The HRT conjecture is an open problem deeply rooted in the study of time-frequency analysis. It was posed about twenty years ago by Chris Heil, Jay Ramanathan, and Pankaj Topiwala. A version of the conjecture states that a function with sufficiently fast decay cannot admit linearly dependent time-frequency shifts. Despite the simple nature of the problem, it has only been resolved partly. In this presentation, I will discuss the ideas behind the various attempts at solving the conjecture. I will also describe what I view as the central obstructions to a complete solution to this problem.

This talk is accessible to an audience of undergraduates. Students interested in undergraduate research are encouraged to attend.