Almost all my work revolves around questions of symmetry, particularly continuous symmetries and the theory of Lie groups and Lie algebras that are the mathematical tools that often carry such symmetry.
I became very interested in the theory of simple Lie groups/algebras, as developed by Cartan, Weyl, Chevalley, Harish-Chandra, and other great mathematicians of the earlier part of the 20th century. I was fortunate enough to be able to generalize this theory to the infinite dimensional setting (simultaneously with V. Kac) where the resulting Lie theory has proven to be of great structural beauty and of great applicability in many areas of mathematics and physics.
Of special and great continuing interest in this research has been the study of vertex operator representations. These representations appear in the theory of solitons, in string theories, conformal field theories, monstrous moonshine, and more recently in efforts to generalize conformal field theories through the tool of the newly invented toroidal Lie algebras. I did quite a bit of work in this area, including quite a bit of work in developing algorithms for computing in finite and infinite dimensional representations. This computational work remains a component in my current research.
For the past 10 years I have also been interested in aperiodic order as manifested in quasicrystals. Briefly, this concerns the types of almost periodic structures that permit normally forbidden symmetries to appear in Nature. The most important manifestation of quasicrystals is their implicit long-range internal order that makes itself apparent in the beautiful and perfect diffraction patterns associated with them.
Over 150 quasicrystal materials are now known, some of them showing perfection that rivals the best crystals. The mathematics used to model such objects and to study their diffractive and their remarkably self-similar internal structures turns out to be a highly interdisciplinary endeavor, including algebraic number theory, the theory of lattices, Fourier analysis, measure theory, the study of self-similar structures and fractal measures, and dynamical systems. I have written papers on all of these things, the main emphasis being to try and create a good context and rigorous foundation for this new area of mathematics.
Present work revolves the long standing question raised by the advent of quasicrystals: given a point set (or more realistically, weighted point set) that diffracts, what can we say about this point set. Since diffractive properties are easily observable, while atomic structure is not, this is a fundamental question. With my co-workers (Michael Baake, Boris Solomyak, and J-Y. Lee) we are making good progress on this. There are three forthcoming papers on the subject that represent the first fruits of this work.
Robert V. Moody