Global Topological and Geometric aspects of AdS/CFT backgrounds.

目的/目标：该项目的主要目的是将爱因斯坦引力理论的10维和11维的所有解分类，其中以某种方式包括反de-Sitter(ADS)子空间。ADS空间是黎曼双曲空间的洛伦兹模拟。这个问题大约在40年前就出现了，并因为Thead/CFT的通信而被赋予了新的冲动。这是引力场理论和共形场理论(CFT)之间的关系。这对应地将规范理论中的强耦合效应与引力理论中ADS背景的某种行为联系起来。博士项目的目标是对特殊情况下的AdS5背景进行分类。由于对偶规范理论是4维的，这是需要考虑的最重要的情况，因为4维理论是我们理解自然的关键。方法论：这是数学物理中的一个问题。尽管这个问题已经确立，而且仍然悬而未决，但已经开发了一种新的方法来解决它。该方法包括使用微分几何，特别是微分流形理论。代数拓扑学和分析，如上同调理论和霍普夫最大值原理，以及其他在流形上全局求解微分方程的技术将被部署。流形理论的其他方面，如自旋结构和广义几何，也将被用来研究解的几何。

Aims/Objectives: The main aim of the project is to classify all solutions of the Einstein equations of gravity theories in 10- and 11-dimensions which include in a certain way a Anti-de-Sitter (AdS) subspace. AdS spaces are the Lorentzian analogue of Riemannian hyperbolic spaces. This question has arisen about 40 years ago and has been given a new impetuous because of theAdS/CFT correspondence. This is a relationship between gravitational and conformal field theories (CFT). This correspondencerelates strong coupling effects in gauge theories with a certain behaviour of AdS backgrounds in gravitational theories. The objective of the PhD project is to classify the special case AdS5 backgrounds. As the dual gauge theories are 4-dimensional, this is the most important case to be considered because 4-dimensional theories are key in our understanding of nature. Methodology: This is a problem in Mathematical Physics. Although the question is well established and remains open, a new approach has been developed to solve it. The methodology includes the use of Differential Geometry and in particular the theory of Differential Manifolds. Algebraic Topology and Analysis, like Cohomology theory and the Hopf maximum principle, and other techniques for solving differential equations globally on Manifolds will be deployed. Other aspects of manifold theory like Spin Structures and Generalized Geometry will also be employed to investigate the geometry of the solutions.