One of the most fundamental developments in modern science has been the theory of quantum mechanics. The attempt to understand both the physics and the underlying mathematics has led directly to current frontiers in two important areas of Modern Analysis, representation theory and operator algebras. Representation theory is a means of exploiting the inherent physical symmetries, while the concept of operator algebras was invented to provide the correct framework for quantization. Recently, there has been a great expansion in our understanding of the mathematics underlying quantization. Representation theory and operator algebras have been brought together recently in a powerful theory that centers around the celebrated Atiyah-Singer index theorem. This evolving field is often referred to as noncommutative differential geometry. The representation theory enters in the group actions on locally symmetric spaces as well as the role that singular algebraic varieties play in representation theory. Current advances point to the movement from finite to infinite discrete groups and from compact manifolds to noncompact singular varieties. These are tied together through the fact that smooth points of singular varieties often arise as quotients of actions of infinite discrete groups on noncompact manifolds. The geometry of the space is reflected in the representation theory of the group and analysis of differential operators on the noncompact manifold. These are the questions that address modern index theory. Professor Haskell is a young researcher who has mastered the broad range of mathematics necessary to contribute to this field. The investigator's prior work, in part collaborative, has laid a solid foundation for undertaking the following work. Professor Haskell will study the index theory of geometric elliptic Fredholm operators on noncompact manifolds. He will then apply this research to equivariant index theory of noncompact group actions, to index theory on the smooth parts of algebraic varieties, and on locally symmetric spaces.
