A couple of weeks ago, I attended a workshop hosted by Darrin Speegle on the HRT Conjecture. First posed twenty years ago by Chris Heil, Jay Ramanathan, and Pankaj Topiwala in this paper, the conjecture states that every finite collection of time-frequency shifts of any nonzero function in is linearly independent. In this post, I will discuss some of the key ideas behind some of the various attempts at chipping away at HRT, and I’ll describe what appear to be fundamental barriers to a complete solution. For more information, see these survey papers: one and two.

First, some notation: Denote the translation-by- and modulation-by- operators by

respectively. Then the formal conjecture statement is as follows:

**The HRT Conjecture.** For every and every finite , the collection is linearly independent.

What follows are some of the popular methods for tackling HRT: