Geometric Structures on Low Dimensional Manifolds

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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.