Geometric Structures on Low Dimensional Manifolds

摘要奖：DMS-0604735主要研究者：Michael T.作者：安德森LeBrun安德森和LeBrun计划联合和独立地研究低维流形上的一组相关几何结构。他们的研究计划侧重于连接几何拓扑结构，代数几何和理论物理的关系。安德森和勒布伦计划共同研究四维流形上正则度规的存在性和模，重点是卡勒的情况。安德森还将研究有关渐近双曲爱因斯坦度规、AdS/CFT对应和广义相对论的数学方面的问题。LeBrun也将研究曲率的4-流形，塞伯格-威滕理论，极值Kahler度量，和twistor几何的holomorphicdisks。 拟议的工作预计将在两个不同的方面产生重大和广泛的影响。一方面，它将促进微分几何与拓扑学、代数几何，特别是理论物理研究者之间的进一步交流。 该项目也对研究生教育产生了重大影响，因为安德森和勒布朗监督了大量研究生，他们正在研究与研究计划密切相关的问题。数学和理论物理之间的关系处于这两个学科研究的前沿，导致了双方显着的新见解和发展。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的时空几何和引力理论（广义相对论）与目前被理解为支配其他基本自然力的量子场论之间的鸿沟。有些研究是关于描述由爱因斯坦引力方程控制的四维宇宙的所有可能几何形状的问题。 其中一个工具是使用赛伯格-威滕方程，它起源于短程核相互作用理论，但在我们理解许多四维宇宙的大尺度结构中发挥了重要作用。计划研究的许多方面都与弦理论的发展密切相关，弦理论可以说是理论物理学中最有前途和最令人兴奋的研究途径，并且对数学的最新进展产生了深刻和多方面的影响。

AbstractAward: DMS-0604735Principal Investigator: Michael T. Anderson, Claude R. LeBrunWorking both jointly and independently, Anderson and LeBrun planto study a cluster of related geometric structures onlow-dimensional manifolds. Their research program focuses onrelationships linking geometry to topological structures,algebraic geometry, and theoretical physics. Jointly, Andersonand LeBrun plan to study the existence and moduli of canonicalmetrics on 4-manifolds, with an emphasis on the Kahler case.Anderson will also investigate problems concerning asympoticallyhyperbolic Einstein metrics, the AdS/CFT correspondence, andmathematical aspects of general relativity. LeBrun will alsostudy the curvature of 4-manifolds, Seiberg-Witten theory,extremal Kahler metrics, and the twistor geometry of holomorphicdisks. The proposed work is expected to have a significantbroader impact on two different fronts. On one hand, it willpromote further interactions between researchers in differentialgeometry and those in topology, in algebraic geometry and,especially, in theoretical physics. The project also has asignificant educational impact at the graduate level, in thatAnderson and LeBrun oversee a large number of graduate studentswho are working on problems closely tied to the researchproposal.The relations between mathematics and theoretical physics are atthe forefront of research in both disciplines, leading on bothsides to remarkable new insights and developments. Much of theplanned research activity takes its inspiration from currentattempts to bridge the gulf between Einstein's theory ofspace-time geometry and gravitation (General Relativity), and thequantum field theories that are currently understood to governthe other fundamental forces of nature. Some of the research hasto do with the problem of describing all possible geometries of4-dimensional universes governed by Einstein's gravitationalequations. One of the toolsto be used involves use of theSeiberg-Witten equations, which originated in the theory ofshort-range nuclear interactions, but nonetheless turn out toplay a major role in our understanding of thelarge-scalestructure of many 4-dimensional universes. Otheraspects of the planned research are closely linked todevelopments in string theory, which is arguably the mostpromising and exciting avenue of research in theoretical physics,and which has had a profound and multi-faceted impact on recentprogress in mathematics.