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?"
Lie groups are fascinating, complicated mathematical objects characterized by algebraic and geometric structures which are compatible with each other. They consist of continuous families of symmetries. For example, the Lie group O(2) is made up of the symmetries of a circle: rotations by arbitrary angles, and reflections about any line passing through the center of the circle will take a circle to itself. This family (rotations and reflections together) is an example of a one-dimensional Lie group. There are Lie groups of any size and dimension. For example, the Lie group E8 has dimension 248. As symmetries, Lie groups appear naturally when studying any number of phenomena, in all areas of mathematics, in physics (in particular quantum mechanics and particle physics), as well as in chemistry. One way to study and try to understand Lie groups is by looking at their "representations", i.e., how they act on other spaces. For example, the group O(2) "acts" on the plane (a two-dimensional space) by moving points through rotations and reflections, but also, in a similar way, it acts on larger spaces. Especially interesting are representations that are length-preserving, the unitary representations. While classical results provide a list of all representation of a given Lie group, determining which of these are unitary is in general a fascinating and difficult open problem. The present project is part of a large collaborative effort, the Atlas of Lie Groups project, with the goal to solve this problem. The Atlas of Lie Groups project may be divided into two tightly connected lines of inquiry: 1) Writing a computer program (the "atlas" software) that will decide whether a given representation is unitary or not; 2) Proving new theorems about unitarity. As part of the collaboration, this (individual) project has been focused on trying to understand the structure of the set of unitary representations ("unitary dual") of two particular infinite families of Lie groups: the metaplectic groups, and the split odd orthogonal groups. We have been able to identify a certain (significant) part of the unitary dual of the metaplectic groups which we call "omega-regular". In addition, we have been able to directly relate the unitarity of representations for the two groups. In particular, we were able to prove both unitarity and non-unitarity for representations of the metaplectic groups from knowledge about representations of the orthogonal groups. We hope that this tight connection between unitary duals of different groups will provide interesting insight into the structure of the unitary dual in general. Additional outcomes of this project concern the software part of the Atlas of Lie Groups. The "atlas" software has become well advanced, and it has the potential to provide (and has been providing) a valuable tool for researchers and students in any field of mathematics or related science that has applications to Lie groups, for finding out a wealth of information on groups and representations, for hypothesis testing, and even for motivating new research questions. One goal is to make it more widely available. While the software may be downloaded by anyone who is interested, using it still requires a fair amount of skill and training. Part of this project has consisted of assisting the software developers by testing the software as it is being written, by creating and trying out new applications (which will also be made available to anyone who has use for them), and by giving software demonstrations and instructions to prospective new users, such as researchers in related fields and graduate students.
