Studies in Riemannian and Complex Geometry

提案0305865 PIs：Michael Anderson，Claude LeBrunss.Justin Sawon研究黎曼和复杂几何Anderson、LeBrun和Sawon将继续研究与数学主流的其他关键领域密切联系的微分几何的各种主题，如数学物理、微分拓扑、偏微分方程组、代数几何和几个复变量。安德森将研究广义相对论、爱因斯坦度量和三维流形的几何化。勒布伦将研究低维流形的黎曼几何，重点是3维和4维的曲率和拓扑，以及2维的Zoll度量。Sawon将从Rozansky-Witten不变量、纽结不变量和拓扑量子场理论的角度研究紧致超Kaehler流形和四元数-Kaehler流形。预计这项研究计划的广泛影响将是显著的。许多正在研究的问题对从事广义相对论、超引力和弦理论的理论物理学家来说都是非常感兴趣的；此外，该小组的成员通过参加会议、研讨会和合作项目，积极发展与在这些领域工作的物理学家的联系。该项目还将直接有助于培养新一代数学研究人员，因为该研究小组的成员积极参与指导许多研究生，包括妇女和少数族裔。

Proposal 0305865PIs: Michael Anderson, Claude LeBrunSr. Assoc.: Justin SawonSTUDIES IN RIEMANNIAN AND COMPLEX GEOMETRYAnderson, LeBrun and Sawon will pursue research on various topics indifferential geometry with close links to other key areas of the mathematicalmainstream, such as mathematical physics, differential topology, partial differential equations, algebraic geometry, and several complex variables. Anderson will carry out research on general relativity, Einstein metrics, and the geometrization of 3-manifolds.LeBrun will work on the Riemannian geometry of low-dimensional manifolds,with a focus on curvature and topology in dimensions 3 and 4, andon Zoll metrics in dimension 2. Sawon will study compact hyper-Kaehlermanifolds and quaternion-Kaehler manifolds from the point of view ofRozansky-Witten invariants, knot invariants, and topological quantum field theories.It is expected that the broader impact of this research program will be significant.Many of the problems under study are of great interestto theoretical physicists working in general relativity, supergravity, andstring theory; moreover, members of the group are actively developinglinks with physicists working in these areas throughparticipation in conferences, seminars and collaborative projects.The project will also directly contribute to the training of anew generation of mathematical researchers, asmembers of the research group are actively engaged in the supervisionof many graduate students, including women and minorities.