The local Langlands conjectures can be viewed as offering two important kinds of connections---first, between the same matrix group, taken with coefficients in two different fields; and second, between two different matrix groups, taken with coefficients in the same field. The conjectures offer an explanation for why the representation theories of real and p-adic, and even of adelic, groups are so similar. The price of this uniformity is that, often, one may no longer speak of individual representations, but rather of finite collections of them (called L-packets). These are expected to encode both number-theoretic (via Galois groups) and algebro-geometric (via many avenues---for example, the theory of stable distributions) information. For some families of representations, the investigators already have clear expectations for what the L-packets should be, but they cannot yet prove that their expectations are correct. For other families, there is not even a reasonable conjecture for what the answer should be. The theory of real groups suggests yet another rewarding perspective, from the point of view of symmetric spaces. For p-adic groups, the serious study of harmonic analysis on such spaces is just beginning to be developed, in large part by the investigators, and the analogues in this setting of the local Langlands conjectures are far from clear. The part of the local Langlands conjectures dealing with functoriality also suggests that the investigators should be able to transfer representation-theoretic information between different matrix groups. A classical realization of this is the theory of lifting, where representations of matrix groups over a large field are related to those of the same group, but with coefficients taken in a smaller field. Most progress in this area has been via somewhat ad hoc methods, but the answers have invariably turned out to be related to natural constructions arising in the symmetric-space setting.
Representation theory, broadly understood, has its origins in two classical problems. The first, investigated by Fourier in the 19th century, was an attempt to understand complicated physical processes, such as heat diffusion, by representing them as combinations of simpler processes. The second, initially studied by Frobenius, Schur, and others, was an attempt to understand the structure of a finite collection of symmetries via an associated polynomial known as its group determinant. The surprising fact that the solutions to these two problems are related has turned out to be just the earliest instance of a family of deep and far-reaching connections that have been formalized in a collection of conjectures known collectively as the (local) Langlands conjectures. The depth and broad reach of these conjectures---for example, they encompass a large part of the celebrated recent proof of the centuries-old Fermat's Last Theorem---has meant that progress has so far been relatively slow. This project brings together a group of mathematicians from a broad variety of related backgrounds, whose combined expertise can be expected to allow significant progress both on these conjectures and on related results in representation theory and harmonic analysis.
