Quaternionic geometry and elliptic genus

Abstract Award: DMS-0405281 Principal Investigator: Haydee Herrera This research proposal focuses on the study of the topology of manifolds with finite second homotopy group and the Differential Geometry of quaternionic- Kaehler manifolds. The emphasis of the project is on the determination of analytical and topological invariants of the manifolds, such as the signature, A-roof-genus and elliptic genera. Following the evidence from our previous research, we propose to apply the aforementioned theory to concrete Geometric Analysis/Differential Geometry problems. Furthermore, this study has applications to Algebraic Geometry (via twistor transform) and to Theoretical Physics, since the manifolds under consideration are of interest in sigma models and in String Theory. My research interests are concerned with the study of certain spaces of large dimension, called quaternion-Kaehler manifolds. In the quest for understanding the universe, physicists have developed many and very sophisticated models, such as Relativity Theory and more recently String Theory, that help explain and predict phenomena observed in the Universe. One of the leading physicists of our times, E. Witten, discovered that quaternion-Kaehler manifolds appear in the formulation of certain Quantum Field Theories. My research focuses on finding the possible shapes and properties of quaternion- Kaehler manifolds.