Principal Investigator: Charles P. Boyer and Krzysztof Galicki
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 Calabi-Yau and Fano varieties and the existence of some special (i.e., Einstein, positive Ricci curvature, transversely Calabi-Yau) 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. Most recently the principal investigators and J. Kollar have solved an open problem in Riemannian geometry. We have proved the existence of Einstein metrics on exotic spheres in a paper to appear in the Annals of Mathematics. Furthermore, we have shown that odd dimensional homotopy spheres that bound parallelizable manifolds admit an enormous number of Einstein metrics. In fact, the number of deformation classes as well as the number of moduli of Sasakian-Einstein metrics grow double exponentially with dimension. 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. 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.
Mathematics is the foundation upon which our modern technology is built, and much of its understanding and development must preceed technological progress. Nevertheless, our research into a particular type of geometry is closely linked to some important problems in modern Theoretical Physics and should provide an important mathematical basis for their understanding. For example, the mathematical models that we are studying are currently being used in supersymmetric string theory which is a model for the unification of gravity with the other fundamental forces of nature. This also has applications to the Physics of black holes.
