This project has two primary goals. The first is to solve the problem of the unitary dual: to describe the irreducible unitary representations of real reductive Lie groups. The primary tool is an algorithm to compute the unitary dual of any given group, which we are implementing inside the "atlas" software. We plan to use this information to prove results about the unitary dual, beginning with the unitarity of Arthur's unipotent representations. The second primary goal is to make information about representation theory of real groups accessible to non-specialists, via the software, a web site, public workshops, and other means. The atlas software is freely available on the atlas web site, and will continue to be maintained there indefinitely.
The idea of using symmetry to study problems in mathematics and science dates back to Fourier's work on heat nearly two hundred years ago. In the hands of Hermann Weyl, Eugene Wigner, and Andre Weil, symmetry has come to play a central role in quantum mechanics and in number theory. Lie groups, named after the Norwegian mathematician Sophus Lie, are the mathematical objects underlying symmetry. Representation theory studies all of the ways a given symmetry, or Lie group, can manifest itself. The problem of understanding all "unitary" representations (in which the symmetry operations preserve lengths) is one of the most important unsolved problems in the subject, and has potential applications in many areas; for example, it is an abstract version of the question, "what quantum mechanical systems can admit a certain kind of symmetry?"
The project falls in the area of representation theory of Lie groups. Lie groups, named after the Norwegian mathematician Sophus Lie (1842-1899), are mathematical objects underlying the symmetries inherent in a system, while their representations, i.e., the ways in which the Lie groups can manifest themselves, have had an important impact in theoretical physics and number theory. The problems considered as part of this project are motivated by one of the most basic questions in representation theory, namely the understanding of the "unitary dual". This can be viewed as a vast generalization of the role of Fourier series in studying periodic functions, and it has been subject to intense research for more than half a century. The project made significant progress resulting in several research papers published in top mathematical journals, generated topics for Ph.D. theses in mathematics and research by graduate and undergraduate students. One of the directions generated by work on this project represents a new instance of Dirac theory for certain algebraic structures. This new direction has already been proved fruitful, and it is very much a subject of continued research. Dirac theory has its roots in the famous Dirac equation in particle physics, and it has been generalized and developed substantially in the framework of representation theory.
