The theory of Lie groups, named in honor of the Norwegian mathematician Sophus Lie, has been one of the major themes in twentieth century mathematics. As the mathematical vehicle for exploiting the symmetries inherent in a system, Lie theory has had a profound impact upon mathematics itself and theoretical physics, especially quantum mechanics and elementary particle physics. The representation theory of a class of such groups called semi-simple Lie groups, is one of the most important and difficult aspects of the subject, as it is interwoven throughout the core of modern mathematics and mathematical physics. At the heart of this representation theory is the profound work of Harish-Chandra from 1955 to 1968, culminating in his Plancherel formula which described the harmonic analysis of square integrable functions. That work left open the major analytic problem of the harmonic analysis for p between 0 and 2. In technical terms this is the problem of characterizing the Fourier transforms of tau-spherical functions for general p. Professor Trombi is one of the few researchers in the world who is currently in pursuit of this difficult and important problem, fraught with imposing technical obstacles, and he has the best results to date. In earlier work, he solved the case p=1, and more recently-under the previous award -- Professor Trombi completely characterized the Schwartz space for general p, a major piece of work, His current investigations concern determining the intersection of these spaces, simplifying the Arthur-Campoli conditions in the Paley-Weiner theorem, and investigating weighted orbital integrals. This is mathematical analysis of the most demanding technical nature.
