Geometry of Low Dimensional Manifolds

Anderson和LeBrun计划在低维流形上研究一组相关的几何结构。他们的研究重点是发现和发展黎曼几何、微分拓扑和理论物理问题之间的基本联系。安德森将研究有关共形紧致爱因斯坦度量和AdS/CFT对应，有界域上爱因斯坦度量的结构和广义相对论的数学方面的问题。同时，LeBrun将研究4流形上正则度量的存在性和模，重点是极值Kahler度量、Seiberg-Witten理论和全纯盘的扭转几何。本研究项目旨在探索微分几何数学领域的基本问题，同时发现数学与理论物理之间的新联系。许多计划中的研究活动的灵感来自于目前试图弥合爱因斯坦引力理论与量子场论之间鸿沟的尝试。量子场论描述了微观尺度上的自然力。一些研究涉及描述由爱因斯坦引力场方程控制的四维宇宙的所有可能几何形状的问题。研究计划的其他方面与弦理论的最新发展密切相关。通过培养一批研究生从事这一领域的研究，该项目还将通过其长期的教育影响，帮助促进数学和物理在人类层面上的直接互动。

Anderson and LeBrun plan to study a cluster of related geometric structures on low-dimensional manifolds. Their research program focuses on the discovery and development of fundamental connections between Riemannian geometry, differential topology, and issues in theoretical physics. Anderson will investigate problems concerning conformally compact Einstein metrics and the AdS/CFT correspondence, the structure of Einstein metrics on bounded domains and mathematical aspects of general relativity. Meanwhile, LeBrun will study the existence and moduli of canonical metrics on 4-manifolds, with an emphasis on extremal Kahler metrics, Seiberg-Witten theory, and the twistor geometry of holomorphic disks.This research program aims both to explore fundamental issues in the mathematical field of differential geometry, and simultaneously to discover new links between mathematics and theoretical physics. Much of the planned research activity takes its inspiration from current attempts to bridge the gulf separating Einstein's theory of gravitation from the quantum field theories that describe the forces of nature on a microscopic scale. Some of the research concerns the problem of describing all possible geometries of 4-dimensional universes governed by Einstein's gravitational field equations. Other aspects of the research program are intimately linked to recent developments in string theory. By also training a group of graduate students to pursue research in this area, the project will additionally help foster interactions between mathematics and physics on an immediate, human scale, through its long-term educational impact.