Summary of Research

My work may be divided into the six periods listed below. For each period, I have highlighted one publication (the numbers in square brackets refer to my List of Publications). All of this research was supported by grants from the National Science Foundation.

1. Lie group topology ([1] - [6])

[6] On the cohomology of homogeneous spaces took a major step towards proving the H. Cartan formula in characteristic p>0. Subsequently, several mathematicians published papers which completed the argument. One of these papers was by my student, J. Wolf, and was based on his Ph.D. thesis.

2. Foliations ([7] - [9], [11]- [13])

On closed manifolds foliations rarely exist. Foliations with singularities, however, exist in great abundance. In [11] Singularities of homomorphic foliations, R. Bott and I proved a formula equating locally defined numbers at the foliation singularities to global invariants. This relates to the Haefliger classifying space and to the general theme of equating local and global invariants as in the vector field index theorem of H. Hopf.

3. Riemann-Roch ([14] - [17], [21])

The Riemann-Roch theorem (of the great nineteenth century mathematician Riemann and his student Roch) is one of the gems of nineteenth century mathematics. Not until the mid-twentieth century did mathematicians understand what the Riemann-Roch problem was in higher dimensions. Once this understanding had been achieved, F. Hirzebruch solved the problem. Hirzebruch's brilliant result was conceptualized and re-proved by A. Grothendieck. The next step in this development was taken in [15] Riemann-Roch for singular varieties, where W. Fulton, R. MacPherson and I extended Grothendieck's theorem to singular varieties.

4. K homology ([18] - [20], [28], [30])

These papers develop the analysis and topology of elliptic operators. In [19] K homology and index theory, R. Douglas and I prove the equality of analytic and topological K homology by a simple and clear construction.

5. K theory for operator algebras ([22], [23], [25] - [27], [34])

I first met Alain Connes at the 1980 AMS summer institute on operator algebras. We began working together almost immediately. Connes had the inspired idea that what R. Douglas and I had done in [19] was just the tip of the iceberg. He believed that some far-reaching general principle waited to be discovered. We then struggled for a dozen years to formulate an equality of geometric-topological K theory and analytic K theory in a non-commutative setting. The standard methods of algebraic topology were not adequate for this task, so new techniques had to be devised. Finally in [34] Classifying space for proper actions and K theory of group C* algebras, Baum, Connes and Higson stated a precise conjecture. This conjecture is unusual in that it cuts across several areas of mathematics and reveals an underlying unity that until now had been completely unknown.

Interest in the conjecture continues and exciting breakthroughs are being made. Several conferences on the conjecture have been held in the U.S., Canada and Europe. Progress on the conjecture has been the subject of talks at the ICM (International Congress of Mathematicians) and at the Seminaire Bourbaki. The conjecture has been proved for linear reductive Lie groups, for p-adic groups, and for a large class of discrete groups. At the present time, there is no discrete group for which the conjecture is known to fail. However, M. Gromov has recently introduced discrete groups which might be counter- examples. The universe of all discrete groups appears too wild for conjectures of Baum-Connes type to be true for all discrete groups.

6. Latest Work: Geometric structure in the representation theory of reductive p-adic groups ([33], [38], [40], [42], [48], [50])

Let G be a reductive p-adic group. One of the main problems in the representation theory of p-adic groups is to describe the smooth dual of G, where smooth dual is the set of equivalence classes of irreducible smooth representations of G. With my co-workers A-M. Aubert, and R. Plymen, I have studied this problem from the point of view of non-commutative algebraic geometry. We conjecture [48] that the smooth dual of G has the structure of a countable disjoint union of complex affine algebraic varieties. We explicitly identify these algebraic varieties and we conjecture that the periodic cyclic homology of the Hecke algebra of G is isomorphic to the direct sum of the cohomology of these varieties. This periodic cyclic homology conjecture is Baum-Connes in the setting of smoth representation theory. If true, it describes the smooth dual at a topological level.

An intriguing question is whether and how this relates to the local Langlands conjecture, which also describes the smooth dual but from a quite different point of view.