Professor Haskell will investigate the index theory of elliptic Fredholm operators on certain noncompact manifolds, and the index theory arising from actions of noncompact groups on manifolds. The former project will be particularly concerned with the case in which the manifold is the smooth part of a singular algebraic variety. A general aim is to relate the behavior of indices to the global topology of the variety and to geometric information about the singularities. The second project will be approached via the crossed product construction from the theory of operator algebras. Elliptic operators in this setting give rise to maps on the K-groups of the crossed product algebra, whose nature Professor Haskell will attempt more precisely to elucidate. A fundamental object in the study of crossed products is Kasparov's representation ring of a connected Lie group. This ring can be realized as equivalence classes of almost intertwining Fredholm operators between unitary representations of the group. Professor Haskell will analyze the structure of this ring for certain semisimple groups by studying intertwining operators with additional structure. Three fundamental subject areas in mathematics are involved in this project, namely differential operators on manifolds, the representation theory of Lie groups, and the theory of operator algebras. The first subject may be thought of as a far-reaching outgrowth of calculus on manifolds (surfaces and their higher- dimensional analogues). The operations of calculus, which are defined in terms of the geometry of the manifold (distance, curvature, and the like on a local scale) turn out to yield information about its topology (overall shape and conformation). When the manifold arises algebraically, say as the zero set of a polynomial, there is even more information that can be harvested from studying differential operators. Lie groups, named after the Norwegian mathematician Sophus Lie, are used by mathematicians and physicists to embody symmetry, roughly the totality of all allowable motions of a given situation that preserve its essential features. Lie groups often act on manifolds in interesting ways, and the important features of such an action can often be detected by certain differential operators. The theory of operator algebras, finally, provides a technical and conceptual framework within which apparently disparate mathematical structures can interact with one another.
