The primary goal of The Atlas of Lie Groups and Representations project is to solve a fundamental problem in representation theory of Lie groups: the classification of the irreducible unitary representations of real reductive Lie groups. The contributors are taking a computational approach to the problem. They have developed the mathematical theory required to reduce the problem to a finite computation. This required rethinking representation theory from the ground up from this new point of view. The key tool is a modified notion of invariant Hermitian form, called the c-Hermitian form, which has critical uniqueness properties lacking in the usual Hermitian form. This allows the formulation of an algorithm to compute the sign of c-Hermitian and Hermitian forms, in terms of a new family of Kazhdan-Lusztig-Vogan polynomials.
Symmetry plays a fundamental role in mathematics and the sciences. In the late 19th century Sophus Lie showed that the symmetry of a system can be captured in an abstract mathematical object; these are now known as Lie (pronounced Lee) groups. The ways in which a particular symmetry (i.e. Lie) group can manifest itself are known as unitary representations. The main goal of the Atlas of Lie Groups and Representations is to understand all such representations. This has applications to physics, as well as many areas of mathematics, including number theory and geometry. The approach is computational. Both the mathematical and computational challenges are great, and the project brings together mathematicians and computer scientists.
