Geometric Structures on Low Dimensional Manifolds

低维流形上的几何结构

• 批准号: 0604735
• 负责人: Michael Anderson
• 金额: $ 56.2万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0604735&HistoricalAwards=false
• 关键词: Geometric; Structures; Low; Dimensional; Manifolds

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.

Abstractaward：DMS-0604735原理研究者：Michael T. Anderson，Claude R. Lebrunworking共同和独立，Anderson和Lebrun Planto研究了一组相关的几何结构，即构成二维歧管。他们的研究计划集中于将几何形状与拓扑结构，代数几何和理论物理学联系起来的跨关系。联合的是，安德森和勒布朗计划研究4个manifolds上规范对称性的存在和模量，重点是卡勒案。安德森还将调查有关非局外毛细血管爱因斯坦指标的问题，ADS/CFT对应关系，并具有一般性相关性。勒布朗（Lebrun）将使4个manifolds，seiberg-witten理论，极端卡勒指标和Holomorphicdisks的扭曲器几何形状的曲率。 拟议的工作预计将对两个不同的战线产生重大影响。一方面，它将在差异几何学中与拓扑，代数几何形状，尤其是理论物理学中的研究人员之间的进一步相互作用。 该项目还在研究生层面上具有很大的教育影响。计划的大部分研究活动都从当前的攻击中启发了它的灵感，以弥合爱因斯坦的空间时间几何学和引力（一般相对论）和目前被理解为管理其他自然基本力量的Quequantum田地理论之间的鸿沟。 Hasto的一些研究解决了由爱因斯坦重力征象支配的4维宇宙的所有可能几何形状的问题。 使用的工具之一涉及使用这些方程式，该方程起源于分子范围的核相互作用理论，但是尽管如此，在我们对许多4维宇宙的thelarge-ScalStructuruct中的理解中，Toplay具有重要作用。计划中的研究的其他方面是弦理论中紧密相关的，这可以说是理论​​物理学研究中最令人兴奋，最令人兴奋的研究途径，并且对近代人的数学影响产生了深远而多方面的影响。