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?"
Mathematics makes constant use of symmetry. Geometry in the plane (and so trigonometry, surveying, and so on) depends on the rotational symmetry of the plane: moving southeast behaves in exactly the same way as moving north by northwest. The sine and cosine functions are characterized by the fact that they change in a very simple way under rotation: the "addition formulas" for sine and cosine from trigonometry. For that reason, sine and cosine are fundamental for understanding anything about measurement in the plane. More complicated kinds of symmetry lead to more complicated versions of sine and cosine. The rotational symmetry of a sphere leads to spherical harmonics, introduced by Laplace more than two hundred years ago in his study of the motion of satellites. These same functions can be used also to describe how the shape of the Earth deviates from a perfect sphere; or indeed for any problem in which the physical laws do not change under rotation of space. When Schrodinger wrote his equation describing the motion of an electron around an atomic nucleus, the equation had this rotational symmetry around the nucleus; so it was possible to write the solutions using Laplace's spherical harmonics from a hundred and forty years in the past. In this way spherical harmonics are used to describe the shape of electron orbitals in atoms, and so to build the foundations of chemistry. This project is joint work with Jeffrey Adams at University of Maryland, Peter Trapa at University of Utah, and Marc van Leeuwen at University of Poitiers in France. It concerns the search for analogues of sines and cosines---the technical name is "irreducible unitary representations"---for still more complicated symmetry groups. Such groups arise in mathematics in problems of number theory, higher dimensional geometry, and differential equations. They arise in physics when one tries (as we do not yet know how to do!) to describe elementary particles and gravity in a common theory. The search for these irreducible unitary representation has occupied many mathematicians for more than a hundred years. There have been many important successes, like work of Hermann Weyl and Elie Cartan in the 1930s which treated all "compact" symmetry groups. (The rotational symmetries of the plane and the sphere are both compact, as are the symmetry groups in the "standard model" of particle physics; that model makes heavy use of the Cartan-Weyl work.) This project concerns "noncompact" symmetry groups. In physics, these are the symmetry groups associated with physical theories incorporating Einstein's theory of gravity. In this setting, previous work of Harish-Chandra and Langlands (around 1970) provided a list of representations that was a little too long: it was guaranteed to include all the interesting (unitary) representations, but there was no way known to decide which representations in the list were unitary (and so could contribute to the description of interesting mathematics or physics). We have found a way to decide this question, and are writing a computer program to carry out the calculations. We anticipate immediate applications to some mathematical problems where these symmetry groups are important, like the number theory field of automorphic forms. We hope eventually to see applications to physics; perhaps even to the question of finding a physical theory incorporating the standard model and gravity.
