In 1996, C. Heil, J. Ramanatha, and P. Topiwala conjectured that the (finite) set $\mathcal{G}(g, \Lambda)=\{e^{2\pi i b_k \cdot}g(\cdot - a_k)\}_{k=1}^N$ is linearly independent for any  non-zero square integrable function $g$ and  subset $\Lambda=\{(a_k, b_k)\}_{k=1}^N \subset \mathbb{R}^2.$ This problem is now known as the HRT  Conjecture and is still largely unresolved.

In the first part of the talk, I will give an overview of the state of the conjecture. I will then survey a few recent attempts to solve it. In particular, I will describe an inductive approach to investigate the conjecture by considering the following question. Suppose the HRT conjecture is true for a function $g$ and a fixed set of $N$ points $\Lambda=\{(a_k, b_k)\}_{k=1}^N \subset \mathbb{R}^2.$ For what other point $(a, b)\in \mathbb{R}^2\setminus \Lambda$ will the HRT remain true for the same function $g$ and the new set of $N+1$ points $\Lambda'=\Lambda \cup \{(a, b)\}$?  I will report on the cases where $\Lambda$ is a subset of a lattice or a line. 

https://univienna.zoom.us/j/62077153839?pwd=T3pxeHNRNEU0RlFoY1J2cnIzbzU5dz09