The two-scale relation in wavelet analysis dictates that a square-integrable function can be written as a linear combination of scaled and shifted copies of itself. This fact is equivalent to the existence of square-integrable functions whose time-scale shifts are linearly dependent. By contrast, by replacing the scaling operator with a modulation operator one would think that the linear dependency of the resulting time-frequency shifts of a square-integrable function might be easily inferred. However, more than two decades after C.~Heil, J.~Ramanatha, and P.~Topiwala conjectured that any such finite collection of time-frequency shifts of a non-zero square-integrable function on the real line is linearly independent, this problem (the HRT Conjecture) remains unresolved.
The talk will give an overview of the HRT conjecture and introduce an inductive approach to investigate it. I will highlight a few methods that have been effective in solving the conjecture in certain special cases. However, despite the origin of the HRT conjecture in Applied and Computational Harmonic Analysis, there is a lack of experimental or numerical methods to resolve it. I will present an attempt to investigate the conjecture numerically.