流形奠定了基础，从而为洛伦兹几何的研究提供了新的思路. 这些发现还将产生一个详细的引力透镜比较当前关键的几何引力模型，如广义相对论，PPN模型和膜世界引力，允许预测，可能会测试这些引力模型。我们开车去上班的路上穿过它们，在游泳池里游泳。 类似的模式也被恒星、星系、黑洞、暗物质和暗能量的引力场投射到整个宇宙中。也许更神秘的是，这些阴影模式与我们物理世界本质的一个深刻问题有关--即物理空间是否比我们熟悉的长、宽和高三维空间更多。 这项研究将利用引力的宇宙阴影模式来探索黑洞和暗物质的性质，以及物理空间的维度。 这项工作也将制定预测，测试爱因斯坦的广义相对论和其他重力模型。 该项目是跨学科的，并创造了数学，物理学和天文学之间的协同作用。

AbstractAward: DMS-0707003Principal Investigator: Arlie O. PettersThe goal of this research project is to develop a mathematicaltheory of the shadow patterns cast in the cosmos by the action ofgravity on light. The project aims to characterize the pointwisetotal magnification and magnification cross section functions ofthese shadow patterns, both in the geometric and wave opticslimits for certain key gravity models. This study requires ageometric analysis of caustic singularities, diffractionintegrals, and asymptotics in a curved spacetime setting. Theproposed research would lay a platform for generalization a studyof shadow patterns to arbitrary Lorentzian manifolds, which inturn would lead to a new exploration in Lorentzian geometry. Thefindings will also yield a detailed gravitational lensingcomparison of current key geometric gravity models like generalrelativity, PPN models, and braneworld gravity, allowing forpredictions that may test these gravity models.Shadow patterns are all around us. We drive through them on theway to work and swim among them in pools. Similar patterns arealso cast throughout the universe by the gravitational fields ofstars, galaxies, black holes, dark matter, and dark energy.Perhaps even more mysterious is that these shadow patterns areconnected with one of the profound questions about the nature ofour physical world -- namely, whether physical space has morethan the familiar three spatial dimensions of length, width, andheight. This research will employ gravity's cosmic shadowpatterns to explore the nature of black holes and dark matter,and the dimensionality of physical space. The work will alsoformulate predictions that test Einstein's general theory ofrelativity and other gravity models. The project isinterdisciplinary and creates a synergistic interaction betweenmathematics, physics, and astronomy.