My work may be divided into the seven periods listed below. All of this research was supported by grants from the National Science Foundation.

### 1. Representation theory of reductive p-adic groups and the local Langlands conjecture [33] [38] [40] [42] [48] [50] [51] [52] [63] [65] [71]

Let `G` be a reductive p-adic group. Examples are GL(n, F) , SL(n, F), etc where n can be any positive integer and F can be any finite extension of the field Q_p of p-adic numbers. The smooth dual of G is the set of (equivalence classes of) smooth representations of G. The representations are on vector spaces over the complex numbers. In a canonical way, the smooth dual of G is the disjoint union of countably many subsets known as the Bernstein components. Results from non-commutative geometry — e.g. Baum-Connes conjecture, periodic cyclic homology of the Hecke algebra of G [40] [42] — indicated that a very simple geometric structure might be present in the smooth dual of G. The ABP (Aubert-Baum-Plymen) conjecture makes this precise by asserting that each Bernstein component in the smooth dual of G is a complex affine variety. These varieties are explicitly identified as certain extended quotients. For split G, my co-workers and I recently proved that ABP is valid for any Bernstein component in the principal series. A corollary is that the local Langlands conjecture is valid throughout the principal series.

### 2. Index formula for non-elliptic operators [64] [66] [67]

In 2010 Erik van Erp and I solved the index problem for a naturally arising class of non-elliptic differential operators. We obtain a topological formula for the index of these operators. This solution was achieved by combining the Heisenberg calculus with the Baum-Douglas isomorphism of topological and analytic K-homology. The solution completes van Erp’s earlier partial result. The theorem suggests that ellipticity might not be the really essential hypothesis needed for formulas like Atiyah-Singer. Such formulas apply well beyond elliptic operators.

### 3. K theory for group C* algebras [22] [23] [25] [26] [27] [34]

I first met Alain Connes at the 1980 AMS summer institute on operator algebras. We began working together almost immediately. Alain had the inspired idea that what Ron 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,” Alain Connes, Nigel Higson, and I stated a precise conjecture. This conjecture is unusual in that it cuts across several areas of mathematics and reveals an underlying unity that previously had been completely unknown.

Interest in the conjecture continues and exciting breakthroughs are being made. Conferences on the conjecture have been held in the U.S., the U. K., Canada, France, Germany, Australia, and Switzerland. 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 almost-connected Lie groups, for reductive p-adic groups, for reductive adelic groups [39] [43] and for large classes of discrete groups

A discrete group introduced by M. Gromov provided a counter-example to an extension of the conjecture known as Baum-Connes with coefficients. However, in a more natural and slightly reformulated version of BC with coefficients, the Gromov example is no longer a counter-example. For exact groups (i.e. all the groups that arise in practice) the reformulated conjecture and the original conjecture are the same. At the present time (January, 2013) there is no known counter-example to the reformulated version of BC and BC with coefficients.

### 4. K homology [18] [19] [20] [28] [30]

These papers develop the analysis and topology of elliptic operators. The relevant topology is K homology i.e. the dual theory to K-theory. In [19] “K homology and index theory,” Ron Douglas and I use Dirac operators to prove the equality of analytic and topological K homology. This established a framework for index theory which had the possibility of extending the Atiyah-Singer index theorem to Fredholm differential operators which are not elliptic. This possibility was realized in 2010 in joint work with Erik van Erp. See 2 above.

### 5. Riemann-Roch [14] [15] [16] [17] [21]

The Riemann-Roch theorem (of the great nineteenth century mathematician Riemann and his very talented co-worker 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. A special issue of *Asterisque* was devoted to this theorem. Uses and applications for the theorem continue to be discovered.

### 6. Foliations [7] [8] [9] [11] [12] [13]

On closed manifolds, foliations rarely exist. Foliations with singularities, however, always exist in great abundance. “Singularities of holomorphic foliations” [11] develops earlier results of Bott, Baum-Bott,[8] [9] and Baum-Cheeger [7]. In [11] Raoul Bott and I proved a formula equating locally defined numbers at foliation singularities to global invariants. This relates to the Haefliger classifying space, to Cheeger-Chern-Simons theory, and to the general theme of equating local and global invariants as in the classical vector field index theorem of H. Hopf [9].

### 7. Lie group topology [1] [2] [3] [4] [5] [6]

Let G be a compact connected Lie group and let H be a compact connected subgroup of G. Henri Cartan proved that the de Rham cohomology of the quotient space G/H can be calculated as the homology of a certain Koszul-type complex. Following-up on joint work with William Browder [2], my Princeton thesis connected Cartan’s theorem to the (new at the time) Eilenberg-Moore spectral sequence. This point of view indicated that Cartan’s theorem might be true for cohomology with coefficients in a finite field. In [6] “On the cohomology of homogeneous spaces” I proved Cartan’s theorem, with coefficients in a finite field, modulo an extra condition which is very often satisfied. The theorem of [6] was featured in John McCleary’s book **A User’s Guide to Spectral Sequences**. Subsequently, several mathematicians published papers which showed that the extra condition is not needed, thus completing the argument and proving the full result. One of these papers was by my student Joel Wolf and was based on his Brown University Ph.D. thesis.