Mathematical Sciences: Operator Algebras, K-Theory and IndexTheory

Professor Higson's project has to do with the use and management of index information in the theory of operator algebras. One major thrust will be in the Dirac operator approach to relative K-homology for manifolds with boundary advocated by Baum, Douglas, and Taylor, with an emphasis on developing techniques for computing one of the connecting homomorphisms in the six-term exact sequence for this theory. Another area of investigation is index theory for the C*-algebra generated by bounded smoothing operators supported in a uniform neighborhood of the diagonal on an open manifold. Certain distinguished subalgebras of this algebra will also be considered. The notion of index pervades a great deal of mathematics. Elementary manifestations can be found in, for instance, the winding number that measures the number of times a point moving in a closed planar orbit winds around a fixed point inside, and the number of arbitrary constants in the general solution of a linear differential equation. An important theorem, now a couple of decades old, relates index behavior for differential equations on a manifold (surface or higher-dimensional analogue) to information about the manifold's general shape, or topology, expressible in terms of things like winding numbers. It has been discovered more recently that richly structured mathematical systems called operator algebras provide an auspicious framework for powerful extensions of the original index theorem. The work of Professor Higson will expand and strengthen this framework.