Professors Boyer and Galicki propose to investigate several projects in geometry and topology. The objective of all the projects is to study fundamental questions in Riemannian Geometry with two main focal points: Contact Geometry of orbifold bundles over Fano varieties and the existence of some special (i.e., Einstein, positive Ricci curvature) metrics on such spaces. The questions and problems proposed here are deeply rooted in the principal investigators' earlier work which exploited a fundamental relationship between contact geometry of Sasakian-Einstein spaces and two kinds of Kaehler geometry, namely Q-factorial Fano varieties with Kaehler-Einstein orbifold metrics with positive scalar curvature, and Calabi-Yau manifolds with their Kaehler Ricci-flat metrics. The most recent work of the principal investigators has led to an important breakthrough in the study of such structures when, using recent results of Demailly and Kollar, the principal investigators were able to construct new examples of compact Einstein manifolds in dimension 5 as well as many positive Einstein metrics on families of rational homology 7-spheres. The techniques used by the principal investigators borrow from several different fields; the algebraic geometry of Mori theory and intersection theory, the analysis of the Calabi Conjecture, and finally the classical differential topology of links of isolated hypersurface singularities. These methods can be extended much further and in various directions. An example that stands out in this respect is the question of the existence of Einstein metrics on exotic spheres, which is one of the main objectives of this proposal. More generally the principal investigators want to address several classification problems concerning compact Sasakian-Einstein manifolds in dimensions 5 and 7. These two dimensions are important for two separate reasons. In view of earlier work higher dimensional examples can be constructed using the join construction. At the same time these two odd dimensions appear to play special role in Superstring Theory. In the context of recent developements in String and M-Theory the principal investigators also propose to investigate some related problems concerning self-dual Einstein metrics in dimension 4 and exceptional holonomy metrics in dimension 7 and 8.
This project is intended to further the understanding of the mathematics of certain important types of geometry. While this endeavor is not directly motivated by advances in technology, the history of mathematics adequately demonstrates that today's pure mathematics often becomes tomorrow's applied mathematics. Indeed the mathematics being considered in this project is currently being used in the field of Elementary Particle Physics, more specifically, in the attempts at understanding a unified description of the fundamental forces of the universe.
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