Groups are the abstract embodiment of symmetry, used in physics to describe physical laws and in communications theory to find superior ways of transmitting and processing information.
Representation theory is the study of how symmetry is expressed, especially through matrices.
The representation theory of groups important to physics, which are continuous in nature, has been modestly well understood, but the representation theory important for the information age has lagged behind. The conjecture in question is one of the fundamental issues in this area. Probably the most interesting cases for applications are the smaller number systems, but a good solution to the Lusztig conjecture could provide insight into these cases as well. Computationally, the larger number systems, such as those dealt with by the Lusztig conjecture, are the most difficult. It is also funded by our UVa Mathematics Department's REU program.