Quaternionic geometry and elliptic genus

项目编号：dms -0405281项目负责人：Haydee herrera本课题主要研究有限次同伦群流形的拓扑结构和四元数- Kaehler流形的微分几何。该项目的重点是确定流形的解析和拓扑不变量，如签名，a -顶属和椭圆属。根据我们先前研究的证据，我们建议将上述理论应用于具体的几何分析/微分几何问题。此外，本研究还可以应用于代数几何（通过扭转或变换）和理论物理，因为所考虑的流形在sigma模型和弦理论中都很有趣。我的研究兴趣是研究某些大维度的空间，称为四元数- kaehler流形。在探索理解宇宙的过程中，物理学家已经发展了许多非常复杂的模型，比如相对论和最近的弦理论，这些模型有助于解释和预测在宇宙中观察到的现象。我们这个时代最重要的物理学家之一E. Witten发现，四元数-凯勒流形出现在某些量子场论的公式中。我的研究重点是寻找四元数- kaehler流形的可能形状和性质。

AbstractAward: DMS-0405281Principal Investigator: Haydee HerreraThis research proposal focuses on the study of the topology ofmanifolds with finite second homotopy group and the DifferentialGeometry of quaternionic- Kaehler manifolds. The emphasis of theproject is on the determination of analytical and topologicalinvariants of the manifolds, such as the signature, A-roof-genusand elliptic genera. Following the evidence from our previousresearch, we propose to apply the aforementioned theory toconcrete Geometric Analysis/Differential Geometryproblems. Furthermore, this study has applications to AlgebraicGeometry (via twistor transform) and to Theoretical Physics,since the manifolds under consideration are of interest in sigmamodels and in String Theory.My research interests are concerned with the study of certainspaces of large dimension, called quaternion-Kaehlermanifolds. In the quest for understanding the universe,physicists have developed many and very sophisticated models,such as Relativity Theory and more recently String Theory, thathelp explain and predict phenomena observed in the Universe. Oneof the leading physicists of our times, E. Witten, discoveredthat quaternion-Kaehler manifolds appear in the formulation ofcertain Quantum Field Theories. My research focuses on findingthe possible shapes and properties of quaternion- Kaehlermanifolds.