A Gabor system consists of the translates and modulates of a window function over a lattice in the time-frequency plane. The Balian-Low Theorem (BLT) is an uncertainty principle for Gabor systems that form orthonormal bases for L^2 (R). The BLT states that the window function for a Gabor orthonormal basis can not be simultaneously too well localized in time and frequency (in a rather strong sense).

We shall survey recent work related to the Balian-Low Theorem (BLT), including issues such as sharpness in the BLT and non-symmetric versions of the BLT. We shall prove a (1, infinity) endpoint version of the BLT.

The theorems presented in this talk involve several collaborators: John Benedetto, Wojtek Czaja, Przemyslaw Gadzinski, and Jacob Sterbenz.