ABSTRACT

Abstract

Award: DMS-0504367
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.

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